Emergy Algebra: Improving Matrix Methods for Calculating Tranformities
Transformity is one of the core concepts in Energy Systems Theory and it is fundamental to the calculation of emergy. Accurate evaluation of transformities and other emergy per unit values is essential for the broad acceptance, application and further development of emergy method...
Set Theory Applied to Uniquely Define the Inputs to Territorial Systems in Emergy Analyses
The language of set theory can be utilized to represent the emergy involved in all processes. In this paper we use set theory in an emergy evaluation to ensure an accurate representation of the inputs to territorial systems. We consider a generic territorial system and we describ...
Emergy, Transformity, the Emergy Unit and the Emergy Baseline.
This paper addresses three important research areas subject to some disagreement within the community of emergy practitioners or to misunderstandings within the scientific community as a whole. Emergy and its sister idea, transformity, are important new concepts derived from the...
Emergy is an important quantity needed for public policy analysis that is based on a complex methodology. The intent of this Environmental Research Brief is to define emergy and its importance in a manner that is accessible to everyone with at least a high school education. Emer...
This web presentation answers basic questions about the relatively new scientific concept, emergy. It dispels some of the confusion surrounding this idea in a PowerPoint presentation. The presentation is written in common language and uses straightforward examples. Emergy indic...
This is the fourth in a series of papers on quantifying the emergy of human services in a manner that is essentially independent of the monetary value of the work performed. In this paper we assembled information on the education and training required to perform the major occupat...
The purpose of this study is to better understand the differences and similarities between emergy and nonmarket economic valuation, when both are applied to value the same policies or development alternatives. The emdollar value of a good or service often exceeds the market value...
EMERGY METHODS: VALUABLE INTEGRATED ASSESSMENT TOOLS
NHEERL's Atlantic Ecology Division is investigating emergy methods as tools for integrated assessment in several projects evaluating environmental impacts, policies, and alternatives for remediation and intervention. Emergy accounting is a methodology that provides a quantitative...
The Geobiosphere Emergy Baseline: A synthesis
Following the Eighth Biennial Emergy Conference (January, 2014), the need for revisiting the procedures and assumptions used to compute the Geobiosphere Emergy Baseline (GEB) emerged as a necessity to strengthen the method of Emergy Accounting and remove sources of ambiguity and ...
CHAPTER 10: CURRENT TECHNICAL PROBLEMS IN EMERGY ANALYSIS
Technical problems related to the determination of the emergy base for self-organization in environmental systems are considered in this paper. The comparability of emergy analysis results depends on emergy analysts making similar choices in determining the emergy base for a part...
The geobiosphere emergy baseline: A synthesis.
The concept of emergy defined as the available energy (or exergy) of one form used up directly and indirectly to produce an item or action (Odum, Environmental Accounting Emergy and Environmental Decision Making, John Wiley & Sons, Inc., 1996) requires the specification of a unif...
EMERGY ANALYSIS OF THE PREHISTORIC NITROGEN CYCLE
Several relationships between the specific emergy or the emergy per unit mass and the mass concentration of nitrogen were shown to exist through an analysis of the global nitrogen cycle. These observed relationships were interpreted by examining the nature of the underlying ener...
Emergy Accounting: A Unified, Comprehensive Triple Bottom Line
Development of the concept of emergy established a medium for accounting that made it possible to express environmental, economic, and social work of all kinds on a common basis as solar emjoules. Emergy accounting can be used to combine both emergy and money accounts on a single...
The Emergy Baseline of the Earth: Is it Arbitrary?
The emergy baseline for the Earth is used in determining the transformities of the products of all planetary processes and through these relationships it influences all emergy evaluations. Estimates of the emergy baseline made in the past have changed depending on the number of i...
Using Toulmin Analysis to Analyse an Instructor's Proof Presentation in Abstract Algebra
ERIC Educational Resources Information Center
Fukawa-Connelly, Timothy
2014-01-01
This paper provides a method for analysing undergraduate teaching of proof-based courses using Toulmin's model (1969) of argumentation. It presents a case study of one instructor's presentation of proofs. The analysis shows that the instructor presents different levels of detail in different proofs; thus, the students have an inconsistent set of…
Comparison of typical mega cities in China using emergy synthesis
NASA Astrophysics Data System (ADS)
Zhang, L. X.; Chen, B.; Yang, Z. F.; Chen, G. Q.; Jiang, M. M.; Liu, G. Y.
2009-06-01
An emergy-based comparison analysis is conducted for three typical mega cities in China, i.e., Beijing, Shanghai and Guangzhou, from 1990 to 2005 in four perspectives including emergy intensity, resource structure, environmental pressure and resource use efficiency. A new index of non-renewable emergy/money ratio is established to indicate the utilization efficiency of the non-renewable resources. The results show that for the three mega urban systems, Beijing, Shanghai and Guangzhou, the total emergy inputs were 3.76E+23, 3.54E+23, 2.52E+23 sej in 2005, of which 64.88%, 91.45% and 72.28% were imported from the outsides, respectively. As to the indicators of emergy intensity involving the total emergy use, emergy density and emergy use per cap, three cities exhibited similar overall increase trends with annual fluctuations from 1990 to 2005. Shanghai achieved the highest level of economic development and non-renewable resource use efficiency, and meanwhile, lower proportion of renewable resource use and higher environmental pressure compared to those of Beijing and Guangzhou. Guangzhou has long term sustainability considering an amount of local renewable resources used, per capita emergy used, energy consumption per unit GDP and the ratio of waste to renewable emergy. It can be concluded that different emergy-based evaluation results arise from different geographical locations, resources endowments, industrial structures and urban orientations of the concerned mega cities.
ERIC Educational Resources Information Center
Wawro, Megan Jean
2011-01-01
In this study, I considered the development of mathematical meaning related to the Invertible Matrix Theorem (IMT) for both a classroom community and an individual student over time. In this particular linear algebra course, the IMT was a core theorem in that it connected many concepts fundamental to linear algebra through the notion of…
Emergy of the Occupations. Chapter 43
In this paper we calculated the emergy contributed to the economy of the United States in the work done by the workers of 558 occupations in 2008. We determined the empower (semj/yr) delivered by an individual engaged in each occupation, the transformity of the occupation’s work ...
CURRENT TECHNICAL PROBLEMS IN EMERGY ANALYSIS
: Emergy Analysis has been a rapidly evolving assessment methodology for the past 30 years. This process of development was primarily driven by the inquiring mind and ceaseless activity of its founder, H.T. Odum, his students, and colleagues. Historically, as new kinds of proble...
Dealing with Emergy Algebra in the Life Cycle Assessment Framework
The Life Cycle Inventory (LCI) represents one of the four steps of the Life Cycle Assessment (LCA) methodology, which is a standardized procedure (ISO 14040:2006) to estimate the environmental impacts generated by the production, use and disposal of goods and services. In this co...
ERIC Educational Resources Information Center
Chiu, Ming Ming
2008-01-01
The micro-time context of group processes (such as argumentation) can affect a group's micro-creativity (new ideas). Eighty high school students worked in groups of four on an algebra problem. Groups with higher mathematics grades showed greater micro-creativity, and both were linked to better problem solving outcomes. Dynamic multilevel analyses…
The Emergy Basis for Formal Education in the United States
The education system of the United States from 1870 to 2006 was evaluated using emergy methods. The system was partitioned into three subsystems, elementary, secondary, and college education and the emergy inputs required to support each subsystem were determined for each year o...
Emergy of the Global Biogeochemical Cycles of Biologically Active Elements
Accurate estimates of the emergy of elemental flows are needed to accurately evaluate the far field effects of anthropogenic wastes. The transformity and specific emergy of the elements and of their different chemical species is also needed to quantify the inputs to many producti...
[Emergy evaluation of power plant eco-industrial park].
Wang, Lingmei; Zhang, Jintun
2004-06-01
In this paper, new emergy indices for the material circulation utilization and energy cascade utilization were presented to evaluate the emergy of power plant eco-industrial park. The common contribution of the members in power plant eco-industrial park should be accounted for emergy yield ratio (PEYR) and emergy investment ratio (PEIR), namely, PEYR = sigma(i = 1)6 (Fi + Ri + Ni)/sigma(i = 1)6 Fi and PEIR = sigma(i = 1)6 Fi/sigma(i = 1)6 (Ni + Ri). Saved renewable and nonrenewable resources and purchased resources and decreased environmental load should be accounted for environmental loading ratio (PELR) and index of sustainability (PESI), namely, PELR = sigma(i = 1)6 Fi/F' and PESI = PEYR/PELR. Case analysis on the Shuozhou power plant eco-industrial park showed that new emergy indices were practical in evaluating the power plant eco-industrial park.
Teaching Algebra without Algebra
ERIC Educational Resources Information Center
Kalman, Richard S.
2008-01-01
Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…
Emergy Synthesis 8 ~ Emergy and environmental accounting: Theories, applications, and methodologies
With the assembly and review of the 12 research papers for this Special Issue of Ecological Modelling, our goal was to continue working with the journal to bring a strong group of papers, indicative of the forefront of emergy research, to the global energy research community and ...
[Emergy value evaluation on rice-duck organic farming mode].
Xi, Yunguan; Qin, Pei
2006-02-01
Employing emergy value evaluation method, this paper compared the ecological and economic benefits of rice-duck organic farming system (mode I) and conventional rice-wheat rotation system (mode II) in Shanghai suburb. The results showed that mode I had better emergy benefits, higher self-organizing ability and sustainability, and higher product safety. The net emergy yield ratio (EYR), feedback ratio of yield emergy (FYE), and emergy sustainable index (ESI) of mode I were respectively 1.57, 14.1, and 8.71 times as much as those of mode II, and the emergy index of product safety (EIPS) was 0 in mode I but -0.66 in mode II. The emergy investment ratio (EIR) and environmental loading ratio (ELR) of mode I were 40.1% and 18.3% of mode II, respectively, suggesting that mode I had less environmental pressure than mode II. Mode I had lower economic benefits than mode II. The economic output, gross income, and net income of mode I were 15.7%, 9.6%, and 29.6% less than those of mode II , respectively. As for Em dollars, the output, gross income, and net income of mode I were respectively 50%, 102.6%, and 136.4% higher than those of mode II. With the system optimization and the development of organic food market, mode I had the potential to improve its economic benefits.
[The Chinese urban metabolisms based on the emergy theory].
Song, Tao; Cai, Jian-Ming; Ni, Pan; Yang, Zhen-Shan
2014-04-01
By using emergy indices of urban metabolisms, this paper analyzed 31 Chinese urban metabolisms' systematic structures and characteristics in 2000 and 2010. The results showed that Chinese urban metabolisms were characterized as resource consumption and coastal external dependency. Non-renewable resource emergy accounted for a higher proportion of the total emergy in the inland cities' urban metabolisms. The emergy of imports and exports accounted for the vast majority of urban metabolic systems in metropolises and coastal cities such as Beijing and Shanghai, showing a significant externally-oriented metabolic characteristic. Based on that, the related policies were put forward: to develop the renewable resource and energy industry; to improve the non-renewable resource and energy utilization efficiencies; to optimize the import and export structure of services, cargo and fuel; and to establish the flexible management mechanism of urban metabolisms.
The Emergy Perspective of Sustainable Trends in Puerto Rico From 1960 to 2013
Emergy analysis quantifies the direct and indirect contributions of nature to human systems providing a sustainability assessment framework, which couples economic growth within biophysical constraints. In this study, Puerto Rico's sustainability was assessed with emergy flow dyn...
[Emergy analysis of ecological-economic system in Liaoning Province].
Liu, Hao; Wang, Qing; Li, Xiu-Juan; Song, Yang; Li, Guang-Jun
2008-03-01
By the methods of emergy analysis, this paper studied the emergy flow in the ecological-economic system in Liaoning Province in 1990-2005, and the relationships between the environmental stress caused by resources' input, output and consumption and the sustainable development of the Province. The results showed that in Liaoning Province, the non-renewable resources occupied over 74% of the total consumed emergy, and the realistic population in 2005 was 3.26 times higher than the supportable population. In 1990-2005, the emergy yield ratio decreased from 65.40 to 10.13, emergy loading ratio increased from 2.72 to 7.18, and emergy sustainable index decreased from 24.03 to 1.41. The rapid economic growth in Liaoning Province was chiefly supported by the consumption of vast non-renewable resources, which caused the pressure of economic development on ecosystem getting more and more intense, the economic development increasingly depending on exogenous resources, and the sustainable development of Liaoning ecological-economic system having a continuing decrease. To realize the sustainable development in Liaoning Province, the principles of reduction, reutilization and recycling should be taken as the guidelines for promoting the reuse of wastes and the closed fine circulation of resources to minimize the discharge of wastes.
Development of the concepts of emergy and transformity established a medium (emergy) for accounting that made it possible to express economic and environmental work of all kinds on a common basis as solar emjoules. Environmental accounting using emdollars, a combined emergy-monet...
Emergy evaluation perspectives of a multipurpose dam proposal in Korea.
Kang, Daeseok; Park, Seok Soon
2002-11-01
The 'emergy' concept was used to evaluate the economy of Korea and the contributions of a multipurpose dam proposal to the real wealth of the Korean economy. Emergy is defined as the available energy of one kind previously used up directly and indirectly to make a product or service. The indices for Korea calculated from the emergy evaluation were close to those of developed countries. Even though its monetary balance of payments was negative in 1997, the economy of Korea showed positive balance in trade when exports and imports were expressed in emergy. The emergy evaluation showed that the Korean economy places a large stress on its environment. Water supply and generation of electricity were the most important contributions of the proposed dam in terms of emergy; 46.5 and 45.1% of the total benefit, respectively. Flood control contributed 8.4% of the total benefit. Major costs associated with the proposed dam were from sediments (33.2% of the total cost), construction services (22.8%), and social disruption of the region (21.6%). In terms of emergy, the ratio of benefits to costs of the proposed dam was 2.78 if sediments are not included, and 1.86 if sediments are included, which result in more benefits than costs in both cases. The benefit to cost ratio of the dam was larger than that of the current system (1.42) without the dam whether sediments are included in the total cost or not. The environmental stress of the proposed dam was considerably lower than that of the Korean economy, but the dam might increase stress on local environment.
Zhang, L X; Ulgiati, S; Yang, Z F; Chen, B
2011-03-01
Emergy and economic methods were used to evaluate and compare three fish production models, i.e., cage fish farming system, pond intensive fish rearing system and semi-natural extensive pond fish rearing system, in Nansi Lake area in China in the year 2007. The goal of this study was to understand the benefits and driving forces of selected fish production models from ecological and economic points of view. The study considered input structure, production efficiency, environmental impacts, economic viability and sustainability. Results show that the main difference among the three production systems was the emergy cost for fish feed associated with their feeding system, i.e., feeding on natural biomass such as plankton and grass or on commercial feedstock. As indicated by EYR, ELR and ESI, it can be clearly shown that the intensive production model with commercial feed is not a sustainable pattern. However, the point is that more environmentally sound patterns do not seem able to provide a competitive net profit in the short run. The intensive pond fish farming system had a net profit of 2.57E+03 $/ha, much higher than 1.27E+03 $/ha for cage fish farming system and slightly higher than 2.37E+03 $/ha for semi-natural fish farming system. With regard to the drivers of local farmer's decisions, the accessibility of land for the required use and investment ability determine the farmer's choice of the production model and the scale of operation, while other factors seem to have little effect. Theoretically, the development of environmentally sustainable production patterns, namely water and land conservation measures, greener feed as well as low waste systems is urgently needed, to keep production activities within the carrying capacity of ecosystems. Coupled emergy and economic analyses can provide better insight into the environmental and economic benefits of fish production systems and help solve the problems encountered during policy making.
A Brief History of the Emergy Society and its Mission in the ...
In 2002, H.T. Odum authorized the formation of a society dedicated to research, communication, and education focused on the new ideas of emergy and transformity and their basis in Energy Systems Theory. From this initial concept and after a 5 period of incubation, the International Society for the Advancement of Emergy Research (ISAER), a.k.a., the Emergy Society, was incorporated as a nonprofit corporation in the United States on July 28, 2007. In this talk, the history of the Emergy Society from its origin to the present time will be presented. The key ideas of emergy and transformity and their derivation and importance will be discussed. The structure of the Emergy Society and how it has carried out its mission over its history will be examined, including the formation of the China Chapter of ISAER. In many ways the history of ISAER is a story about its Presidents and the Executive Council members, who worked hard to promote the growth of the Emergy Society and to expand its services to its membership and the world. The development of the Emergy Society website as the major interface to communicate with its members and the general public will be recounted along with the story of the development of the Emergy Society logo and its esoteric meaning. Finally, the accomplishments of ISAER in the United States, China, Brazil and Europe to date will be considered and the future of the Emergy Society explored. This talk was delivered as a plenary address to the asse
Emergy accounting for regional studies: case study of Canada and its provinces.
Hossaini, Navid; Hewage, Kasun
2013-03-30
Sustainable regional management (development) requires an understanding of interactions between the social, economic, and ecological systems within the boundaries of a region. In this paper, application of emergy (an environmental accounting method) for regional planning is discussed through a case study. Emergy (spelled with an "m") methodology is an environmental accounting technique that evaluates the energy system for the thermodynamics of an open system. Major renewable and non-renewable resource fluxes to a region, including energy, matter, human activities, and money can be converted to emergy by using corresponding transformity functions. As a case study, this paper discusses the emergy accounting of Canada and its provinces with various emergy-based indicators. Moreover, emergy maps were generated in a form of emergy geography. These maps are multi-dimensional illustrations that show resource consumption, emergy per person, and emergy density across Canada under two parameters: (1) the quantities of resources consumed and (2) the location of consumption. Emergy analysis also highlights concentrations of renewable and natural resources in Canada and distinguishes the provinces with the highest resource consumption. Analysis of emergy indicator for Canadian provinces shows that Alberta with the highest EYR (7.35) provides energy to the economy of Canada. However, ELR value of Alberta (8.5) indicates that the province's current economic approach is not sustainable as it relies mainly on non-renewable emergy inputs (mainly from fossil fuels). ELR of British Columbia and Manitoba indicates that these two provinces created a firm balance between emergy use of renewable and non-renewable resources. The characterizations of regions provided in this paper can be used for future land planning and management both in federal and provincial levels.
What is the role of a system's size in an emergy evaluation, when we have to evaluate nested systems? To answer this question, we consider a simple nested system with three levels of organization and then examine the relationships among the emergy flows at each level and among th...
AN EMERGY AUDIT OF WEST VIRGINIA: STRENGTHS AND VULNERABILITIES
Emergy provides a general accounting mechanism that allows us to view the economy of humanity and the economy of nature on the same income and balance sheets. In this manner we can verify the financial picture given by economic analysis by checking it against a similar represen...
Emergy Evaluation of Educational Attainment in the United States
The emergy of educational attainment in the United States was quantified over the period for which the necessary data were available, i.e., 1948 to 2006. We propose that the portion of the knowledge in educational attainment delivered by the active workforce be considered as part...
Brazilian Soybean Production: Emergy Analysis with an Expanded Scope
ERIC Educational Resources Information Center
Ortega, Enrique; Cavalett, Otavio; Bonifacio, Robert; Watanabe, Marcos
2005-01-01
This article offers the results of emergy analysis used to evaluate four different soybean production systems in Brazil that were divided into two main categories: biological models (organic and ecological farms) and industrial models (green-revolution chemical farms and herbicide with no-tillage farms). The biological models show better…
Integrating Emergy into LCA: potential added value and lingering obstacles
Emergy attempts to measure the environmental work required to generate (ecosystem) goods and services that can be used by humans. It is claimed that the use of inventory modelling principles behind the Life Cycle Assessment (LCA) method (European Commission, 2010a) may improve th...
A thermodynamic geography: night-time satellite imagery as a proxy measure of emergy.
Coscieme, Luca; Pulselli, Federico M; Bastianoni, Simone; Elvidge, Christopher D; Anderson, Sharolyn; Sutton, Paul C
2014-11-01
Night-time satellite imagery enables the measurement, visualization, and mapping of energy consumption in an area. In this paper, an index of the "sum of lights" as observed by night-time satellite imagery within national boundaries is compared with the emergy of the nations. Emergy is a measure of the solar energy equivalent used, directly or indirectly, to support the processes that characterize the economic activity in a country. Emergy has renewable and non-renewable components. Our results show that the non-renewable component of national emergy use is positively correlated with night-time satellite imagery. This relationship can be used to produce emergy density maps which enable the incorporation of spatially explicit representations of emergy in geographic information systems. The region of Abruzzo (Italy) is used to demonstrate this relationship as a spatially disaggregate case.
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
Development of the concept of emergy established a medium for accounting that made it possible to express economic and environmental work of all kinds on a common basis as solar emjoules. Environmental accounting using emdollars, a combined emergy-monetary unit, can be used to pr...
Emergy baseline for the Earth: A historical review of the science and a new calculation
Quantifying the emergy baseline of the Earth is a practical necessity for emergy evaluations, because it serves as a unified basis for determining transformities of the available energy storages and flows of the geobiosphere. The current debate over the value and significance of ...
Using the Language of Sets to Describe Nested Systems in Emergy Evaluations
The language of set theory has been recently used to describe the emergy evaluation of a process. In this paper this mathematical language is used as a guide to evaluate the emergy of nested systems. We analyze a territorial system on multiple scales as an example of hierarchical...
Emergy Evaluation of Formal Education in the United States: 1870 to 2011
We evaluated the education system of the United States from 1870 to 2011 using emergy methods. The system was partitioned into three subsystems (elementary, secondary and college education) and the emergy inputs required to support each subsystem were determined for every year ov...
Methods for Estimating the Uncertainty in Emergy Table-Form Models
Emergy studies have suffered criticism due to the lack of uncertainty analysis and this shortcoming may have directly hindered the wider application and acceptance of this methodology. Recently, to fill this gap, the sources of uncertainty in emergy analysis were described and an...
The Emergy of Money Ratio of the United States from 1990 to 2007
The emergy to money ratio of a system is a key index used in emergy evaluations, because it represents the buying power of money and it can be used as an estimator of the average value of human service. Since both the material and energy inputs to societies and the money circulat...
Estimates of the emergy carried by the flows of biologically active elements (BAE) and compounds are needed to accurately evaluate the near and far field effects of anthropogenic wastes. The transformities and specific emergies of these elements and of their different chemical sp...
Emergy synthesis of tourism-based urban ecosystem.
Lei, Kampeng; Wang, Zhishi
2008-09-01
Macao is a tourist city with a dense population, but is short on natural resources. Almost all of the life-support resources of the city depend on imports from outside. During the past 20 years, Macao has experienced an economic boom accompanied by rapid social development. The tourism industry (including gambling, hotel accommodation, restaurant dining, and shows) have become the city's main economic activity since 1991. This paper uses emergy flow analysis to investigate and characterize the urban evolution and city development that have occurred in Macao from 1983 to 2003. Macao's tourism industry has existed almost from the establishment of the city, with the legalization of gambling in Macao occurring in 1850. Tourism has become the biggest industry in Macao, contributing more than half of the city's revenues since 1995. The emergy flow related to tourism was tracked and analyzed to measure its contribution to Macao. In addition, we used statistical analysis to divide the various emergy-based indicators into three categories: positive, negative, and insensitive indicators.
ERIC Educational Resources Information Center
Hitt, Fernando; Saboya, Mireille; Cortés Zavala, Carlos
2016-01-01
This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…
Keeping the books for environmental systems: an emergy analysis of West Virginia.
Campbell, Daniel; Meisch, Maria; Demoss, Thomas; Pomponio, John; Bradley, M Patricia
2004-06-01
Emergy provides a general accounting mechanism that allows us to view the economy and the environment on the same income statement and balance sheet. This allows an auditor to verify the economic picture by checking it against a more complete representation of the flows and storages of real wealth as measured by emergy. In this study, we constructed emergy accounts for the state of West Virginia in 1997. The income statement showed annual production, consumption, and flows of emergy and dollars into and out of the state. The balance sheet evaluated the storage of emergy in some of the state's assets. Emergy indices were used to answer questions posed by managers and gain insight into the state's economic and environmental strengths and weaknesses. West Virginia has great wealth in nonrenewable resources (9E14 sej m-2 or 17 times the U.S. average). The investment ratio of emergy purchased outside to indigenous renewable and nonrenewable emergy was 2.2:1, which indicates a high potential for future development. However, the environmental loading ratio (14:1) was already 1.5 times higher than that found at an average location in the U.S. Twice as much emergy was exported as received and standard of living indicators showed that people have largely failed to benefit from their state's wealth. We propose that, just as in business, where decisions made using financial accounts ensure solvency; decisions governing the environment should be made based on an emergy accounting of activities, assets, and liabilities for the combined system of humanity and nature.
NASA Technical Reports Server (NTRS)
Mulligan, Jeffrey B.
2017-01-01
A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.
NASA Astrophysics Data System (ADS)
Mikhalev, A. V.; Pinchuk, I. A.
2005-06-01
The structure of Steinberg conformal algebras is studied; these are analogues of Steinberg groups (algebras, superalgebras).A Steinberg conformal algebra is defined as an abstract algebra by a system of generators and relations between the generators. It is proved that a Steinberg conformal algebra is the universal central extension of the corresponding conformal Lie algebra; the kernel of this extension is calculated.
[Emergy of agro-ecosystem in Hunan Province: evolution and trend].
Zhu, Yu-Lin; Li, Ming-Jie
2012-02-01
By using emergy analysis method, a trend analysis was made on the total emergy, its input-output structure, and emergy indices of the agro-ecosystem in Hunan Province of South-central China from 1999 to 2008. In the study period, the available total emergy input of the ecosystem was basically maintained at a stable level, but the input structure changed with the input of non-renewable industrial auxiliary emergy increased from 4.00E+22 sej in 1999 to 5.53E+22 sej in 2008, while that of renewable organic emergy decreased from 1.32E+23 sej to 1.20E+23 sej. Both the total emergy output and the output efficiency of the ecosystem had a great increase, with the total output reached 1.69E+23 sej in 2008, which was 23.8% higher than that in 1999, and the net output ratio increased from 0.79 to 0.96. Owing to the ever-increasing trend of the environmental loading ratio which was from 1.12 to 1.79, the sustainable development index of the ecosystem presented a decreasing trend, from 0.71 to 0.54, indicating that the agriculture in Hunan Province was overall belonged to the type of ecosystem driven by high consumption, and had relatively apparent extensive development characteristics.
Emergy Evaluation of a Production and Utilization Process of Irrigation Water in China
Chen, Dan; Luo, Zhao-Hui; Chen, Jing; Kong, Jun; She, Dong-Li
2013-01-01
Sustainability evaluation of the process of water abstraction, distribution, and use for irrigation can contribute to the policy of decision making in irrigation development. Emergy theory and method are used to evaluate a pumping irrigation district in China. A corresponding framework for its emergy evaluation is proposed. Its emergy evaluation shows that water is the major component of inputs into the irrigation water production and utilization systems (24.7% and 47.9% of the total inputs, resp.) and that the transformities of irrigation water and rice as the systems' products (1.72E + 05 sej/J and 1.42E + 05 sej/J, resp.; sej/J = solar emjoules per joule) represent their different emergy efficiencies. The irrigated agriculture production subsystem has a higher sustainability than the irrigation water production subsystem and the integrated production system, according to several emergy indices: renewability ratio (%R), emergy yield ratio (EYR), emergy investment ratio (EIR), environmental load ratio (ELR), and environmental sustainability index (ESI). The results show that the performance of this irrigation district could be further improved by increasing the utilization efficiencies of the main inputs in both the production and utilization process of irrigation water. PMID:24082852
Emergy evaluation of a production and utilization process of irrigation water in China.
Chen, Dan; Luo, Zhao-Hui; Chen, Jing; Kong, Jun; She, Dong-Li
2013-01-01
Sustainability evaluation of the process of water abstraction, distribution, and use for irrigation can contribute to the policy of decision making in irrigation development. Emergy theory and method are used to evaluate a pumping irrigation district in China. A corresponding framework for its emergy evaluation is proposed. Its emergy evaluation shows that water is the major component of inputs into the irrigation water production and utilization systems (24.7% and 47.9% of the total inputs, resp.) and that the transformities of irrigation water and rice as the systems' products (1.72E + 05 sej/J and 1.42E + 05 sej/J, resp.; sej/J = solar emjoules per joule) represent their different emergy efficiencies. The irrigated agriculture production subsystem has a higher sustainability than the irrigation water production subsystem and the integrated production system, according to several emergy indices: renewability ratio (%R), emergy yield ratio (EYR), emergy investment ratio (EIR), environmental load ratio (ELR), and environmental sustainability index (ESI). The results show that the performance of this irrigation district could be further improved by increasing the utilization efficiencies of the main inputs in both the production and utilization process of irrigation water.
ERIC Educational Resources Information Center
Capani, Antonio; De Dominicis, Gabriel
This paper proposes a model for a general interface between people and Computer Algebra Systems (CAS). The main features in the CAS interface are data navigation and the possibility of accessing powerful remote machines. This model is based on the idea of session management, in which the main engine of the tool enables interactions with the…
Chen, Chun-feng; Wang, Hong-yan; Xiao, Du-ning; Wang, Da-qing
2008-11-01
By using traditional ecological footprint method and its modification, emergy ecological footprint method, the sustainable development status of Heilongjiang Province in 2005 was analyzed. The results showed that the ecological deficits of Heilongjiang Province in 2005 based on emergy and conventional ecological footprint methods were 1.919 and 0.6256 hm2 x cap(-1), respectively. The ecological footprint value based on the two methods both exceeded its carrying capacity, which indicated that the social and economic development of the study area was not sustainable. Emergy ecological footprint method was used to discuss the relationships between human's material demand and ecosystem resources supply, and more stable parameters such as emergy transformity and emergy density were introduced into emergy ecological footprint method, which overcame some of the shortcomings of conventional ecological method.
Emergy and Eco-exergy Evaluation of Four Forest Restoration Modes
Four different forest restoration modes (Acacia mangium plantation, mixed-native species plantation, conifer plantation and Eucalyptus plantation) were evaluated using Energy System Theory and the emergy synthesis method. In addition, the eco-exergies of the four forest restorati...
Sustainable Biofuel Project: Emergy Analysis of South Florida Energy Crops
Amponsah, Nana Yaw; Izursa, Jose-Luis; Hanlon, Edward A.; Capece, John C.
2012-11-15
This study evaluates the sustainability of various farming systems, namely (1) sugarcane on organic and mineral soils and (2) energycane and sweet sorghum on mineral soils. The primary objective of the study is to compare the relative sustainability matrices of these energy crops and their respective farming systems. These matrices should guide decision and policy makers to determine the overall sustainability of an intended or proposed bioethanol project related to any of these studied crops. Several different methods of energy analysis have been proposed to assess the feasibility or sustainability of projects exploiting natural resources (such as (Life Cycle Analysis, Energy Analysis, Exergy Analysis, Cost Benefit Analysis, Ecological Footprint, etc.). This study primarily focused on the concept of Emergy Analysis, a quantitative analytical technique for determining the values of nonmonied and monied resources, services and commodities in common units of the solar energy it took to make them. With this Emergy Analysis study, the Hendry County Sustainable Biofuels Center intends to provide useful perspective for different stakeholder groups to (1) assess and compare the sustainability levels of above named crops cultivation on mineral soils and organic soils for ethanol production and (2) identify processes within the cultivation that could be targeted for improvements. The results provide as much insight into the assumptions inherent in the investigated approaches as they do into the farming systems in this study.
NASA Astrophysics Data System (ADS)
Wang, Guodong; Dong, Shuanglin; Tian, Xiangli; Gao, Qinfeng; Wang, Fang
2015-06-01
Emergy analysis is effective for analyzing ecological economic systems. However, the accuracy of the approach is affected by the diversity of economic level, meteorological and hydrological parameters in different regions. The present study evaluated the economic benefits, environmental impact, and sustainability of indoor, semi-intensive and extensive farming systems of sea cucumber ( Apostichopus japonicus) in the same region. The results showed that A. japonicus indoor farming system was high in input and output (yield) whereas pond extensive farming system was low in input and output. The output/input ratio of indoor farming system was lower than that of pond extensive farming system, and the output/input ratio of semi-intensive farming system fell in between them. The environmental loading ratio of A. japonicus extensive farming system was lower than that of indoor farming system. In addition, the emergy yield and emergy exchange ratios, and emergy sustainability and emergy indexes for sustainable development were higher in extensive farming system than those in indoor farming system. These results indicated that the current extensive farming system exerted fewer negative influences on the environment, made more efficient use of available resources, and met more sustainable development requirements than the indoor farming system. A. japonicus farming systems showed more emergy benefits than fish farming systems. The pond farming systems of A. japonicus exploited more free local environmental resources for production, caused less potential pressure on the local environment, and achieved higher sustainability than indoor farming system.
Guan, Fa-Chun; Sha, Zhi-Peng; Zhang, Yu-Yang; Wang, Jun-Feng; Wang, Chao
2016-08-01
Home courtyard agriculture is an important model of agricultural production on the Tibetan plateau. Because of the sensitive and fragile plateau environment, it needs to have optimal performance characteristics, including high sustainability, low environmental pressure, and high economic benefit. Emergy analysis is a promising tool for evaluation of the environmental-economic performance of these production systems. In this study, emergy analysis was used to evaluate three courtyard agricultural production models: Raising Geese in Corn Fields (RGICF), Conventional Corn Planting (CCP), and Pea-Wheat Rotation (PWR). The results showed that the RGICF model produced greater economic benefits, and had higher sustainability, lower environmental pressure, and higher product safety than the CCP and PWR models. The emergy yield ratio (EYR) and emergy self-support ratio (ESR) of RGICF were 0.66 and 0.11, respectively, lower than those of the CCP production model, and 0.99 and 0.08, respectively, lower than those of the PWR production model. The impact of RGICF (1.45) on the environment was lower than that of CCP (2.26) and PWR (2.46). The emergy sustainable indices (ESIs) of RGICF were 1.07 and 1.02 times higher than those of CCP and PWR, respectively. With regard to the emergy index of product safety (EIPS), RGICF had a higher safety index than those of CCP and PWR. Overall, our results suggest that the RGICF model is advantageous and provides higher environmental benefits than the CCP and PWR systems.
Guan, Fa-chun; Sha, Zhi-peng; Zhang, Yu-yang; Wang, Jun-feng; Wang, Chao
2016-01-01
Home courtyard agriculture is an important model of agricultural production on the Tibetan plateau. Because of the sensitive and fragile plateau environment, it needs to have optimal performance characteristics, including high sustainability, low environmental pressure, and high economic benefit. Emergy analysis is a promising tool for evaluation of the environmental-economic performance of these production systems. In this study, emergy analysis was used to evaluate three courtyard agricultural production models: Raising Geese in Corn Fields (RGICF), Conventional Corn Planting (CCP), and Pea-Wheat Rotation (PWR). The results showed that the RGICF model produced greater economic benefits, and had higher sustainability, lower environmental pressure, and higher product safety than the CCP and PWR models. The emergy yield ratio (EYR) and emergy self-support ratio (ESR) of RGICF were 0.66 and 0.11, respectively, lower than those of the CCP production model, and 0.99 and 0.08, respectively, lower than those of the PWR production model. The impact of RGICF (1.45) on the environment was lower than that of CCP (2.26) and PWR (2.46). The emergy sustainable indices (ESIs) of RGICF were 1.07 and 1.02 times higher than those of CCP and PWR, respectively. With regard to the emergy index of product safety (EIPS), RGICF had a higher safety index than those of CCP and PWR. Overall, our results suggest that the RGICF model is advantageous and provides higher environmental benefits than the CCP and PWR systems. PMID:27487808
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
Derive Workshop Matrix Algebra and Linear Algebra.
ERIC Educational Resources Information Center
Townsley Kulich, Lisa; Victor, Barbara
This document presents the course content for a workshop that integrates the use of the computer algebra system Derive with topics in matrix and linear algebra. The first section is a guide to using Derive that provides information on how to write algebraic expressions, make graphs, save files, edit, define functions, differentiate expressions,…
Brandt-Williams, Sherry; Wigand, Cathleen; Campbell, Daniel E
2013-02-01
This study evaluated the link between watershed activities and salt marsh structure, function, and condition using spatial emergy flow density (areal empower density) in the watershed and field data from 10 tidal salt marshes in Narragansett Bay, RI, USA. The field-collected data were obtained during several years of vegetation, invertebrate, soil, and water quality sampling. The use of emergy as an accounting mechanism allowed disparate factors (e.g., the amount of building construction and the consumption of electricity) to be combined into a single landscape index while retaining a uniform quantitative definition of the intensity of landscape development. It expanded upon typical land use percentage studies by weighting each category for the intensity of development. At the RI salt marsh sites, an impact index (watershed emergy flow normalized for marsh area) showed significant correlations with mudflat infauna species richness, mussel density, plant species richness, the extent and density of dominant plant species, and denitrification potential within the high salt marsh. Over the 4-year period examined, a loading index (watershed emergy flow normalized for watershed area) showed significant correlations with nitrite and nitrate concentrations, as well as with the nitrogen to phosphorus ratios in stream discharge into the marshes. Both the emergy impact and loading indices were significantly correlated with a salt marsh condition index derived from intensive field-based assessments. Comparison of the emergy indices to calculated nitrogen loading estimates for each watershed also produced significant positive correlations. These results suggest that watershed emergy flow is a robust index of human disturbance and a potential tool for rapid assessment of coastal wetland condition.
Energy and resource basis of an Italian coastal resort region integrated using emergy synthesis.
Vassallo, Paolo; Paoli, Chiara; Tilley, David R; Fabiano, Mauro
2009-10-01
Sustainable development of coastal zones must balance economic development that encourages human visitation from a larger population with desires that differ from the local residents with the need to maintain opportunities for the local resident society and conserve ecological capital, which may serve as the basis for residents. We present a case study in which the sustainability level of a coastal zone (Riviera del Beigua), located along the Ligurian coast of north-western Italy, was assessed through the lens of systems ecology using emergy synthesis to integrate across economic, social and environmental sub-systems. Our purposes were (1) to quantify the environmental sustainability level of this coastal zone, (2) to evaluate the role of tourism in affecting the economy, society and environment, and (3) to compare emergy synthesis to Butler's Tourism Area Life Cycle model (TALC). Results showed that 81% of the total emergy consumption in the coastal zone was derived from external sources, indicating that this tourist-heavy community was not sustainable. Tourism, as the dominant economic sub-system, consumed 42% of the total emergy budget, while local residents used the remaining 58%. The progressive stages of the TALC model were found to parallel the dynamic changes in the ratio of external emergy inputs to local emergy inputs, suggesting that emergy synthesis could be a useful tool for detecting a tourist region's TALC stage. Use of such a quantitative tool could expedite sustainability assessment to allow administrative managers to understand the complex relationship between a region's economy, environment and resident society so sound policies can be developed to improve overall sustainability.
An energy systems view of sustainability: emergy evaluation of the San Luis Basin, Colorado.
Campbell, Daniel E; Garmestani, Ahjond S
2012-03-01
Energy Systems Theory (EST) provides a framework for understanding and interpreting sustainability. EST implies that "what is sustainable" for a system at any given level of organization is determined by the cycles of change originating in the next larger system and within the system of concern. The pulsing paradigm explains the ubiquitous cycles of change that apparently govern ecosystems, rather than succession to a steady state that is then sustainable. Therefore, to make robust decisions among environmental policies and alternatives, decision-makers need to know where their system resides in the cycles of change that govern it. This theory was examined by performing an emergy evaluation of the sustainability of a regional system, the San Luis Basin (SLB), CO. By 1980, the SLB contained a climax stage agricultural system with well-developed crop and livestock production along with food and animal waste processing. The SLB is also a hinterland in that it exports raw materials and primary products (exploitation stage) to more developed areas. Emergy indices calculated for the SLB from 1995 to 2005 revealed changes in the relative sustainability of the system over this time. The sustainability of the region as indicated by the renewable emergy used as a percent of total use declined 4%, whereas, the renewable carrying capacity declined 6% over this time. The Emergy Sustainability Index (ESI) showed the largest decline (27%) in the sustainability of the region. The total emergy used by the SLB, a measure of system well-being, was fairly stable (CV = 0.05). In 1997, using renewable emergy alone, the SLB could support 50.7% of its population at the current standard of living, while under similar conditions the U.S. could support only 4.8% of its population. In contrast to other indices of sustainability, a new index, the Emergy Sustainable Use Index (ESUI), which considers the benefits gained by the larger system compared to the potential for local environmental
Geng, Yong; Liu, Zuoxi; Xue, Bing; Dong, Huijuan; Fujita, Tsuyoshi; Chiu, Anthony
2014-12-01
Industrial symbiosis is the sharing of services, utility, and by-product resources among industries. This is usually made in order to add value, reduce costs, and improve the environment, and therefore has been taken as an effective approach for developing an eco-industrial park, improving resource efficiency, and reducing pollutant emission. Most conventional evaluation approaches ignored the contribution of natural ecosystem to the development of industrial symbiosis and cannot reveal the interrelations between economic development and environmental protection, leading to a need of an innovative evaluation method. Under such a circumstance, we present an emergy analysis-based evaluation method by employing a case study at Shenyang Economic and Technological Development Zone (SETDZ). Specific emergy indicators on industrial symbiosis, including emergy savings and emdollar value of total emergy savings, were developed so that the holistic picture of industrial symbiosis can be presented. Research results show that nonrenewable inputs, imported resource inputs, and associated services could be saved by 89.3, 32.51, and 15.7 %, and the ratio of emergy savings to emergy of the total energy used would be about 25.58 %, and the ratio of the emdollar value of total emergy savings to the total gross regional product (GRP) of SETDZ would be 34.38 % through the implementation of industrial symbiosis. In general, research results indicate that industrial symbiosis could effectively reduce material and energy consumption and improve the overall eco-efficiency. Such a method can provide policy insights to industrial park managers so that they can raise appropriate strategies on developing eco-industrial parks. Useful strategies include identifying more potential industrial symbiosis opportunities, optimizing energy structure, increasing industrial efficiency, recovering local ecosystems, and improving public and industrial awareness of eco-industrial park policies.
Emergy evaluation of contrasting dairy systems at multiple levels.
Vigne, Mathieu; Peyraud, Jean-Louis; Lecomte, Philippe; Corson, Michael S; Wilfart, Aurélie
2013-11-15
Emergy accounting (EmA) was applied to a range of dairy systems, from low-input smallholder systems in South Mali (SM), to intermediate-input systems in two regions of France, Poitou-Charentes (PC) and Bretagne (BR), to high-input systems on Reunion Island (RI). These systems were studied at three different levels: whole-farm (dairy system and cropping system), dairy-system (dairy herd and forage land), and herd (animals only). Dairy farms in SM used the lowest total emergy at all levels and was the highest user of renewable resources. Despite the low quality of resources consumed (crop residues and natural pasture), efficiency of their use was similar to that of industrialised inputs by intensive systems in RI, PC and BR. In addition, among the systems studied, SM dairy farms lay closest to environmental sustainability, contradicting the usual image of high environmental impact of cattle production in developing countries. EmA also revealed characteristics of the three intensive systems. Systems from RI and PC had lower resource transformation efficiency and higher environmental impacts than those from BR, due mainly to feeding strategies that differed due to differing socio-climatic constraints. Application of EmA at multiple levels revealed the importance of a multi-level analysis. While the whole-farm level assesses the overall contribution of the system to its environment, the dairy-system level is suitable for comparison of multi-product systems. In contrast, the herd level focuses on herd management and bypasses debates about definition of system boundaries by excluding land management. Combining all levels highlights the contribution of livestock to the global agricultural system and identifies inefficiencies and influences of system components on the environment.
Using emergy to quantify an economic hierarchy between four U.S. states - Maryland, Delaware, New Jersey and Pennsylvania
A regional emergy evaluation was completed for each of four adjacent states with differing levels and sources of economic productivity ? Maryland, Delaware...
Mao, De-Hua; Hu, Guang-Wei; Liu, Hui-Jie; Li, Zheng-Zui; Li, Zhi-Long; Tan, Zi-Fang
2014-02-01
The annual emergy and currency value of the main ecological service value of returning cropland to lake in Dongting Lake region from 1999 to 2010 was calculated based on emergy analysis. The calculation method of ecological compensation standard was established by calculating annual total emergy of ecological service function increment since the starting year of returning cropland to lake, and the annual ecological compensation standard and compensation area were analyzed from 1999 to 2010. The results indicated that ecological compensation standard from 1999 to 2010 was 40.31-86.48 yuan x m(-2) with the mean of 57.33 yuan x m(-2). The ecological compensation standard presented an increase trend year by year due to the effect of eco-recovery of returning cropland to lake. The ecological compensation standard in the research area presented a swift and steady growth trend after 2005 mainly due to the intensive economy development of Hunan Province, suggesting the value of natural ecological resources would increase along with the development of society and economy. Appling the emergy analysis to research the ecological compensation standard could reveal the dynamics of annual ecological compensation standard, solve the abutment problem of matter flow, energy flow and economic flow, and overcome the subjective and arbitrary of environment economic methods. The empirical research of ecological compensation standard in Dongting Lake region showed that the emergy analysis was feasible and advanced.
Quantum cluster algebras and quantum nilpotent algebras
Goodearl, Kenneth R.; Yakimov, Milen T.
2014-01-01
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197
[Evaluation of ecosystem service and emergy of Wanshan Waters in Zhuhai, Guangdong Province, China].
Qin, Chuan-xin; Chen, Pi-mao; Zhang, An-kai; Yuan, Hua; Li, Guo-ying; Shu, Li-ming; Zhou, Yan-bo; Li, Xiao-guo
2015-06-01
The method for monetary value and emergy value analysis of ecosystem service was used in this paper to analyze the change in value of marine ecosystem service of Wanshan District, Zhuhai from 2007 to 2012. The result showed that the monetary value and emergy value of marine ecosystem service of Wanshan District, Zhuhai rose to 11512840000 yuan and 1.97 x 10(22) sej from 7721630000 yuan and 1.04 x 10(22) sej, respectively. Both monetary value and emergy value could forecast the change in the value of marine ecosystem service, but they reflected different value structures and ecological energy, which could be used to more objectively evaluate the ecosystem service. Ecological civilization development, as an inherent driving force to impel the development of marine ecosystem service structure, was important for rational exploitation of marine resources and optimization of marine ecosystem service.
Emergy Evaluation of the Natural Value of Water Resources in Chinese Rivers
NASA Astrophysics Data System (ADS)
Chen, Dan; Chen, Jing; Luo, Zhaohui; Lv, Zhuwu
2009-08-01
Emergy theory and method were used to evaluate the economy of China and the contributions of water resources in Chinese rivers to the real wealth of the Chinese economy. The water cycle and energy conversion were reviewed, and an emergy method for evaluating the natural value of water resources in a river watershed was developed. The indices for China calculated from the emergy evaluation were close to those of developing countries. Despite a small surplus in its balance of payments, China had a net emergy loss from its trade in 2002. The efficiency of Chinese natural resource use was still not high and did not match its economic growth rate. Furthermore, the Chinese economy placed a stress on its ecological environment and natural resources. Several indices of Chinese rivers from the emergy evaluation were close to those of average global river water. The main average indices of Chinese rivers were transformity (4.17 × 104 sej/J), emergy per volume (2.05 × 1011 sej/m3), and emdollar per volume (0.06 /m3). The total value of all the rivers’ water made up 13.0% of the GDP of China in 2002, and that of water consumption accounted for 2.1%. The value of the water resources in the Haihe-luanhe River (11.39 × 104 sej/J) was the highest, followed by the Yellow River (10.27 × 104 sej/J), while the rivers in Southwest China had the lowest values (2.92 × 104 sej/J).
Emergy evaluation of the natural value of water resources in Chinese rivers.
Chen, Dan; Chen, Jing; Luo, Zhaohui; Lv, Zhuwu
2009-08-01
Emergy theory and method were used to evaluate the economy of China and the contributions of water resources in Chinese rivers to the real wealth of the Chinese economy. The water cycle and energy conversion were reviewed, and an emergy method for evaluating the natural value of water resources in a river watershed was developed. The indices for China calculated from the emergy evaluation were close to those of developing countries. Despite a small surplus in its balance of payments, China had a net emergy loss from its trade in 2002. The efficiency of Chinese natural resource use was still not high and did not match its economic growth rate. Furthermore, the Chinese economy placed a stress on its ecological environment and natural resources. Several indices of Chinese rivers from the emergy evaluation were close to those of average global river water. The main average indices of Chinese rivers were transformity (4.17 x 10(4) sej/J), emergy per volume (2.05 x 10(11) sej/m(3)), and emdollar per volume (0.06 $/m(3)). The total value of all the rivers' water made up 13.0% of the GDP of China in 2002, and that of water consumption accounted for 2.1%. The value of the water resources in the Haiheluanhe River (11.39 x 10(4) sej/J) was the highest, followed by the Yellow River (10.27 x 10(4) sej/J), while the rivers in Southwest China had the lowest values (2.92 x 10(4) sej/J).
Emergy analysis of biogas systems based on different raw materials.
Wang, Yang; Lin, Cong; Li, Jing; Duan, Na; Li, Xue; Fu, Yanyan
2013-01-01
Environmental pollution and energy crisis restrict the development of China, and the utilization of renewable technology is an effective strategy to alleviate the damage. Biogas engineering has rapidly developed attributes to solve environmental problems and create a renewable energy product biogas. In this paper, two different biogas plants' materials were analyzed by emergy method. One of them is a biogas project whose degraded material is feces (BPF system), and the other is the one whose degraded material is corn straw (BPC system). As a result, the ecological-economic values of BPF and BPC are $28,300/yr and $8,100/yr, respectively. Considering currency, environment, and human inputs, both of the biogas projects have the ability of disposing waste and potential for development. The proportion of biogas output is much more than fertilizer output; so, fertilizer utilization should be emphasized in the future. In comparison, BPF is better than BPC in the aspects of ecological-economic benefits, environmental benefits, and sustainability. The reason is the difficulty of corn straw seasonal collection and degradation. Thus it is proposed that BPC should be combined with the other raw materials.
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
ERIC Educational Resources Information Center
Miller, L. Diane; England, David A.
1989-01-01
Describes a study in a large metropolitan high school to ascertain what influence the use of regular writing in algebra classes would have on students' attitudes towards algebra and their skills in algebra. Reports the simpler and more direct the writing topics the better. (MVL)
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Applied Algebra Curriculum Modules.
ERIC Educational Resources Information Center
Texas State Technical Coll., Marshall.
This collection of 11 applied algebra curriculum modules can be used independently as supplemental modules for an existing algebra curriculum. They represent diverse curriculum styles that should stimulate the teacher's creativity to adapt them to other algebra concepts. The selected topics have been determined to be those most needed by students…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Ternary Virasoro - Witt algebra.
Zachos, C.; Curtright, T.; Fairlie, D.; High Energy Physics; Univ. of Miami; Univ. of Durham
2008-01-01
A 3-bracket variant of the Virasoro-Witt algebra is constructed through the use of su(1,1) enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.
The Algebra of the NODEN Analyser
1989-08-01
c, c, E, 0, 01 If the range of r is not 0 to upb, but lwb to upb, then the input value is derived by making a value for an input in the range 0 to upb... lwb as described above, and adding a literal value lwb to it (see section 6.3). So the value of an input with a range 1 to 6 (mapped onto three...outside the range of the constrained type. If F1 is a function defined to map between unconstrained numeric values and values of a type in the range lwb
Computer algebra and operators
NASA Technical Reports Server (NTRS)
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
Dean, Marleah; Gill, Rebecca; Barbour, Joshua B
2016-12-01
Communication is key to hospital emergency department (ED) caregiving. Interventions in ED processes (and health care organizing in general) have struggled when they have ignored the professional role expectations that enable and constrain providers with patients and each other. Informed by a communication as design (CAD) approach, this study explored the intersections of professional roles, physical space, and communication at EmergiCare-an academic medical center and level-1 trauma center hospital. Based on an ethnographic analysis of field notes from 70 hours of shadowing at the EmergiCare ED, this study identified two specific communication patterns, "case talk" and "comfort talk," that reflect different logics for communication in health care organizing. The findings indicate (a) that case and comfort talk have different status and therefore different influence in EmergiCare ED interprofessional communication and (b) that the arrangement of physical space at EmergiCare ED reflects the requirements of case talk more so than comfort talk. These findings have important implications for theory and practice, including the importance of considering the macro-discursive construction of professional roles reified in the arrangement of work space.
A reexamination of the emergy input to a system from the wind
With the establishment of a new, rigorously-determined, solar equivalence baseline for the geobiosphere, 12.0E+24 seJ y-1, it is now appropriate to reexamine the calculation of the emergy delivered by the major secondary products of the geobiosphere, e.g., wind and rainfall, whi...
Relationships Among the Energy, Emergy, and Money Flows of the United States From 1900 to 2011
In this paper, we examine the relationships among the energy, emergy, and money flows of the United States from 1900 to 2011. To establish a theoretical basis for understanding these relationships, Energy Systems Language models of the resource base for the World System and of e...
This study evaluated the link between watershed activities and salt marsh structure, function, and condition using spatial emergy flow density (areal empower density) in the watershed and field data from 10 tidal salt marshes in Narragansett Bay, RI. The field-collected data wer...
The idea that the methods and models of accounting and bookkeeping might be useful in describing, understanding, and managing environmental systems is implicit in the title of H.T. Odum's book, Environmental Accounting: Emergy and Environmental Decision Making. In this paper, I ...
Integrated Emergy and Economic Evaluation of Tea Production Chains in Anxi, China
Emergy and economic methods were used to evaluate and compare tea production systems in Anxi, China. Tea production was classified into three phases, i.e., the nursery, the plantation and tea processing, and each phase was evaluated. The results showed that the nursery subsystems...
A Brief History of the Emergy Society and its Mission in the World
In 2002, H.T. Odum authorized the formation of a society dedicated to research, communication, and education focused on the new ideas of emergy and transformity and their basis in Energy Systems Theory. From this initial concept and after a 5 period of incubation, the Internation...
KEEPING THE BOOKS FOR ENVIRONMENTAL SYSTEMS: AN EMERGY ANALYSIS OF WEST VIRGINIA
Emergy provides a general accounting mechanism that allows us to view the economy and the environment on the same income statement and balance sheet. This allows an auditor to verify the economic picture by checking it against a more complete representation of the flows and stora...
Emergy evaluation of management measures for derelict fishing gears in Korea
NASA Astrophysics Data System (ADS)
Hong, Sunwook; Lee, Jongmyoung; Kang, Daeseok
2015-09-01
Derelict fishing gears (DFGs) have been recognized as an important marine debris issue that requires urgent action in Korea. This study used the emergy methodology to evaluate the cost efficiency of three DFG management measures of Korea from a biophysical perspective. Selected measures were the cleanup-using-ships (CL) program, the buyback (BB) program, and the floating reception barge (FB) program that ranged from more preventive approaches to curative ones in the management spectrum of DFGs. Among three measures, the CL program was the costliest one to implement both in emergy and monetary terms. The FB program that is closest to the preventive side of the spectrum required the least emergy input and monetary cost. The emergy requirements per ton of DFGs collected for the CL and BB programs were 4.4 and 3.6 times that of the FB program, respectively. The CL and BB program also required more inputs in monetary terms - 8.6 and 6.8 times higher, respectively, than the FB program did. The closer a measure is to the preventive side of the management spectrum, the less costly its implementation. This suggests that in terms of cost to implement, the current DFG management policy needs to be reorganized so that preventive measures can play a larger role in dealing with DFG issues in Korea.
Relationships among the energy, emergy, and money flows of the United States from 1900 to 2011.
Energy Systems Language models of the resource base for the U.S. economy and of economic exchange were used, respectively, (1) to show how energy consumption and emergy use contribute to real and nominal gross domestic product (GDP) and (2) to propose a model of coupled flows tha...
Emergy and Evaluating Ecosystem Services in a Sumatran Peat Swamp, Indonesia. Chapter 13
The purpose of this article is to document preliminary investigations into valuing peat swamp ecosystem function and services in Indonesia and to compare these values with the current development alternative, Acacia pulp wood plantation, using the emergy methodology. The Zamrud N...
LCA and emergy accounting of aquaculture systems: towards ecological intensification.
Wilfart, Aurélie; Prudhomme, Jehane; Blancheton, Jean-Paul; Aubin, Joël
2013-05-30
An integrated approach is required to optimise fish farming systems by maximising output while minimising their negative environmental impacts. We developed a holistic approach to assess the environmental performances by combining two methods based on energetic and physical flow analysis. Life Cycle Assessment (LCA) is a normalised method that estimates resource use and potential impacts throughout a product's life cycle. Emergy Accounting (EA) refers the amount of energy directly or indirectly required by a product or a service. The combination of these two methods was used to evaluate the environmental impacts of three contrasting fish-farming systems: a farm producing salmon in a recirculating system (RSF), a semi-extensive polyculture pond (PF1) and an extensive polyculture pond (PF2). The RSF system, with a low feed-conversion ratio (FCR = 0.95), had lower environmental impacts per tonne of live fish produced than did the two pond farms, when the effects on climate change, acidification, total cumulative energy demand, land competition and water dependence were considered. However, RSF was clearly disconnected from the surrounding environment and depended highly on external resources (e.g. nutrients, energy). Ponds adequately incorporated renewable natural resources but had higher environmental impacts due to incomplete use of external inputs. This study highlighted key factors necessary for the successful ecological intensification of fish farming, i.e., minimise external inputs, lower the FCR, and increase the use of renewable resources from the surrounding environment. The combination of LCA and EA seems to be a practical approach to address the complexity of optimising biophysical efficiency in aquaculture systems.
Prediction of Algebraic Instabilities
NASA Astrophysics Data System (ADS)
Zaretzky, Paula; King, Kristina; Hill, Nicole; Keithley, Kimberlee; Barlow, Nathaniel; Weinstein, Steven; Cromer, Michael
2016-11-01
A widely unexplored type of hydrodynamic instability is examined - large-time algebraic growth. Such growth occurs on the threshold of (exponentially) neutral stability. A new methodology is provided for predicting the algebraic growth rate of an initial disturbance, when applied to the governing differential equation (or dispersion relation) describing wave propagation in dispersive media. Several types of algebraic instabilities are explored in the context of both linear and nonlinear waves.
Connecting Algebra and Chemistry.
ERIC Educational Resources Information Center
O'Connor, Sean
2003-01-01
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
Bicovariant quantum algebras and quantum Lie algebras
NASA Astrophysics Data System (ADS)
Schupp, Peter; Watts, Paul; Zumino, Bruno
1993-10-01
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun(mathfrak{G}_q ) to U q g, given by elements of the pure braid group. These operators—the “reflection matrix” Y≡L + SL - being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for Y in SO q (N).
ERIC Educational Resources Information Center
Cavanagh, Sean
2008-01-01
A popular humorist and avowed mathphobe once declared that in real life, there's no such thing as algebra. Kathie Wilson knows better. Most of the students in her 8th grade class will be thrust into algebra, the definitive course that heralds the beginning of high school mathematics, next school year. The problem: Many of them are about three…
Parastatistics Algebras and Combinatorics
NASA Astrophysics Data System (ADS)
Popov, T.
2005-03-01
We consider the algebras spanned by the creation parafermionic and parabosonic operators which give rise to generalized parastatistics Fock spaces. The basis of such a generalized Fock space can be labelled by Young tableaux which are combinatorial objects. By means of quantum deformations a nice combinatorial structure of the algebra of the plactic monoid that lies behind the parastatistics is revealed.
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
Learning Activity Package, Algebra.
ERIC Educational Resources Information Center
Evans, Diane
A set of ten teacher-prepared Learning Activity Packages (LAPs) in beginning algebra and nine in intermediate algebra, these units cover sets, properties of operations, number systems, open expressions, solution sets of equations and inequalities in one and two variables, exponents, factoring and polynomials, relations and functions, radicals,…
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this dissertation,…
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
Ren, Jingzheng; Liang, Hanwei; Dong, Liang; Sun, Lu; Gao, Zhiqiu
2016-08-15
Industrial symbiosis provides novel and practical pathway to the design for the sustainability. Decision support tool for its verification is necessary for practitioners and policy makers, while to date, quantitative research is limited. The objective of this work is to present an innovative approach for supporting decision-making in the design for the sustainability with the implementation of industrial symbiosis in chemical complex. Through incorporating the emergy theory, the model is formulated as a multi-objective approach that can optimize both the economic benefit and sustainable performance of the integrated industrial system. A set of emergy based evaluation index are designed. Multi-objective Particle Swarm Algorithm is proposed to solve the model, and the decision-makers are allowed to choose the suitable solutions form the Pareto solutions. An illustrative case has been studied by the proposed method, a few of compromises between high profitability and high sustainability can be obtained for the decision-makers/stakeholders to make decision.
The joint use of LCA and emergy evaluation for the analysis of two Italian wine farms.
Pizzigallo, A C I; Granai, C; Borsa, S
2008-01-01
The aim of this paper is to evaluate two agroindustrial productive processes in their entirety (one organic and one semi-industrial), focusing on the comparison of impacts derived from the inputs and outputs of the system (life cycle assessment, LCA), integrated with a physical evaluation of the resources and natural services, on a common basis (emergy). Methods based on the joint use of LCA and emergy evaluation are useful, as they measure the contribution of environmental services and products to the productive process thus focusing primarily on the environmental impact of emissions and non-renewable energy inputs. The complementarity of the methods used in this paper contributes important elements and information useful for the comprehension of the organization of agriculture within Siena's territory. The results show important elements and useful information: (1) for the comprehension of the two agroecosystems' organization; (2) for the use of the energy flows that determine their development. Moreover, the combined use of emergy and LCA gives a comparative thermodynamic performance evaluation between organic and semi-industrial farming.
Algebraic Nonlinear Collective Motion
NASA Astrophysics Data System (ADS)
Troupe, J.; Rosensteel, G.
1998-11-01
Finite-dimensional Lie algebras of vector fields determine geometrical collective models in quantum and classical physics. Every set of vector fields on Euclidean space that generates the Lie algebra sl(3, R) and contains the angular momentum algebra so(3) is determined. The subset of divergence-free sl(3, R) vector fields is proven to be indexed by a real numberΛ. TheΛ=0 solution is the linear representation that corresponds to the Riemann ellipsoidal model. The nonlinear group action on Euclidean space transforms a certain family of deformed droplets among themselves. For positiveΛ, the droplets have a neck that becomes more pronounced asΛincreases; for negativeΛ, the droplets contain a spherical bubble of radius |Λ|1/3. The nonlinear vector field algebra is extended to the nonlinear general collective motion algebra gcm(3) which includes the inertia tensor. The quantum algebraic models of nonlinear nuclear collective motion are given by irreducible unitary representations of the nonlinear gcm(3) Lie algebra. These representations model fissioning isotopes (Λ>0) and bubble and two-fluid nuclei (Λ<0).
Li, Chun-fa; Cao, Ying-ying; Yang, Jian-cho; Yang, Qi-qi
2015-08-01
Dynamic evaluation of sustainable development is one of the key fundamental parts of the success of Sino-Singapore Tianjin Eco-city, which is the first eco-city in China constructed by international cooperation. Based on the analysis of nature and economy, function and structure, planning control indices and so on, we constructed a sustainable development evaluation index system and a system dynamics model of Sino-Singapore Tianjin Eco-city to explore dynamic trends of its population, material and currency by comprehensive utilization of emergy analysis and system dynamics method. Five scenarios were set up and simulated, including inertial scenario, scientific and technological scenario, economic scenario, environmental scenario and harmonious development scenario. Then, the sustainability of the 5 scenarios was evaluated and compared. The results showed that in the economy and environment sustainable development scenario, there was a steady growth trend of GDP, accumulation of both emergy and currency, and relatively lower values in emergy waste ratio, emergy ratio of waste, and emergy loading ratio. Although both sustainable evaluation indices, such as ESI and UEI, were relatively low, the economy and environment sustainable development scenario was still the best development scenario which was more active than others.
Algebraic invariants for homotopy types
NASA Astrophysics Data System (ADS)
Blanc, David
1999-11-01
We define a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the [Pi]-algebra [pi][low asterisk]X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract [Pi]-algebra can be realized as the homotopy [Pi]-algebra of a space.
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number,…
Emergy-based comparative analysis of energy intensity in different industrial systems.
Liu, Zhe; Geng, Yong; Wang, Hui; Sun, Lu; Ma, Zhixiao; Tian, Xu; Yu, Xiaoman
2015-12-01
With the rapid economic development, energy consumption of China has been the second place in the world next to the USA. Usually, measuring energy consumption intensity or efficiency applies heat unit which is joule per gross domestic production (GDP) or coal equivalent per GDP. However, this measuring approach is only oriented by the conversion coefficient of heat combustion which does not match the real value of the materials during their formation in the ecological system. This study applied emergy analysis to evaluate the energy consumption intensity to fill this gap. Emergy analysis is considered as a bridge between ecological system and economic system, which can evaluate the contribution of ecological products and services as well as the load placed on environmental systems. In this study, emergy indicator for performing energy consumption intensity of primary energy was proposed. Industrial production is assumed as the main contributor of energy consumption compared to primary and tertiary industries. Therefore, this study validated this method by investigating the two industrial case studies which were Dalian Economic Development Area (DEDA) and Fuzhou economic and technological area (FETA), to comparatively study on their energy consumption intensity between the different kinds of industrial systems and investigate the reasons behind the differences. The results show that primary energy consumption (PEC) of DEDA was much higher than that of FETA during 2006 to 2010 and its primary energy consumption ratio (PECR) to total emergy involvement had a dramatically decline from year 2006 to 2010. In the same time, nonrenewable energy of PEC in DEDA was also much higher than that in FETA. The reason was that industrial structure of DEDA was mainly formed by heavy industries like petro-chemistry industry, manufacturing industries, and high energy-intensive industries. However, FETA was formed by electronic business, food industry, and light industries. Although
Pseudo-Riemannian Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2008-08-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.
NASA Astrophysics Data System (ADS)
Markarian, Nikita
2017-03-01
We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must be perturbative Chern-Simons invariants.
Developing Algebraic Thinking.
ERIC Educational Resources Information Center
Alejandre, Suzanne
2002-01-01
Presents a teaching experience that resulted in students getting to a point of full understanding of the kinesthetic activity and the algebra behind it. Includes a lesson plan for a traffic jam activity. (KHR)
Jordan Algebraic Quantum Categories
NASA Astrophysics Data System (ADS)
Graydon, Matthew; Barnum, Howard; Ududec, Cozmin; Wilce, Alexander
2015-03-01
State cones in orthodox quantum theory over finite dimensional complex Hilbert spaces enjoy two particularly essential features: homogeneity and self-duality. Orthodox quantum theory is not, however, unique in that regard. Indeed, all finite dimensional formally real Jordan algebras -- arenas for generalized quantum theories with close algebraic kinship to the orthodox theory -- admit homogeneous self-dual positive cones. We construct categories wherein these theories are unified. The structure of composite systems is cast from universal tensor products of the universal C*-algebras enveloping ambient spaces for the constituent state cones. We develop, in particular, a notion of composition that preserves the local distinction of constituent systems in quaternionic quantum theory. More generally, we explicitly derive the structure of hybrid quantum composites with subsystems of arbitrary Jordan algebraic type.
Accounting Equals Applied Algebra.
ERIC Educational Resources Information Center
Roberts, Sondra
1997-01-01
Argues that students should be given mathematics credits for completing accounting classes. Demonstrates that, although the terminology is different, the mathematical concepts are the same as those used in an introductory algebra class. (JOW)
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Algebraic quantum gravity (AQG): II. Semiclassical analysis
NASA Astrophysics Data System (ADS)
Giesel, K.; Thiemann, T.
2007-05-01
In the previous paper (Giesel and Thiemann 2006 Conceptual setup Preprint gr-qc/0607099) a new combinatorial and thus purely algebraical approach to quantum gravity, called algebraic quantum gravity (AQG), was introduced. In the framework of AQG, existing semiclassical tools can be applied to operators that encode the dynamics of AQG such as the master constraint operator. In this paper, we will analyse the semiclassical limit of the (extended) algebraic master constraint operator and show that it reproduces the correct infinitesimal generators of general relativity. Therefore, the question of whether general relativity is included in the semiclassical sector of the theory, which is still an open problem in LQG, can be significantly improved in the framework of AQG. For the calculations, we will substitute SU(2) with U(1)3. That this substitution is justified will be demonstrated in the third paper (Giesel and Thiemann 2006 Semiclassical perturbation theory Preprint gr-qc/0607101) of this series.
Algebraic mesh quality metrics
KNUPP,PATRICK
2000-04-24
Quality metrics for structured and unstructured mesh generation are placed within an algebraic framework to form a mathematical theory of mesh quality metrics. The theory, based on the Jacobian and related matrices, provides a means of constructing, classifying, and evaluating mesh quality metrics. The Jacobian matrix is factored into geometrically meaningful parts. A nodally-invariant Jacobian matrix can be defined for simplicial elements using a weight matrix derived from the Jacobian matrix of an ideal reference element. Scale and orientation-invariant algebraic mesh quality metrics are defined. the singular value decomposition is used to study relationships between metrics. Equivalence of the element condition number and mean ratio metrics is proved. Condition number is shown to measure the distance of an element to the set of degenerate elements. Algebraic measures for skew, length ratio, shape, volume, and orientation are defined abstractly, with specific examples given. Combined metrics for shape and volume, shape-volume-orientation are algebraically defined and examples of such metrics are given. Algebraic mesh quality metrics are extended to non-simplical elements. A series of numerical tests verify the theoretical properties of the metrics defined.
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
NASA Astrophysics Data System (ADS)
Durka, R.
2017-04-01
The S-expansion framework is analyzed in the context of a freedom in closing the multiplication tables for the abelian semigroups. Including the possibility of the zero element in the resonant decomposition, and associating the Lorentz generator with the semigroup identity element, leads to a wide class of the expanded Lie algebras introducing interesting modifications to the gauge gravity theories. Among the results, we find all the Maxwell algebras of type {{B}m} , {{C}m} , and the recently introduced {{D}m} . The additional new examples complete the resulting generalization of the bosonic enlargements for an arbitrary number of the Lorentz-like and translational-like generators. Some further prospects concerning enlarging the algebras are discussed, along with providing all the necessary constituents for constructing the gravity actions based on the obtained results.
NASA Astrophysics Data System (ADS)
Roytenberg, Dmitry
2007-11-01
A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and certain "up to homotopy" structures on the complex; for strictly skew-symmetric Lie 2-algebras these are L∞-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer-Cartan equation in some differential graded Lie algebras and L∞-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
A Holistic Approach to Algebra.
ERIC Educational Resources Information Center
Barbeau, Edward J.
1991-01-01
Described are two examples involving recursive mathematical sequences designed to integrate a holistic approach to learning algebra. These examples promote pattern recognition with algebraic justification, full class participation, and mathematical values that can be transferred to other situations. (MDH)
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
An algebra of reversible computation.
Wang, Yong
2016-01-01
We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.
Liu, Zhe; Geng, Yong; Zhang, Pan; Dong, Huijuan; Liu, Zuoxi
2014-09-01
In China, local governments of many areas prefer to give priority to the development of heavy industrial clusters in pursuit of high value of gross domestic production (GDP) growth to get political achievements, which usually results in higher costs from ecological degradation and environmental pollution. Therefore, effective methods and reasonable evaluation system are urgently needed to evaluate the overall efficiency of industrial clusters. Emergy methods links economic and ecological systems together, which can evaluate the contribution of ecological products and services as well as the load placed on environmental systems. This method has been successfully applied in many case studies of ecosystem but seldom in industrial clusters. This study applied the methodology of emergy analysis to perform the efficiency of industrial clusters through a series of emergy-based indices as well as the proposed indicators. A case study of Shenyang Economic Technological Development Area (SETDA) was investigated to show the emergy method's practical potential to evaluate industrial clusters to inform environmental policy making. The results of our study showed that the industrial cluster of electric equipment and electronic manufacturing produced the most economic value and had the highest efficiency of energy utilization among the four industrial clusters. However, the sustainability index of the industrial cluster of food and beverage processing was better than the other industrial clusters.
Emergy and economic methods were used to evaluate and compare a traditional tropical fruit cultivation system, for bananas, and three newly introduced fruit cultivation systems, for papaya, guava and wampee, on reclaimed wetlands of the Pearl River Estuary, China. The evaluations...
Algebraic parameters identification of DC motors: methodology and analysis
NASA Astrophysics Data System (ADS)
Becedas, J.; Mamani, G.; Feliu, V.
2010-10-01
A fast, non-asymptotic, algebraic parameter identification method is applied to an uncertain DC motor to estimate the uncertain parameters: viscous friction coefficient and inertia. In this work, the methodology is developed and analysed, its convergence, a comparative study between the traditional recursive least square method and the algebraic identification method is carried out, and an analysis of the estimator in a noisy system is presented. Computer simulations were carried out to validate the suitability of the identification algorithm.
ERIC Educational Resources Information Center
Ketterlin-Geller, Leanne R.; Jungjohann, Kathleen; Chard, David J.; Baker, Scott
2007-01-01
Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems.…
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
Algebraic Thinking through Origami.
ERIC Educational Resources Information Center
Higginson, William; Colgan, Lynda
2001-01-01
Describes the use of paper folding to create a rich environment for discussing algebraic concepts. Explores the effect that changing the dimensions of two-dimensional objects has on the volume of related three-dimensional objects. (Contains 13 references.) (YDS)
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
NASA Astrophysics Data System (ADS)
Chen, Dan; Luo, Zhaohui; Webber, Michael; Chen, Jing; Wang, Weiguang
2014-09-01
Emergy theory and method are used to evaluate the contribution of irrigation water, and the process of its utilization, in three agricultural systems. The agricultural systems evaluated in this study were rice, wheat, and oilseed rape productions in an irrigation pumping district of China. A corresponding framework for emergy evaluation and sensitivity analysis methods was proposed. Two new indices, the fraction of irrigation water ( FIW), and the irrigation intensity of agriculture ( IIA), were developed to depict the contribution of irrigation water. The calculated FIW indicated that irrigation water used for the rice production system (34.7%) contributed more than irrigation water used for wheat (5.3%) and oilseed rape (11.2%) production systems in a typical dry year. The wheat production with an IIA of 19.0 had the highest net benefit from irrigation compared to the rice (2.9) and oilseed rape (8.9) productions. The transformities of the systems' products represented different energy efficiencies for rice (2.50E + 05 sej·J-1), wheat (1.66E + 05 sej·J-1) and oilseed rape (2.14E + 05 sej·J-1) production systems. According to several emergy indices, of the three systems evaluated, the rice system had the greatest level of sustainability. However, all of them were less sustainable than the ecological agricultural systems. A sensitivity analysis showed that the emergy inputs of irrigation water and nitrogenous fertilizer were the highest sensitivity factors influencing the emergy ratios. Best Management Practices, and other agroecological strategies, could be implemented to make further improvements in the sustainability of the three systems.
Chen, Han-Shen
2017-01-30
In this paper, the overall ecological and environmental sustainability in the Cing-Jing region in Taiwan is examined. As land use and cover change has been found to be an important analysis method, an emergy ecological footprint model was applied and the eco-security assessed to ensure authorities maintain a balance between ecological preservation and tourism development. While the ecological environment in the Cing-Jing region from 2008 to 2014 was found to be within safe levels, all related indices had increased considerably. A Grey model was used to predict the 2015-2024 ecological carrying capacities, from which it was found that there is expected to be a large increase in per capita ecological footprints (EFs), meaning that in the future there is going to be a larger ecological deficit and a higher ecological pressure index (EFI), with the eco-security predicted to reach a Grade 2 intermediate level in 2022. As the Cing-Jing region is predicted to become ecologically unsustainable, local, regional, and national governments need to implement regulations to strictly control the land use in the Cing-Jing region. This study demonstrated that emergy EF (EEF) theory application can give objective guidance to decision-makers to ensure that recreational non-urban eco-security can be maintained at a safe level.
Emergy analysis of the recycling options for construction and demolition waste.
Yuan, Fang; Shen, Li-yin; Li, Qi-ming
2011-12-01
Construction and demolition (C&D) waste is becoming a major contributor to environmental pollution. In Shanghai, China, the quantity of C&D waste is 2.11E+07 t/yr, which accounts for 45% of the total quantity of solid waste. There has been a growing promotion of recycling C&D waste as an effective way to solve this waste problem. However, the evaluation of the efficiency of recycling C&D waste as a potential source of resources is largely based on traditional economic analysis. The economic analysis emphasizes money instead of the harmony between economic benefit and environmental effects. There is a need for a new strategic approach to investigate the efficiency of recycling C&D waste to achieve the integration between economic, social and environmental effects. Emergy theory can be employed to analyze different recycling options for C&D waste. With reference to the Chinese construction industry, this paper demonstrates that the close-loop recycling option is better than the open-loop recycling option for C&D waste in terms of the integration of social, environmental and sustainable aspects. To evaluate different technology solutions for C&D waste recycling, the emergy theory and method is not limited to a cost-benefit balance but can include economic, social, environmental and sustainable effects.
Chen, Han-Shen
2017-01-01
In this paper, the overall ecological and environmental sustainability in the Cing-Jing region in Taiwan is examined. As land use and cover change has been found to be an important analysis method, an emergy ecological footprint model was applied and the eco-security assessed to ensure authorities maintain a balance between ecological preservation and tourism development. While the ecological environment in the Cing-Jing region from 2008 to 2014 was found to be within safe levels, all related indices had increased considerably. A Grey model was used to predict the 2015–2024 ecological carrying capacities, from which it was found that there is expected to be a large increase in per capita ecological footprints (EFs), meaning that in the future there is going to be a larger ecological deficit and a higher ecological pressure index (EFI), with the eco-security predicted to reach a Grade 2 intermediate level in 2022. As the Cing-Jing region is predicted to become ecologically unsustainable, local, regional, and national governments need to implement regulations to strictly control the land use in the Cing-Jing region. This study demonstrated that emergy EF (EEF) theory application can give objective guidance to decision-makers to ensure that recreational non-urban eco-security can be maintained at a safe level. PMID:28146086
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
On Dunkl angular momenta algebra
NASA Astrophysics Data System (ADS)
Feigin, Misha; Hakobyan, Tigran
2015-11-01
We consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl( N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.
EMERGY ANALYSIS AND ECONOMIC ANALYSIS A COMPARATIVE STUDY
Our mission at USEPA is to protect human health and safeguard the natural environment. We aim to base our environmental regulations and policies on sound scientific and, where appropriate, economic analyses. Although EPA has conducted analysis of the impact of regulations on ...
Marquette, Ian
2013-07-15
We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
Clifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras
NASA Astrophysics Data System (ADS)
Catto, Sultan; Gürcan, Yasemin; Khalfan, Amish; Kurt, Levent; Kato La, V.
2016-10-01
We discuss a construction scheme for Clifford numbers of arbitrary dimension. The scheme is based upon performing direct products of the Pauli spin and identity matrices. Conjugate fermionic algebras can then be formed by considering linear combinations of the Clifford numbers and the Hermitian conjugates of such combinations. Fermionic algebras are important in investigating systems that follow Fermi-Dirac statistics. We will further comment on the applications of Clifford algebras to Fueter analyticity, twistors, color algebras, M-theory and Leech lattice as well as unification of ancient and modern geometries through them.
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.
Quantum computation using geometric algebra
NASA Astrophysics Data System (ADS)
Matzke, Douglas James
This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.
ERIC Educational Resources Information Center
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
Applications of algebraic grid generation
NASA Technical Reports Server (NTRS)
Eiseman, Peter R.; Smith, Robert E.
1990-01-01
Techniques and applications of algebraic grid generation are described. The techniques are univariate interpolations and transfinite assemblies of univariate interpolations. Because algebraic grid generation is computationally efficient, the use of interactive graphics in conjunction with the techniques is advocated. A flexible approach, which works extremely well in an interactive environment, called the control point form of algebraic grid generation is described. The applications discussed are three-dimensional grids constructed about airplane and submarine configurations.
Ren, Jingzheng; Manzardo, Alessandro; Mazzi, Anna; Fedele, Andrea; Scipioni, Antonio
2013-01-01
Biodiesel as a promising alternative energy resource has been a hot spot in chemical engineering nowadays, but there is also an argument about the sustainability of biodiesel. In order to analyze the sustainability of biodiesel production systems and select the most sustainable scenario, various kinds of crop-based biodiesel including soybean-, rapeseed-, sunflower-, jatropha- and palm-based biodiesel production options are studied by emergy analysis; soybean-based scenario is recognized as the most sustainable scenario that should be chosen for further study in China. DEA method is used to evaluate the sustainability efficiencies of these options, and the biodiesel production systems based on soybean, sunflower, and palm are considered as DEA efficient, whereas rapeseed-based and jatropha-based scenarios are needed to be improved, and the improved methods have also been specified.
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
An Example of Competence-Based Learning: Use of Maxima in Linear Algebra for Engineers
ERIC Educational Resources Information Center
Diaz, Ana; Garcia, Alfonsa; de la Villa, Agustin
2011-01-01
This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is…
What ''Counts'' as Algebra in the Eyes of Preservice Elementary Teachers?
ERIC Educational Resources Information Center
Stephens, Ana C.
2008-01-01
This study examined conceptions of algebra held by 30 preservice elementary teachers. In addition to exploring participants' general ''definitions'' of algebra, this study examined, in particular, their analyses of tasks designed to engage students in relational thinking or a deep understanding of the equal sign as well as student work on these…
Zhang, Yiran; Liu, Jian; Zhang, Jian; Wang, Renqing
2013-07-01
Constructed wetland has been widely adopted to deal with degraded natural wetlands and water bodies; thus, more attention should be focused on ecological-economic sustainability and ecological efficiency of these projects for long-term success. Emergy accounting was conducted to investigate the energy and resource flows in constructed wetlands during the restoration process. Emergy-based indexes were adopted to evaluate the sustainability of a pilot large-scale constructed wetland in a large wetland restoration project in North China, carried out to enhance the river water quality and offset the degradation of natural wetland. Emergy and emdollar values for ecosystem services and natural capital were also calculated. The results showed that when outflow was considered as the product, the studied large-scale constructed wetland was more self-supporting and could be operated with lesser financial investment, although the waste treatment efficiency and the sustainability index were lower than conventional small-scale treatment constructed wetlands. Compared with natural wetlands, more visits from tourists and lesser financial investment coming in as feedback into the wetland would reduce system environment loading and promote system self-support ability, ultimately generating sustainability. In addition, the studied large-scale constructed wetland can effectively simulate energy and resource flows of natural wetland ecosystem and contribute a roughly equal value of ecosystem services in term of gross primary production. The studied large-scale constructed wetland can successfully achieve ecosystem functions as replacement for natural wetland and hasten the restoration process, although the restoration effectiveness of ecosystem structures in terms of living biomass and water using emergy-value accounting is still inconclusive.
The worth of land use: a GIS-emergy evaluation of natural and human-made capital.
Mellino, Salvatore; Buonocore, Elvira; Ulgiati, Sergio
2015-02-15
Natural systems make their natural capital and ecosystem services available to human economy. A careful analysis of the interplay between natural and human-made capital is needed to prevent natural capital being overexploited for present economic benefits, affecting lifestyles and wellbeing of future generations. In this study, the emergy synthesis is used to evaluate the natural and the human-made capital of Campania region (southern Italy) by accounting for the environmental support directly and indirectly provided by nature to resource generation. Furthermore, geographic information system (GIS) models are integrated with the emergy accounting procedure to generate maps of the spatial patterns of both natural and human-made capital distribution. Regional storages of natural and human-made capital are identified and evaluated in emergy units (seJ). The human-made capital of the Campania region (6.29E+24seJ) results to be about 11 times higher than the natural capital (5.69E+23seJ) due to the past and present exploitation of the natural resources needed to generate it over time. Moreover, by overlaying the total natural capital map and the total human-made capital map with a map of the protected areas within the region, only the 19% of the regional natural capital appears to be concentrated within protected areas, while most of it (81%) is concentrated outside. These findings suggest that the conservation of natural resources is also necessary outside protected areas by means of suitable policies, directives and investments. The human-made capital is mainly concentrated (88%) inside non-protected areas and interacts with the local natural capital. A management of the interactions between the two categories of wealth is crucial to prevent that the growth of human-made storages degrades the natural ecosystems and the environment. The proposed emergy-GIS framework reveals to be a useful tool for environmental planning and resource management aimed to conserve and
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
Viterbi/algebraic hybrid decoder
NASA Technical Reports Server (NTRS)
Boyd, R. W.; Ingels, F. M.; Mo, C.
1980-01-01
Decoder computer program is hybrid between optimal Viterbi and optimal algebraic decoders. Tests have shown that hybrid decoder outperforms any strictly Viterbi or strictly algebraic decoder and effectively handles compound channels. Algorithm developed uses syndrome-detecting logic to direct two decoders to assume decoding load alternately, depending on real-time channel characteristics.
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
Linear Algebra and Image Processing
ERIC Educational Resources Information Center
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
Learning Algebra from Worked Examples
ERIC Educational Resources Information Center
Lange, Karin E.; Booth, Julie L.; Newton, Kristie J.
2014-01-01
For students to be successful in algebra, they must have a truly conceptual understanding of key algebraic features as well as the procedural skills to complete a problem. One strategy to correct students' misconceptions combines the use of worked example problems in the classroom with student self-explanation. "Self-explanation" is the…
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
The Algebra of Complex Numbers.
ERIC Educational Resources Information Center
LePage, Wilbur R.
This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…
Thermodynamics. [algebraic structure
NASA Technical Reports Server (NTRS)
Zeleznik, F. J.
1976-01-01
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
Symplectic Clifford Algebraic Field Theory.
NASA Astrophysics Data System (ADS)
Dixon, Geoffrey Moore
We develop a mathematical framework on which is built a theory of fermion, scalar, and gauge vector fields. This field theory is shown to be equivalent to the original Weinberg-Salam model of weak and electromagnetic interactions, but since the new framework is more rigid than that on which the original Weinberg-Salam model was built, a concomitant reduction in the number of assumptions lying outside of the framework has resulted. In particular, parity violation is actually hiding within our framework, and with little difficulty we are able to manifest it. The mathematical framework upon which we build our field theory is arrived at along two separate paths. The first is by the marriage of a Clifford algebra and a Lie superalgebra, the result being called a super Clifford algebra. The second is by providing a new characterization for a Clifford algebra employing its generators and a symmetric array of metric coefficients. Subsequently we generalize this characterization to the case of an antisymmetric array of metric coefficients, and we call the algebra which results a symplectic Clifford algebra. It is upon one of these that we build our field theory, and it is shown that this symplectic Clifford algebra is a particular subalgebra of a super Clifford algebra. The final ingredient is the operation of bracketing which involves treating the elements of our algebra as endomorphisms of a particular inner product space, and employing this space and its inner product to provide us with maps from our algebra to the reals. It is this operation which enables us to manifest the parity violation hiding in our algebra.
ERIC Educational Resources Information Center
Gonzalez-Vega, Laureano
1999-01-01
Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
Chinese Whispers - Algebra Style: Grammatical, Notational, Mathematical and Activity Tensions
ERIC Educational Resources Information Center
Hewitt, Dave
2005-01-01
This paper analyses students' written work from an activity based on two well known games in the UK: Chinese Whispers and Consequences. Within this activity students were asked to translate formal algebraic equations into word statements and vice versa. Using the framework of affordances and constraints to offer an account for what the students'…
Investigating Teacher Noticing of Student Algebraic Thinking
ERIC Educational Resources Information Center
Walkoe, Janet Dawn Kim
2013-01-01
Learning algebra is critical for students in the U.S. today. Algebra concepts provide the foundation for much advanced mathematical content. In addition, algebra serves as a gatekeeper to opportunities such as admission to college. Yet many students in the U.S. struggle in algebra classes. Researchers claim that one reason for these difficulties…
Central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, M.; Sheinman, O. K.
2008-08-01
Lax operator algebras were introduced by Krichever and Sheinman as a further development of Krichever's theory of Lax operators on algebraic curves. These are almost-graded Lie algebras of current type. In this paper local cocycles and associated almost-graded central extensions of Lax operator algebras are classified. It is shown that in the case when the corresponding finite-dimensional Lie algebra is simple the two-cohomology space is one-dimensional. An important role is played by the action of the Lie algebra of meromorphic vector fields on the Lax operator algebra via suitable covariant derivatives.
Asymptotic aspect of derivations in Banach algebras.
Roh, Jaiok; Chang, Ick-Soon
2017-01-01
We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.
Computing Matrix Representations of Filiform Lie Algebras
NASA Astrophysics Data System (ADS)
Ceballos, Manuel; Núñez, Juan; Tenorio, Ángel F.
In this paper, we compute minimal faithful unitriangular matrix representations of filiform Lie algebras. To do it, we use the nilpotent Lie algebra, g_n, formed of n ×n strictly upper-triangular matrices. More concretely, we search the lowest natural number n such that the Lie algebra g_n contains a given filiform Lie algebra, also computing a representative of this algebra. All the computations in this paper have been done using MAPLE 9.5.
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
GCD, LCM, and Boolean Algebra?
ERIC Educational Resources Information Center
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
NASA Technical Reports Server (NTRS)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
Coherent States for Hopf Algebras
NASA Astrophysics Data System (ADS)
Škoda, Zoran
2007-07-01
Families of Perelomov coherent states are defined axiomatically in the context of unitary representations of Hopf algebras. A global geometric picture involving locally trivial noncommutative fibre bundles is involved in the construction. If, in addition, the Hopf algebra has a left Haar integral, then a formula for noncommutative resolution of identity in terms of the family of coherent states holds. Examples come from quantum groups.
Multiplier operator algebras and applications
Blecher, David P.; Zarikian, Vrej
2004-01-01
The one-sided multipliers of an operator space X are a key to “latent operator algebraic structure” in X. We begin with a survey of these multipliers, together with several of the applications that they have had to operator algebras. We then describe several new results on one-sided multipliers, and new applications, mostly to one-sided M-ideals. PMID:14711990
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
Algebraic approximations for transcendental equations with applications in nanophysics
NASA Astrophysics Data System (ADS)
Barsan, Victor
2015-09-01
Using algebraic approximations of trigonometric or hyperbolic functions, a class of transcendental equations can be transformed in tractable, algebraic equations. Studying transcendental equations this way gives the eigenvalues of Sturm-Liouville problems associated to wave equation, mainly to Schroedinger equation; these algebraic approximations provide approximate analytical expressions for the energy of electrons and phonons in quantum wells, quantum dots (QDs) and quantum wires, in the frame of one-particle models of such systems. The advantage of this approach, compared to the numerical calculations, is that the final result preserves the functional dependence on the physical parameters of the problem. The errors of this method, situated between some few percentages and ?, are carefully analysed. Several applications, for quantum wells, QDs and quantum wires, are presented.
An ecological compensation standard based on emergy theory for the Xiao Honghe River Basin.
Guan, Xinjian; Chen, Moyu; Hu, Caihong
2015-01-01
The calculation of an ecological compensation standard is an important, but also difficult aspect of current ecological compensation research. In this paper, the factors affecting the ecological-economic system in the Xiao Honghe River Basin, China, including the flow of energy, materials, and money, were calculated using the emergy analysis method. A consideration of the relationships between the ecological-economic value of water resources and ecological compensation allowed the ecological-economic value to be calculated. On this basis, the amount of water needed for dilution was used to develop a calculation model for the ecological compensation standard of the basin. Using the Xiao Honghe River Basin as an example, the value of water resources and the ecological compensation standard were calculated using this model according to the emission levels of the main pollutant in the basin, chemical oxygen demand. The compensation standards calculated for the research areas in Xipin, Shangcai, Pingyu, and Xincai were 34.91 yuan/m3, 32.97 yuan/m3, 35.99 yuan/m3, and 34.70 yuan/m3, respectively, and such research output would help to generate and support new approaches to the long-term ecological protection of the basin and improvement of the ecological compensation system.
A reexamination of the emergy input to a system from the wind ...
With the establishment of a new, rigorously-determined, solar equivalence baseline for the geobiosphere, 12.0E+24 seJ y-1, it is now appropriate to reexamine the calculation of the emergy delivered by the major secondary products of the geobiosphere, e.g., wind and rainfall, which are derived from the transformation of the solar equivalent joules supplied through the baseline. In this study, the methods for calculating the available energy of the wind dissipated in the planetary boundary layer are revisited and further elucidated. Particular consideration is given to the method used to estimate the geostrophic and gradient wind from measurements of surface wind and to the role of the drag coefficient in determining the available wind energy dissipated in the boundary layer (900 to 1000 mb or the lower 1000 m of the atmosphere) as it passes over various surfaces. In addition, we made a more rigorous estimate of the transformity of the available wind energy dissipated in the planetary boundary layer based on a synthesis of the results from three evaluations of a model of the general circulation of the atmosphere. The rounded estimate of the transformity of the wind from these combined studies was 1230 sej J-1. We consider the variability of the transformity of the wind dissipated in the boundary layer between summer and winter and between the Northern and Southern hemispheres. We conclude that the properties of the system and its spatial and temporal boundarie
Novikov algebras with associative bilinear forms
NASA Astrophysics Data System (ADS)
Zhu, Fuhai; Chen, Zhiqi
2007-11-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. The goal of this paper is to study Novikov algebras with non-degenerate associative symmetric bilinear forms, which we call quadratic Novikov algebras. Based on the classification of solvable quadratic Lie algebras of dimension not greater than 4 and Novikov algebras in dimension 3, we show that quadratic Novikov algebras up to dimension 4 are commutative. Furthermore, we obtain the classification of transitive quadratic Novikov algebras in dimension 4. But we find that not every quadratic Novikov algebra is commutative and give a non-commutative quadratic Novikov algebra in dimension 6.
NASA Astrophysics Data System (ADS)
Siche, Raúl; Pereira, Lucas; Agostinho, Feni; Ortega, Enrique
2010-10-01
In the last decade, two scientific tools have been extensively used worldwide to measure the human impact on nature: ecological footprint (EF) and emergy analysis (EA). Papers trying to combine the strong points of EF and EA, and obtain more accurate results have appeared in scientific literature, in which Zhao's et al. (2005) [61] approach is an important one. Unfortunately, some weak points of the original methods still remain on the new approaches proposed. The aim of this present work is to discuss some weak points found in Zhao's approach, trying to overcome them through a new approach called emergetic ecological footprint (EEF). The main difference between Zhao's approach and EEF is that the last one accounted for the internal storage of capital natural in the biocapacity calculation. Besides that, soil loss and water for human consume were considered as additional categories in the footprint calculation. After discussing it through comparisons with other approaches, EEF was used to assess Peru as a case study, resulting in a biocapacity of 51.76 gha capita-1 and a footprint of 12.23 gha capita-1, with 2004 data; that resulted in an ecological surplus of 39.53 gha capita-1. The load capacity factor obtained was 4.23, meaning that Peru can support a population 4.23 times bigger considering the life style of 2004. The main limitations of the EEF are: (i) it is impossible to make comparisons between the biocapacity and footprint for each category; (ii) a need for a handbook with emergy intensity factors with good quality. On the other hand, the main positive points are: (i) its easiness of application in global and national scales; (ii) its final indicators account for all the previous energy (or emergy) used to make something; (iii) internal natural capital storage was accounted for in the biocapacity calculation, which can be a valid step towards the evaluation and assess of services provided by nature.
Quantum Q systems: from cluster algebras to quantum current algebras
NASA Astrophysics Data System (ADS)
Di Francesco, Philippe; Kedem, Rinat
2017-02-01
This paper gives a new algebraic interpretation for the algebra generated by the quantum cluster variables of the A_r quantum Q-system (Di Francesco and Kedem in Int Math Res Not IMRN 10:2593-2642, 2014). We show that the algebra can be described as a quotient of the localization of the quantum algebra U_{√{q}}({n}[u,u^{-1}])subset U_{√{q}}(widehat{{sl}}_2), in the Drinfeld presentation. The generating current is made up of a subset of the cluster variables which satisfy the Q-system, which we call fundamental. The other cluster variables are given by a quantum determinant-type formula, and are polynomials in the fundamental generators. The conserved quantities of the discrete evolution (Di Francesco and Kedem in Adv Math 228(1):97-152, 2011) described by quantum Q-system generate the Cartan currents at level 0, in a non-standard polarization. The rest of the quantum affine algebra is also described in terms of cluster variables.
Dang, Xiaohu; Liu, Guobin
2012-07-15
Evaluating the sustainability of a target region for an ecological restoration programme is challenging because it involves different aspects of human society and environment as well as multiple disciplines. Carrying capacity provides a useful measure of the sustainability of a given region where an ecological restoration programme is implemented. In this article, the Yangou catchment, a geomorphic unit, was used as a case study in the Loess Hilly Region of China, where emergy synthesis was used to measure the environmental resources base. The specific standard of living in terms of emergy was employed to calculate carrying capacity over the period 1998-2005 and to assess the sustainability of the Yangou catchment where an ecological restoration programme was carried out. The results of the evaluation indicated that after implementing the ecological restoration programme, there was some improvement in the environmental aspects of the Yangou catchment during the study period, suggesting that the ecological restoration programme alleviated ecological degradation. However, several emergy-based indices and the support areas also illustrated that the ecological restoration programme was not successful enough in terms of preservation and utilisation of environmental resources to enhance sustainability. This indicates that further actions are necessary on conserving environmental resources, improving the emergy input structure for agricultural production and in lifestyle changes for the local people in living in the Yangou catchment.
Moving frames and prolongation algebras
NASA Technical Reports Server (NTRS)
Estabrook, F. B.
1982-01-01
Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.
Algebraic Lattices in QFT Renormalization
NASA Astrophysics Data System (ADS)
Borinsky, Michael
2016-07-01
The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the sets of subdivergences of Feynman diagrams form algebraic lattices. This class of QFTs includes the standard model. In kinematic renormalization schemes, in which tadpole diagrams vanish, these lattices are semimodular. This implies that the Hopf algebra of Feynman diagrams is graded by the coradical degree or equivalently that every maximal forest has the same length in the scope of BPHZ renormalization. As an application of this framework, a formula for the counter terms in zero-dimensional QFT is given together with some examples of the enumeration of primitive or skeleton diagrams.
Colored Quantum Algebra and Its Bethe State
NASA Astrophysics Data System (ADS)
Wang, Jin-Zheng; Jia, Xiao-Yu; Wang, Shi-Kun
2014-12-01
We investigate the colored Yang—Baxter equation. Based on a trigonometric solution of colored Yang—Baxter equation, we construct a colored quantum algebra. Moreover we discuss its algebraic Bethe ansatz state and highest wight representation.
Using Number Theory to Reinforce Elementary Algebra.
ERIC Educational Resources Information Center
Covillion, Jane D.
1995-01-01
Demonstrates that using the elementary number theory in algebra classes helps students to use acquired algebraic skills as well as helping them to more clearly understand concepts that are presented. Discusses factoring, divisibility rules, and number patterns. (AIM)
Algebraic orbifold conformal field theories
Xu, Feng
2000-01-01
The unitary rational orbifold conformal field theories in the algebraic quantum field theory and subfactor theory framework are formulated. Under general conditions, it is shown that the orbifold of a given unitary rational conformal field theory generates a unitary modular category. Many new unitary modular categories are obtained. It is also shown that the irreducible representations of orbifolds of rank one lattice vertex operator algebras give rise to unitary modular categories and determine the corresponding modular matrices, which has been conjectured for some time. PMID:11106383
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Eco-exergy and emergy based self-organization of three forest plantations in lower subtropical China
Lu, Hongfang; Fu, Fangyan; Li, Hao; Campbell, Daniel E.; Ren, Hai
2015-01-01
The bio-thermodynamic structures of a mixed native species plantation, a conifer plantation and an Acacia mangium plantation in Southern China were quantified over a period of 15 years based on eco-exergy methods. The efficiencies of structural development and maintenance were quantified through an integrated application of eco-exergy and emergy methods. The results showed that the storage of eco-exergy increased over 3 times in all three plantations, as predicted by the maximum eco-exergy principle. This trend was primarily seen due to the accumulation of biomass, instead of an increase in the specific eco-exergy (eco-exergy per unit biomass), although species richness did increase. The eco-exergy to emergy and eco-exergy to empower ratios of the three plantations generally increased during the study period, but the rate of increase slowed down after 20 years. The dominant trees are the largest contributors to the eco-exergy stored in the plantations, and thus, the introduction of suitable indigenous tree species should be considered after the existing trees pass through their period of most rapid growth or around 20 years after planting. The combined application of C-values and suggested weighting factors in the eco-exergy calculation can imply opposite results, but may also supply useful information for forest management. PMID:26486821
NASA Astrophysics Data System (ADS)
Ingwersen, Wesley W.
Life cycle assessment (LCA) is an internationally standardized framework for assessing the environmental impacts of products that is rapidly evolving to improve understanding and quantification of how complex product systems depend upon and affect the environment. This dissertation contributes to that evolution through the development of new methods for measuring impacts, estimating the uncertainty of impacts, and measuring ranges of environmental performance, with a focus on product systems in non-OECD countries that have not been well characterized. The integration of a measure of total energy use, emergy, is demonstrated in an LCA of gold from the Yanacocha mine in Peru in the second chapter. A model for estimating the accuracy of emergy results is proposed in the following chapter. The fourth chapter presents a template for LCA-based quantification of the range of environmental performance for tropical agricultural products using the example of fresh pineapple production for export in Costa Rica that can be used to create product labels with environmental information. The final chapter synthesizes how each methodological contribution will together improve the science of measuring product environmental performance.
Lu, Hongfang; Fu, Fangyan; Li, Hao; Campbell, Daniel E; Ren, Hai
2015-10-21
The bio-thermodynamic structures of a mixed native species plantation, a conifer plantation and an Acacia mangium plantation in Southern China were quantified over a period of 15 years based on eco-exergy methods. The efficiencies of structural development and maintenance were quantified through an integrated application of eco-exergy and emergy methods. The results showed that the storage of eco-exergy increased over 3 times in all three plantations, as predicted by the maximum eco-exergy principle. This trend was primarily seen due to the accumulation of biomass, instead of an increase in the specific eco-exergy (eco-exergy per unit biomass), although species richness did increase. The eco-exergy to emergy and eco-exergy to empower ratios of the three plantations generally increased during the study period, but the rate of increase slowed down after 20 years. The dominant trees are the largest contributors to the eco-exergy stored in the plantations, and thus, the introduction of suitable indigenous tree species should be considered after the existing trees pass through their period of most rapid growth or around 20 years after planting. The combined application of C-values and suggested weighting factors in the eco-exergy calculation can imply opposite results, but may also supply useful information for forest management.
Symmetry algebras of linear differential equations
NASA Astrophysics Data System (ADS)
Shapovalov, A. V.; Shirokov, I. V.
1992-07-01
The local symmetries of linear differential equations are investigated by means of proven theorems on the structure of the algebra of local symmetries of translationally and dilatationally invariant differential equations. For a nonparabolic second-order equation, the absence of nontrivial nonlinear local symmetries is proved. This means that the local symmetries reduce to the Lie algebra of linear differential symmetry operators. For the Laplace—Beltrami equation, all local symmetries reduce to the enveloping algebra of the algebra of the conformal group.
Spatial-Operator Algebra For Robotic Manipulators
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Milman, Mark H.
1991-01-01
Report discusses spatial-operator algebra developed in recent studies of mathematical modeling, control, and design of trajectories of robotic manipulators. Provides succinct representation of mathematically complicated interactions among multiple joints and links of manipulator, thereby relieving analyst of most of tedium of detailed algebraic manipulations. Presents analytical formulation of spatial-operator algebra, describes some specific applications, summarizes current research, and discusses implementation of spatial-operator algebra in the Ada programming language.
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
Applications of Algebraic Logic and Universal Algebra to Computer Science
1989-06-21
conference, with roughly equal representation from Mathematics and Computer Science . The conference consisted of eight invited lectures (60 minutes...each) and 26 contributed talks (20-40 minutes each). There was also a round-table discussion on the role of algebra and logic in computer science . Keywords
A Balancing Act: Making Sense of Algebra
ERIC Educational Resources Information Center
Gavin, M. Katherine; Sheffield, Linda Jensen
2015-01-01
For most students, algebra seems like a totally different subject than the number topics they studied in elementary school. In reality, the procedures followed in arithmetic are actually based on the properties and laws of algebra. Algebra should be a logical next step for students in extending the proficiencies they developed with number topics…
Algebra? A Gate! A Barrier! A Mystery!
ERIC Educational Resources Information Center
Mathematics Educatio Dialogues, 2000
2000-01-01
This issue of Mathematics Education Dialogues focuses on the nature and the role of algebra in the K-14 curriculum. Articles on this theme include: (1) "Algebra For All? Why?" (Nel Noddings); (2) "Algebra For All: It's a Matter of Equity, Expectations, and Effectiveness" (Dorothy S. Strong and Nell B. Cobb); (3) "Don't Delay: Build and Talk about…
Unifying the Algebra for All Movement
ERIC Educational Resources Information Center
Eddy, Colleen M.; Quebec Fuentes, Sarah; Ward, Elizabeth K.; Parker, Yolanda A.; Cooper, Sandi; Jasper, William A.; Mallam, Winifred A.; Sorto, M. Alejandra; Wilkerson, Trena L.
2015-01-01
There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the "Algebra for All" movement. In light of this…
UCSMP Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…
Constraint-Referenced Analytics of Algebra Learning
ERIC Educational Resources Information Center
Sutherland, Scot M.; White, Tobin F.
2016-01-01
The development of the constraint-referenced analytics tool for monitoring algebra learning activities presented here came from the desire to firstly, take a more quantitative look at student responses in collaborative algebra activities, and secondly, to situate those activities in a more traditional introductory algebra setting focusing on…
Embedding Algebraic Thinking throughout the Mathematics Curriculum
ERIC Educational Resources Information Center
Vennebush, G. Patrick; Marquez, Elizabeth; Larsen, Joseph
2005-01-01
This article explores the algebra that can be uncovered in many middle-grades mathematics tasks that, on first inspection, do not appear to be algebraic. It shows connections to the other four Standards that occur in traditional algebra problems, and it offers strategies for modifying activities so that they can be used to foster algebraic…
Teaching Strategies to Improve Algebra Learning
ERIC Educational Resources Information Center
Zbiek, Rose Mary; Larson, Matthew R.
2015-01-01
Improving student learning is the primary goal of every teacher of algebra. Teachers seek strategies to help all students learn important algebra content and develop mathematical practices. The new Institute of Education Sciences[IES] practice guide, "Teaching Strategies for Improving Algebra Knowledge in Middle and High School Students"…
Build an Early Foundation for Algebra Success
ERIC Educational Resources Information Center
Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela
2016-01-01
Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…
Teacher Actions to Facilitate Early Algebraic Reasoning
ERIC Educational Resources Information Center
Hunter, Jodie
2015-01-01
In recent years there has been an increased emphasis on integrating the teaching of arithmetic and algebra in primary school classrooms. This requires teachers to develop links between arithmetic and algebra and use pedagogical actions that facilitate algebraic reasoning. Drawing on findings from a classroom-based study, this paper provides an…
Difficulties in Initial Algebra Learning in Indonesia
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-01-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was…
Cyclic homology for Hom-associative algebras
NASA Astrophysics Data System (ADS)
Hassanzadeh, Mohammad; Shapiro, Ilya; Sütlü, Serkan
2015-12-01
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
Algebra for All: California’s Eighth-Grade Algebra Initiative as Constrained Curricula
Domina, Thurston; Penner, Andrew M.; Penner, Emily K.; Conley, Annemarie
2015-01-01
prior achievement. Findings/Results We find that students’ odds of taking higher level mathematics courses increased as this district implemented the state’s Algebra mandate. However, even as the district implemented a constrained curriculum strategy, mathematics achievement growth between 6th sixth and 10th grade slowed and the achievement advantages associated with 8th eighth grade Algebra declined. Conclusions/Recommendations Our analyses suggest that curricular intensification increased the inclusiveness and decreased the selectivity of the mathematics tracking regime in Towering Pines middle schools. However, the findings suggest that this constrained curriculum strategy may have may have unintended negative consequences for student achievement. PMID:26120219
Carry Groups: Abstract Algebra Projects
ERIC Educational Resources Information Center
Miller, Cheryl Chute; Madore, Blair F.
2004-01-01
Carry Groups are a wonderful collection of groups to introduce in an undergraduate Abstract Algebra course. These groups are straightforward to define but have interesting structures for students to discover. We describe these groups and give examples of in-class group projects that were developed and used by Miller.
Algebra, Home Mortgages, and Recessions
ERIC Educational Resources Information Center
Mariner, Jean A. Miller; Miller, Richard A.
2009-01-01
The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…
Exploring Algebraic Misconceptions with Technology
ERIC Educational Resources Information Center
Sakow, Matthew; Karaman, Ruveyda
2015-01-01
Many students struggle with algebra, from simplifying expressions to solving systems of equations. Students also have misconceptions about the meaning of variables. In response to the question "Can x + y + z ever equal x + p + z?" during a student interview, the student claimed, "Never . . . because p has to have a different value…
Easing Students' Transition to Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2006-01-01
Traditionally, students learn arithmetic throughout their primary schooling, and this is seen as the ideal preparation for the learning of algebra in the junior secondary school. The four operations are taught and rehearsed in the early years and from this, it is assumed, "children will induce the fundamental structure of arithmetic" (Warren &…
Algebra for All. Research Brief
ERIC Educational Resources Information Center
Bleyaert, Barbara
2009-01-01
The call for "algebra for all" is not a recent phenomenon. Concerns about the inadequacy of math (and science) preparation in America's high schools have been a steady drumbeat since the 1957 launch of Sputnik; a call for raising standards and the number of math (and science) courses required for graduation has been a part of countless…
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary…
Adventures in Flipping College Algebra
ERIC Educational Resources Information Center
Van Sickle, Jenna
2015-01-01
This paper outlines the experience of a university professor who implemented flipped learning in two sections of college algebra courses for two semesters. It details how the courses were flipped, what technology was used, advantages, challenges, and results. It explains what students do outside of class, what they do inside class, and discusses…
Elementary Algebra Connections to Precalculus
ERIC Educational Resources Information Center
Lopez-Boada, Roberto; Daire, Sandra Arguelles
2013-01-01
This article examines the attitudes of some precalculus students to solve trigonometric and logarithmic equations and systems using the concepts of elementary algebra. With the goal of enticing the students to search for and use connections among mathematical topics, they are asked to solve equations or systems specifically designed to allow…
Celestial mechanics with geometric algebra
NASA Technical Reports Server (NTRS)
Hestenes, D.
1983-01-01
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.
Math Sense: Algebra and Geometry.
ERIC Educational Resources Information Center
Howett, Jerry
This book is designed to help students gain the range of math skills they need to succeed in life, work, and on standardized tests; overcome math anxiety; discover math as interesting and purposeful; and develop good number sense. Topics covered in this book include algebra and geometry. Lessons are organized around four strands: (1) skill lessons…
Weaving Geometry and Algebra Together
ERIC Educational Resources Information Center
Cetner, Michelle
2015-01-01
When thinking about student reasoning and sense making, teachers must consider the nature of tasks given to students along with how to plan to use the tasks in the classroom. Students should be presented with tasks in a way that encourages them to draw connections between algebraic and geometric concepts. This article focuses on the idea that it…
Algebraic Activities Aid Discovery Lessons
ERIC Educational Resources Information Center
Wallace-Gomez, Patricia
2013-01-01
After a unit on the rules for positive and negative numbers and the order of operations for evaluating algebraic expressions, many students believe that they understand these principles well enough, but they really do not. They clearly need more practice, but not more of the same kind of drill. Wallace-Gomez provides three graphing activities that…
Teachers' Understanding of Algebraic Generalization
NASA Astrophysics Data System (ADS)
Hawthorne, Casey Wayne
Generalization has been identified as a cornerstone of algebraic thinking (e.g., Lee, 1996; Sfard, 1995) and is at the center of a rich conceptualization of K-8 algebra (Kaput, 2008; Smith, 2003). Moreover, mathematics teachers are being encouraged to use figural-pattern generalizing tasks as a basis of student-centered instruction, whereby teachers respond to and build upon the ideas that arise from students' explorations of these activities. Although more and more teachers are engaging their students in such generalizing tasks, little is known about teachers' understanding of generalization and their understanding of students' mathematical thinking in this domain. In this work, I addressed this gap, exploring the understanding of algebraic generalization of 4 exemplary 8th-grade teachers from multiple perspectives. A significant feature of this investigation is an examination of teachers' understanding of the generalization process, including the use of algebraic symbols. The research consisted of two phases. Phase I was an examination of the teachers' understandings of the underlying quantities and quantitative relationships represented by algebraic notation. In Phase II, I observed the instruction of 2 of these teachers. Using the lens of professional noticing of students' mathematical thinking, I explored the teachers' enacted knowledge of algebraic generalization, characterizing how it supported them to effectively respond to the needs and queries of their students. Results indicated that teachers predominantly see these figural patterns as enrichment activities, disconnected from course content. Furthermore, in my analysis, I identified conceptual difficulties teachers experienced when solving generalization tasks, in particular, connecting multiple symbolic representations with the quantities in the figures. Moreover, while the teachers strived to overcome the challenges of connecting different representations, they invoked both productive and unproductive
A graph algebra for scalable visual analytics.
Shaverdian, Anna A; Zhou, Hao; Michailidis, George; Jagadish, Hosagrahar V
2012-01-01
Visual analytics (VA), which combines analytical techniques with advanced visualization features, is fast becoming a standard tool for extracting information from graph data. Researchers have developed many tools for this purpose, suggesting a need for formal methods to guide these tools' creation. Increased data demands on computing requires redesigning VA tools to consider performance and reliability in the context of analysis of exascale datasets. Furthermore, visual analysts need a way to document their analyses for reuse and results justification. A VA graph framework encapsulated in a graph algebra helps address these needs. Its atomic operators include selection and aggregation. The framework employs a visual operator and supports dynamic attributes of data to enable scalable visual exploration of data.
Algebraically special Einstein-Maxwell fields
NASA Astrophysics Data System (ADS)
Van den Bergh, Norbert
2017-01-01
The Geroch-Held-Penrose formalism is used to re-analyse algebraically special non-null Einstein-Maxwell fields, aligned as well as non-aligned, in the presence of a possible non-vanishing cosmological constant. A new invariant characterization is given of the García-Plebański and Plebański-Hacyan metrics within the family of aligned solutions and of the Griffiths metrics within the family of the non-aligned solutions. As a corollary also the double alignment of the Debever-McLenaghan `class D' metrics with non-vanishing cosmological constant is shown to be equivalent with the shear-free and geodesic behavior of their Debever-Penrose vectors.
Explicit field realizations of W algebras
NASA Astrophysics Data System (ADS)
Wei, Shao-Wen; Liu, Yu-Xiao; Zhang, Li-Jie; Ren, Ji-Rong
2009-06-01
The fact that certain nonlinear W2,s algebras can be linearized by the inclusion of a spin-1 current can provide a simple way to realize W2,s algebras from linear W1,2,s algebras. In this paper, we first construct the explicit field realizations of linear W1,2,s algebras with double scalar and double spinor, respectively. Then, after a change of basis, the realizations of W2,s algebras are presented. The results show that all these realizations are Romans-type realizations.
Array algebra estimation in signal processing
NASA Astrophysics Data System (ADS)
Rauhala, U. A.
A general theory of linear estimators called array algebra estimation is interpreted in some terms of multidimensional digital signal processing, mathematical statistics, and numerical analysis. The theory has emerged during the past decade from the new field of a unified vector, matrix and tensor algebra called array algebra. The broad concepts of array algebra and its estimation theory cover several modern computerized sciences and technologies converting their established notations and terminology into one common language. Some concepts of digital signal processing are adopted into this language after a review of the principles of array algebra estimation and its predecessors in mathematical surveying sciences.
On special classes of n-algebras
NASA Astrophysics Data System (ADS)
Vainerman, L.; Kerner, R.
1996-05-01
We define n-algebras as linear spaces on which the internal composition law involves n elements: m:V⊗n■V. It is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics. Using the notion of a commutant of two subalgebras of an n-algebra, we distinguish certain classes of n-algebras with reasonable properties: semisimple, Abelian, nilpotent, solvable. We also consider a few examples of n-algebras of different types, and show their properties.
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.
Powell, Sarah R.; Fuchs, Lynn S.
2014-01-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2nd- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty. PMID:25309044
Recursion and feedback in image algebra
NASA Astrophysics Data System (ADS)
Ritter, Gerhard X.; Davidson, Jennifer L.
1991-04-01
Recursion and feedback are two important processes in image processing. Image algebra, a unified algebraic structure developed for use in image processing and image analysis, provides a common mathematical environment for expressing image processing transforms. It is only recently that image algebra has been extended to include recursive operations [1]. Recently image algebra was shown to incorporate neural nets [2], including a new type of neural net, the morphological neural net [3]. This paper presents the relationship of the recursive image algebra to the field of fractions of the ring of matrices, and gives the two dimensional moving average filter as an example. Also, the popular multilayer perceptron with back propagation and a morphology neural network with learning rule are presented in image algebra notation. These examples show that image algebra can express these important feedback concepts in a succinct way.
Deformed Kac Moody and Virasoro algebras
NASA Astrophysics Data System (ADS)
Balachandran, A. P.; Queiroz, A. R.; Marques, A. M.; Teotonio-Sobrinho, P.
2007-07-01
Whenever the group {\\bb R}^n acts on an algebra {\\cal A} , there is a method to twist \\cal A to a new algebra {\\cal A}_\\theta which depends on an antisymmetric matrix θ (θμν = -θνμ = constant). The Groenewold-Moyal plane {\\cal A}_\\theta({\\bb R}^{d+1}) is an example of such a twisted algebra. We give a general construction to realize this twist in terms of {\\cal A} itself and certain 'charge' operators Qμ. For {\\cal A}_\\theta({\\bb R}^{d+1}), Q_\\mu are translation generators. This construction is then applied to twist the oscillators realizing the Kac-Moody (KM) algebra as well as the KM currents. They give different deformations of the KM algebra. From one of the deformations of the KM algebra, we construct, via the Sugawara construction, the Virasoro algebra. These deformations have an implication for statistics as well.
Algebraic complexities and algebraic curves over finite fields
Chudnovsky, D. V.; Chudnovsky, G. V.
1987-01-01
We consider the problem of minimal (multiplicative) complexity of polynomial multiplication and multiplication in finite extensions of fields. For infinite fields minimal complexities are known [Winograd, S. (1977) Math. Syst. Theory 10, 169-180]. We prove lower and upper bounds on minimal complexities over finite fields, both linear in the number of inputs, using the relationship with linear coding theory and algebraic curves over finite fields. PMID:16593816
DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J
2015-05-01
To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions.
Alternative algebraic approaches in quantum chemistry
Mezey, Paul G.
2015-01-22
Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.
Algebraic Methods to Design Signals
2015-08-27
group theory are employed to investigate the theory of their construction methods leading to new families of these arrays and some generalizations...sequences and arrays with desirable correlation properties. The methods used are very algebraic and number theoretic. Many new families of sequences...context of optical quantum computing, we prove that infinite families of anticirculant block weighing matrices can be obtained from generic weighing
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
BLAS- BASIC LINEAR ALGEBRA SUBPROGRAMS
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1994-01-01
The Basic Linear Algebra Subprogram (BLAS) library is a collection of FORTRAN callable routines for employing standard techniques in performing the basic operations of numerical linear algebra. The BLAS library was developed to provide a portable and efficient source of basic operations for designers of programs involving linear algebraic computations. The subprograms available in the library cover the operations of dot product, multiplication of a scalar and a vector, vector plus a scalar times a vector, Givens transformation, modified Givens transformation, copy, swap, Euclidean norm, sum of magnitudes, and location of the largest magnitude element. Since these subprograms are to be used in an ANSI FORTRAN context, the cases of single precision, double precision, and complex data are provided for. All of the subprograms have been thoroughly tested and produce consistent results even when transported from machine to machine. BLAS contains Assembler versions and FORTRAN test code for any of the following compilers: Lahey F77L, Microsoft FORTRAN, or IBM Professional FORTRAN. It requires the Microsoft Macro Assembler and a math co-processor. The PC implementation allows individual arrays of over 64K. The BLAS library was developed in 1979. The PC version was made available in 1986 and updated in 1988.
Introduction to Image Algebra Ada
NASA Astrophysics Data System (ADS)
Wilson, Joseph N.
1991-07-01
Image Algebra Ada (IAA) is a superset of the Ada programming language designed to support use of the Air Force Armament Laboratory's image algebra in the development of computer vision application programs. The IAA language differs from other computer vision languages is several respects. It is machine independent, and an IAA translator has been implemented in the military standard Ada language. Its image operands and operations can be used to program a range of both low- and high-level vision algorithms. This paper provides an overview of the image algebra constructs supported in IAA and describes the embodiment of these constructs in the IAA extension of Ada. Examples showing the use of IAA for a range of computer vision tasks are given. The design of IAA as a superset of Ada and the implementation of the initial translator in Ada represent critical choices. The authors discuss the reasoning behind these choices as well as the benefits and drawbacks associated with them. Implementation strategies associated with the use of Ada as an implementation language for IAA are also discussed. While one can look on IAA as a program design language (PDL) for specifying Ada programs, it is useful to consider IAA as a separate language superset of Ada. This admits the possibility of directly translating IAA for implementation on special purpose architectures. This paper explores strategies for porting IAA to various architectures and notes the critical language and implementation features for porting to different architectures.
Algebra: A Challenge at the Crossroads of Policy and Practice
ERIC Educational Resources Information Center
Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.
2011-01-01
The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…
Bilinear forms on fermionic Novikov algebras
NASA Astrophysics Data System (ADS)
Chen, Zhiqi; Zhu, Fuhai
2007-05-01
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. Fermionic Novikov algebras correspond to a certain Hamiltonian super-operator in a super-variable. In this paper, we show that there is a remarkable geometry on fermionic Novikov algebras with non-degenerate invariant symmetric bilinear forms, which we call pseudo-Riemannian fermionic Novikov algebras. They are related to pseudo-Riemannian Lie algebras. Furthermore, we obtain a procedure to classify pseudo-Riemannian fermionic Novikov algebras. As an application, we give the classification in dimension <=4. Motivated by the one in dimension 4, we construct some examples in high dimensions.
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
NASA Astrophysics Data System (ADS)
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Numerical linear algebra algorithms and software
NASA Astrophysics Data System (ADS)
Dongarra, Jack J.; Eijkhout, Victor
2000-11-01
The increasing availability of advanced-architecture computers has a significant effect on all spheres of scientific computation, including algorithm research and software development in numerical linear algebra. Linear algebra - in particular, the solution of linear systems of equations - lies at the heart of most calculations in scientific computing. This paper discusses some of the recent developments in linear algebra designed to exploit these advanced-architecture computers. We discuss two broad classes of algorithms: those for dense, and those for sparse matrices.
Symbolic Lie algebras manipulations using COMMON LISP
NASA Astrophysics Data System (ADS)
Cecchini, R.; Tarlini, M.
1989-01-01
We present a description and an implementation of a program in COMMON LISP to perform symbolic computations in a given Lie algebra. Using the general definitions of vector space Lie algebra and enveloping algebra, the program is able to compute commutators, to evaluate similarity transformations and the general Baker-Campbell-Hausdorff formula. All the computations are exact, including numerical coefficients. For the interactive user an optional menu facility and online help are available. LISP knowledge is unnecessary.
Lie algebras of classical and stochastic electrodynamics
NASA Astrophysics Data System (ADS)
Neto, J. J. Soares; Vianna, J. D. M.
1994-03-01
The Lie algebras associated with infinitesimal symmetry transformations of third-order differential equations of interest to classical electrodynamics and stochastic electrodynamics have been obtained. The structure constants for a general case are presented and the Lie algebra for each particular application is easily achieved. By the method used here it is not necessary to know the explicit expressions of the infinitesimal generators in order to determine the structure constants of the Lie algebra.
NASA Astrophysics Data System (ADS)
Manerowska, Anna; Nieznański, Edward; Mulawka, Jan
2013-10-01
Our aim is to present the algebra of concepts in two formal languages. First, after introducing a primary relation between concepts, which is subsumption, we shall specify in a language that uses quantifiers, the Boolean algebra of general concepts. Next, we shall note down the same algebra in simplified non-quantifying language, in order to use it as basis for two specific implementations, i.e. to create the Boolean algebras of deontic concepts and axiological concepts.
Wang, Chengdong; Zhang, Shenyan; Yan, Wanglin; Wang, Renqing; Liu, Jian; Wang, Yutao
2016-01-01
Renewable natural resources, such as solar radiation, rainfall, wind, and geothermal heat, together with ecosystem services, provide the elementary supports for the sustainable development of human society. To improve regional sustainability, we studied the spatial distributions and quantities of renewable natural resources and net primary productivity (NPP) in Hokkaido, which is the second largest island of Japan. With the help of Geographic Information System (GIS) software, distribution maps for each type of renewable natural resource were generated by kriging interpolation based on statistical records. A composite map of the flow of all types of renewable natural resources was also generated by map layer overlapping. Additionally, we utilized emergy analysis to convert each renewable flow with different attributes into a unified unit (i.e., solar equivalent joules [sej]). As a result, the spatial distributions of the flow of renewable natural resources of the Hokkaido region are presented in the form of thematic emergy maps. Thus, the areas with higher renewable emergy can be easily visualized and identified. The dominant renewable flow in certain areas can also be directly distinguished. The results can provide useful information for regional sustainable development, environmental conservation and ecological management. PMID:27857230
NASA Astrophysics Data System (ADS)
Wang, Chengdong; Zhang, Shenyan; Yan, Wanglin; Wang, Renqing; Liu, Jian; Wang, Yutao
2016-11-01
Renewable natural resources, such as solar radiation, rainfall, wind, and geothermal heat, together with ecosystem services, provide the elementary supports for the sustainable development of human society. To improve regional sustainability, we studied the spatial distributions and quantities of renewable natural resources and net primary productivity (NPP) in Hokkaido, which is the second largest island of Japan. With the help of Geographic Information System (GIS) software, distribution maps for each type of renewable natural resource were generated by kriging interpolation based on statistical records. A composite map of the flow of all types of renewable natural resources was also generated by map layer overlapping. Additionally, we utilized emergy analysis to convert each renewable flow with different attributes into a unified unit (i.e., solar equivalent joules [sej]). As a result, the spatial distributions of the flow of renewable natural resources of the Hokkaido region are presented in the form of thematic emergy maps. Thus, the areas with higher renewable emergy can be easily visualized and identified. The dominant renewable flow in certain areas can also be directly distinguished. The results can provide useful information for regional sustainable development, environmental conservation and ecological management.
Wang, Chengdong; Zhang, Shenyan; Yan, Wanglin; Wang, Renqing; Liu, Jian; Wang, Yutao
2016-11-18
Renewable natural resources, such as solar radiation, rainfall, wind, and geothermal heat, together with ecosystem services, provide the elementary supports for the sustainable development of human society. To improve regional sustainability, we studied the spatial distributions and quantities of renewable natural resources and net primary productivity (NPP) in Hokkaido, which is the second largest island of Japan. With the help of Geographic Information System (GIS) software, distribution maps for each type of renewable natural resource were generated by kriging interpolation based on statistical records. A composite map of the flow of all types of renewable natural resources was also generated by map layer overlapping. Additionally, we utilized emergy analysis to convert each renewable flow with different attributes into a unified unit (i.e., solar equivalent joules [sej]). As a result, the spatial distributions of the flow of renewable natural resources of the Hokkaido region are presented in the form of thematic emergy maps. Thus, the areas with higher renewable emergy can be easily visualized and identified. The dominant renewable flow in certain areas can also be directly distinguished. The results can provide useful information for regional sustainable development, environmental conservation and ecological management.
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-04-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Dispersion Operators Algebra and Linear Canonical Transformations
NASA Astrophysics Data System (ADS)
Andriambololona, Raoelina; Ranaivoson, Ravo Tokiniaina; Hasimbola Damo Emile, Randriamisy; Rakotoson, Hanitriarivo
2017-02-01
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is given.
Banach Algebras Associated to Lax Pairs
NASA Astrophysics Data System (ADS)
Glazebrook, James F.
2015-04-01
Lax pairs featuring in the theory of integrable systems are known to be constructed from a commutative algebra of formal pseudodifferential operators known as the Burchnall- Chaundy algebra. Such pairs induce the well known KP flows on a restricted infinite-dimensional Grassmannian. The latter can be exhibited as a Banach homogeneous space constructed from a Banach *-algebra. It is shown that this commutative algebra of operators generating Lax pairs can be associated with a commutative C*-subalgebra in the C*-norm completion of the *-algebra. In relationship to the Bose-Fermi correspondence and the theory of vertex operators, this C*-algebra has an association with the CAR algebra of operators as represented on Fermionic Fock space by the Gelfand-Naimark-Segal construction. Instrumental is the Plücker embedding of the restricted Grassmannian into the projective space of the associated Hilbert space. The related Baker and tau-functions provide a connection between these two C*-algebras, following which their respective state spaces and Jordan-Lie-Banach algebras structures can be compared.
Difficulties in initial algebra learning in Indonesia
NASA Astrophysics Data System (ADS)
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-12-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.
Multicloning and Multibroadcasting in Operator Algebras
NASA Astrophysics Data System (ADS)
Kaniowski, Krzysztof; Lubnauer, Katarzyna; Łuczak, Andrzej
2015-12-01
We investigate multicloning and multibroadcasting in the general operator algebra framework in arbitrary dimension, generalizing thus results obtained in this framework for simple cloning and broadcasting.
ERIC Educational Resources Information Center
Ozgun-Koca, S. Ash
2010-01-01
Although growing numbers of secondary school mathematics teachers and students use calculators to study graphs, they mainly rely on paper-and-pencil when manipulating algebraic symbols. However, the Computer Algebra Systems (CAS) on computers or handheld calculators create new possibilities for teaching and learning algebraic manipulation. This…
Classification of filiform Lie algebras of order 3
NASA Astrophysics Data System (ADS)
Navarro, Rosa María
2016-12-01
Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne's result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.
The Algebra of Lexical Semantics
NASA Astrophysics Data System (ADS)
Kornai, András
The current generative theory of the lexicon relies primarily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the generative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of describing discrepancies between syntactic type (lexical category) and semantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR.
Strengthening Effect Algebras in a Logical Perspective: Heyting-Wajsberg Algebras
NASA Astrophysics Data System (ADS)
Konig, Martinvaldo
2014-10-01
Heyting effect algebras are lattice-ordered pseudoboolean effect algebras endowed with a pseudocomplementation that maps on the center (i.e. Boolean elements). They are the algebraic counterpart of an extension of both Łukasiewicz many-valued logic and intuitionistic logic. We show that Heyting effect algebras are termwise equivalent to Heyting-Wajsberg algebras where the two different logical implications are defined as primitive operators. We prove this logic to be decidable, to be strongly complete and to have the deduction-detachment theorem.
Algebraic Thinking: A Problem Solving Approach
ERIC Educational Resources Information Center
Windsor, Will
2010-01-01
Algebraic thinking is a crucial and fundamental element of mathematical thinking and reasoning. It initially involves recognising patterns and general mathematical relationships among numbers, objects and geometric shapes. This paper will highlight how the ability to think algebraically might support a deeper and more useful knowledge, not only of…
Learning from Student Approaches to Algebraic Proofs
ERIC Educational Resources Information Center
D'Ambrosio, Beatriz S.; Kastberg, Signe E.; Viola dos Santos, Joao Ricardo
2010-01-01
Many mathematics teachers struggle to support their students' developing understanding of proof as an essential element in investigations of mathematics. The area of mathematics where the development of an understanding of proof is most challenging is algebra. In the case of algebraic proof, analysis of student written work on tasks that demand…
From operator algebras to superconformal field theory
Kawahigashi, Yasuyuki
2010-01-15
We survey operator algebraic approach to (super)conformal field theory. We discuss representation theory, classification results, full and boundary conformal field theories, relations to supervertex operator algebras and Moonshine, connections to subfactor theory of Jones, and certain aspects of noncommutative geometry of Connes.
A Technology-Intensive Approach to Algebra.
ERIC Educational Resources Information Center
Heid, M. Kathleen; Zbiek, Rose Mary
1995-01-01
Computer-Intensive Algebra (CIA) focuses on the use of technology to help develop a rich understanding of fundamental algebraic concepts in real-world settings using computing tools for easy access to numerical, graphical, and symbolic representations of mathematical ideas. (MKR)
Focus on Fractions to Scaffold Algebra
ERIC Educational Resources Information Center
Ooten, Cheryl Thomas
2013-01-01
Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…
Post-Lie Algebras and Isospectral Flows
NASA Astrophysics Data System (ADS)
Ebrahimi-Fard, Kurusch; Lundervold, Alexander; Mencattini, Igor; Munthe-Kaas, Hans Z.
2015-11-01
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.
Teaching Modeling and Axiomatization with Boolean Algebra.
ERIC Educational Resources Information Center
De Villiers, Michael D.
1987-01-01
Presented is an alternative approach to the traditional teaching of Boolean algebra for secondary school mathematics. The main aim of the approach is to use Boolean algebra to teach pupils such mathematical processes as modeling and axiomatization. A course using the approach is described. (RH)
Arithmetic and Cognitive Contributions to Algebra
ERIC Educational Resources Information Center
Cirino, Paul T.; Tolar, Tammy D.; Fuchs, Lynn S.
2013-01-01
Algebra is a prerequisite for access to STEM careers and occupational success (NMAP, 2008a), yet algebra is difficult for students through high school (US DOE, 2008). Growth in children's conceptual and procedural arithmetical knowledge is reciprocal, although conceptual knowledge has more impact on procedural knowledge than the reverse…
Algebraic Thinking through Koch Snowflake Constructions
ERIC Educational Resources Information Center
Ghosh, Jonaki B.
2016-01-01
Generalizing is a foundational mathematical practice for the algebra classroom. It entails an act of abstraction and forms the core of algebraic thinking. Kinach (2014) describes two kinds of generalization--by analogy and by extension. This article illustrates how exploration of fractals provides ample opportunity for generalizations of both…
Calif. Laws Shift Gears on Algebra, Textbooks
ERIC Educational Resources Information Center
Robelen, Erik W.
2012-01-01
New laws in California have set the state on a course for some potentially significant changes to the curriculum, including a measure that revisits the matter of teaching Algebra 1 in 8th grade and another that revamps the state's textbook-adoption process and hands districts greater leeway in choosing instructional materials. The algebra-related…
Using Students' Interests as Algebraic Models
ERIC Educational Resources Information Center
Whaley, Kenneth A.
2012-01-01
Fostering algebraic thinking is an important goal for middle-grades mathematics teachers. Developing mathematical reasoning requires that teachers cultivate students' habits of mind. Teachers develop students' understanding of algebra by engaging them in tasks that involve modeling and representation. This study was designed to investigate how…
An algebraic approach to the scattering equations
NASA Astrophysics Data System (ADS)
Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui
2015-12-01
We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.
THE RADICAL OF A JORDAN ALGEBRA
McCrimmon, Kevin
1969-01-01
In this paper we define a Jacobson radical for Jordan algebras analogous to that for associative algebras and show that it enjoys many of the properties of the associative radical. We then relate the corresponding notion of “semisimplicity” to the previously defined notion of “nondegeneracy” (Jacobson, N., these Proceedings, 55, 243-251 (1966)). PMID:16591736
Cartan calculus on quantum Lie algebras
Schupp, P.; Watts, P.; Zumino, B.
1993-12-09
A generalization of the differential geometry of forms and vector fields to the case of quantum Lie algebras is given. In an abstract formulation that incorporates many existing examples of differential geometry on quantum spaces we combine an exterior derivative, inner derivations, Lie derivatives, forms and functions au into one big algebra, the ``Cartan Calculus.``
The operator algebra approach to quantum groups
Kustermans, Johan; Vaes, Stefaan
2000-01-01
A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory. PMID:10639116
Using the Internet To Investigate Algebra.
ERIC Educational Resources Information Center
Sherwood, Walter
The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…
Teaching Algebra to Students with Learning Disabilities
ERIC Educational Resources Information Center
Impecoven-Lind, Linda S.; Foegen, Anne
2010-01-01
Algebra is a gateway to expanded opportunities, but it often poses difficulty for students with learning disabilities. Consequently, it is essential to identify evidence-based instructional strategies for these students. The authors begin by identifying three areas of algebra difficulty experienced by students with disabilities: cognitive…
Gary M. Klingler Algebra Teacher Assistance Packages
ERIC Educational Resources Information Center
Klingler, Gary
2005-01-01
Several packages designed by Elizabeth Marquez for mathematics teachers of grades 6-12, officially entitled the Teacher Assistance Package in Support of Better Algebra Assessment, is a series of resources developed to accompany ET's End-of-Course Algebra Assessment. It is designed to enhance teachers classroom assessment by providing examples of…
Just Say Yes to Early Algebra!
ERIC Educational Resources Information Center
Stephens, Ana; Blanton, Maria; Knuth, Eric; Isler, Isil; Gardiner, Angela Murphy
2015-01-01
Mathematics educators have argued for some time that elementary school students are capable of engaging in algebraic thinking and should be provided with rich opportunities to do so. Recent initiatives like the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) have taken up this call by reiterating the place of early algebra in…
Symbolic Notations and Students' Achievements in Algebra
ERIC Educational Resources Information Center
Peter, Ebiendele E.; Olaoye, Adetunji A.
2013-01-01
This study focuses on symbolic notations and its impact on students' achievement in Algebra. The main reason for this study rests on the observation from personal and professional experiences on students' increasing hatred for Algebra. One hundred and fifty (150) Senior Secondary School Students (SSS) from Ojo Local Education District, Ojo, Lagos,…
SAYD Modules over Lie-Hopf Algebras
NASA Astrophysics Data System (ADS)
Rangipour, Bahram; Sütlü, Serkan
2012-11-01
In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.
Algebraic Formulas for Areas between Curves.
ERIC Educational Resources Information Center
Gabai, Hyman
1982-01-01
Korean secondary school students preparing for college learn about a simple algebraic formula for area bounded by a parabola and line. The approach does not seem well-known among American students. It is noted that, while the formula derivations rely on integration, algebra students could use the formulas without proofs. (MP)
Some Applications of Algebraic System Solving
ERIC Educational Resources Information Center
Roanes-Lozano, Eugenio
2011-01-01
Technology and, in particular, computer algebra systems, allows us to change both the way we teach mathematics and the mathematical curriculum. Curiously enough, unlike what happens with linear system solving, algebraic system solving is not widely known. The aim of this paper is to show that, although the theory lying behind the "exact…
An Inquiry-Based Linear Algebra Class
ERIC Educational Resources Information Center
Wang, Haohao; Posey, Lisa
2011-01-01
Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…
Practicing Algebraic Skills: A Conceptual Approach
ERIC Educational Resources Information Center
Friedlander, Alex; Arcavi, Abraham
2012-01-01
Traditionally, a considerable part of teaching and learning algebra has focused on routine practice and the application of rules, procedures, and techniques. Although today's computerized environments may have decreased the need to master algebraic skills, procedural competence is still a central component in any mathematical activity. However,…
Success in Algebra among Community College Students
ERIC Educational Resources Information Center
Reyes, Czarina
2010-01-01
College algebra is a required course for most majors, but is viewed by many as a gatekeeper course for degree completion by students. With almost half a million students taking college algebra each year, faculty are experimenting with new course lengths of time that might result in higher success, completion, and retention rates for college…
Is Algebra Really Difficult for All Students?
ERIC Educational Resources Information Center
Egodawatte, Gunawardena
2009-01-01
Research studies have shown that students encounter difficulties in transitioning from arithmetic to algebra. Errors made by high school students were analyzed for patterns and their causes. The origins of errors were: intuitive assumptions, failure to understand the syntax of algebra, analogies with other familiar symbol systems such as the…
A systematic investigation of the link between rational number processing and algebra ability.
Hurst, Michelle; Cordes, Sara
2017-02-27
Recent research suggests that fraction understanding is predictive of algebra ability; however, the relative contributions of various aspects of rational number knowledge are unclear. Furthermore, whether this relationship is notation-dependent or rather relies upon a general understanding of rational numbers (independent of notation) is an open question. In this study, college students completed a rational number magnitude task, procedural arithmetic tasks in fraction and decimal notation, and an algebra assessment. Using these tasks, we measured three different aspects of rational number ability in both fraction and decimal notation: (1) acuity of underlying magnitude representations, (2) fluency with which symbols are mapped to the underlying magnitudes, and (3) fluency with arithmetic procedures. Analyses reveal that when looking at the measures of magnitude understanding, the relationship between adults' rational number magnitude performance and algebra ability is dependent upon notation. However, once performance on arithmetic measures is included in the relationship, individual measures of magnitude understanding are no longer unique predictors of algebra performance. Furthermore, when including all measures simultaneously, results revealed that arithmetic fluency in both fraction and decimal notation each uniquely predicted algebra ability. Findings are the first to demonstrate a relationship between rational number understanding and algebra ability in adults while providing a clearer picture of the nature of this relationship.
MODEL IDENTIFICATION AND COMPUTER ALGEBRA.
Bollen, Kenneth A; Bauldry, Shawn
2010-10-07
Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods.
LINPACK. Simultaneous Linear Algebraic Equations
Miller, M.A.
1990-05-01
LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).
LINPACK. Simultaneous Linear Algebraic Equations
Dongarra, J.J.
1982-05-02
LINPACK is a collection of FORTRAN subroutines which analyze and solve various classes of systems of simultaneous linear algebraic equations. The collection deals with general, banded, symmetric indefinite, symmetric positive definite, triangular, and tridiagonal square matrices, as well as with least squares problems and the QR and singular value decompositions of rectangular matrices. A subroutine-naming convention is employed in which each subroutine name consists of five letters which represent a coded specification (TXXYY) of the computation done by that subroutine. The first letter, T, indicates the matrix data type. Standard FORTRAN allows the use of three such types: S REAL, D DOUBLE PRECISION, and C COMPLEX. In addition, some FORTRAN systems allow a double-precision complex type: Z COMPLEX*16. The second and third letters of the subroutine name, XX, indicate the form of the matrix or its decomposition: GE General, GB General band, PO Positive definite, PP Positive definite packed, PB Positive definite band, SI Symmetric indefinite, SP Symmetric indefinite packed, HI Hermitian indefinite, HP Hermitian indefinite packed, TR Triangular, GT General tridiagonal, PT Positive definite tridiagonal, CH Cholesky decomposition, QR Orthogonal-triangular decomposition, SV Singular value decomposition. The final two letters, YY, indicate the computation done by the particular subroutine: FA Factor, CO Factor and estimate condition, SL Solve, DI Determinant and/or inverse and/or inertia, DC Decompose, UD Update, DD Downdate, EX Exchange. The LINPACK package also includes a set of routines to perform basic vector operations called the Basic Linear Algebra Subprograms (BLAS).
Hexagonal tessellations in image algebra
NASA Astrophysics Data System (ADS)
Eberly, David H.; Wenzel, Dennis J.; Longbotham, Harold G.
1990-11-01
In image algebra '' the concept of a coordinate set X is general in that such a set is simply a subset of ndimensional Euclidean space . The standard applications in 2-dimensional image processing use coordinate sets which are rectangular arrays X 72 x ZZm. However some applications may require other geometries for the coordinate set. We look at three such related applications in the context of image algebra. The first application is the modeling of photoreceptors in primate retinas. These receptors are inhomogeneously distributed on the retina. The largest receptor density occurs in the center of the fovea and decreases radially outwards. One can construct a hexagonal tessellation of the retina such that each hexagon contains approximately the same number of receptors. The resulting tessellation called a sunflower heart2 consists of concentric rings of hexagons whose sizes increase as the radius of the ring increases. The second application is the modeling of the primary visual . The neurons are assumed to be uniformly distributed as a regular hexagonal lattice. Cortical neural image coding is modeled by a recursive convolution of the retinal neural image using a special set of filters. The third application involves analysis of a hexagonally-tessellated image where the pixel resolution is variable .
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
ERIC Educational Resources Information Center
Van Galen, Jane, Ed.; And Others
1992-01-01
This theme issue of the serial "Educational Foundations" contains four articles devoted to the topic of "Sociopolitical Analyses." In "An Interview with Peter L. McLaren," Mary Leach presented the views of Peter L. McLaren on topics of local and national discourses, values, and the politics of difference. Landon E.…
Light polarization: A geometric-algebra approach
NASA Astrophysics Data System (ADS)
Baylis, W. E.; Bonenfant, J.; Derbyshire, J.; Huschilt, J.
1993-06-01
The geometric algebra of three-dimensional space (the ``Pauli algebra'') is known to provide an efficient geometric description of electromagnetic phenomena. Here, it is applied to the three-dimensional Stokes subspace to describe the polarization of an approximately monochromatic collimated beam of electromagnetic radiation. The coherency density ρ is a real element of the algebra whose components are the four Stokes parameters: a scalar representing the total photon flux density plus a three-dimensional vector whose direction and length in the Poincaré sphere give the type and degree of polarization. The detection of the radiation and the incoherent and coherent modification of the polarization by various optical elements are calculated by algebraic multiplication which has faithful representations in 2×2 matrices. One matrix representation of ρ is the coherency matrix with which Jones and Mueller matrices are related whereas another representation is the spin density matrix. However, the calculations are simplest to perform and interpret in the algebraic form independent of any particular matrix representation. It is shown that any possible change in the Stokes parameters can be treated algebraically by a combination of attenuation, depolarization, polarization, and rotation transformations of ρ. The geometric algebra thus unifies Stokes parameters, the Poincaré sphere, Jones and Mueller matrices, and the coherency and density matrices in a single, simple formalism.
Conformal current algebra in two dimensions
NASA Astrophysics Data System (ADS)
Ashok, Sujay K.; Benichou, Raphael; Troost, Jan
2009-06-01
We construct a non-chiral current algebra in two dimensions consistent with conformal invariance. We show that the conformal current algebra is realized in non-linear sigma-models on supergroup manifolds with vanishing Killing form, with or without a Wess-Zumino term. The current algebra is computed using two distinct methods. First we exploit special algebraic properties of supergroups to compute the exact two- and three-point functions of the currents and from them we infer the current algebra. The algebra is also calculated by using conformal perturbation theory about the Wess-Zumino-Witten point and resumming the perturbation series. We also prove that these models realize a non-chiral Kac-Moody algebra and construct an infinite set of commuting operators that is closed under the action of the Kac-Moody generators. The supergroup models that we consider include models with applications to statistical mechanics, condensed matter and string theory. In particular, our results may help to systematically solve and clarify the quantum integrability of PSU(n|n) models and their cosets, which appear prominently in string worldsheet models on anti-deSitter spaces.
Working memory, worry, and algebraic ability.
Trezise, Kelly; Reeve, Robert A
2014-05-01
Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.
Three-dimensional polarization algebra.
R Sheppard, Colin J; Castello, Marco; Diaspro, Alberto
2016-10-01
If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.
A Study of the Use of a Handheld Computer Algebra System in Discrete Mathematics
ERIC Educational Resources Information Center
Powers, Robert A.; Allison, Dean E.; Grassl, Richard M.
2005-01-01
This study investigated the impact of the TI-92 handheld Computer Algebra System (CAS) on student achievement in a discrete mathematics course. Specifically, the researchers examined the differences between a CAS section and a control section of discrete mathematics on students' in-class examinations. Additionally, they analysed student approaches…
A Research on Future Mathematics Teachers' Instructional Explanations: The case of Algebra
ERIC Educational Resources Information Center
Guler, Mustafa; Celik, Derya
2016-01-01
In this study, explanations of future mathematics teachers about algebra were analysed according to the levels of understanding used by Kinach (2002). The participants for the study were 101 teacher candidates attending the final semester of a teacher training program. For data collection, a form containing four scenario-type items were…
ERIC Educational Resources Information Center
Ma, Tingting; Brown, Irving A.; Kulm, Gerald; Davis, Trina J.; Lewis, Chance W.; Allen, G. Donald
2016-01-01
From the perspectives of Graduate Research Assistants (GRAs), this study examines the design and implementation of a simulated teaching environment in "Second Life" (SL) for prospective teachers to teach algebra for diverse learners. Drawing upon the Learning-for-Use framework, the analyses provide evidence on the development of student…
Jucys-Murphy elements for Birman-Murakami-Wenzl algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Ogievetsky, O. V.
2011-05-01
The Burman-Wenzl-Murakami algebra, considered as the quotient of the braid group algebra, possesses the commutative set of Jucys-Murphy elements. We show that the set of Jucys-Murphy elements is maximal commutative for the generic Birman-Wenzl-Murakami algebra and reconstruct the representation theory of the tower of Birman-Wenzl-Murakami algebras.
Kinematical superalgebras and Lie algebras of order 3
Campoamor-Stursberg, R.; Rausch de Traubenberg, M.
2008-06-15
We study and classify kinematical algebras which appear in the framework of Lie superalgebras or Lie algebras of order 3. All these algebras are related through generalized Inonue-Wigner contractions from either the orthosymplectic superalgebra or the de Sitter Lie algebra of order 3.
Becchi-Rouet-Stora-Tyutin operators for W algebras
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2008-07-15
The study of quantum Lie algebras motivates a use of noncanonical ghosts and antighosts for nonlinear algebras, such as W-algebras. This leads, for the W{sub 3} and W{sub 3}{sup (2)} algebras, to the Becchi-Rouet-Stora-Tyutin operator having the conventional cubic form.
On computational complexity of Clifford algebra
NASA Astrophysics Data System (ADS)
Budinich, Marco
2009-05-01
After a brief discussion of the computational complexity of Clifford algebras, we present a new basis for even Clifford algebra Cl(2m) that simplifies greatly the actual calculations and, without resorting to the conventional matrix isomorphism formulation, obtains the same complexity. In the last part we apply these results to the Clifford algebra formulation of the NP-complete problem of the maximum clique of a graph introduced by Budinich and Budinich ["A spinorial formulation of the maximum clique problem of a graph," J. Math. Phys. 47, 043502 (2006)].
Gauged Ads-Maxwell Algebra and Gravity
NASA Astrophysics Data System (ADS)
Durka, R.; Kowalski-Glikman, J.; Szczachor, M.
We deform the anti-de Sitter algebra by adding additional generators {Z}ab, forming in this way the negative cosmological constant counterpart of the Maxwell algebra. We gauge this algebra and construct a dynamical model with the help of a constrained BF theory. It turns out that the resulting theory is described by the Einstein-Cartan action with Holst term, and the gauge fields associated with the Maxwell generators {Z}ab appear only in topological terms that do not influence dynamical field equations. We briefly comment on the extension of this construction, which would lead to a nontrivial Maxwell fields dynamics.
Homomorphisms between C*-algebras and linear derivations on C*-algebras
NASA Astrophysics Data System (ADS)
Park, Choonkil; Boo, Deok-Hoon; An, Jong Su
2008-01-01
It is shown that every almost unital almost linear mapping of a unital C*-algebra to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all unitaries , all , and all , and that every almost unital almost linear continuous mapping of a unital C*-algebra of real rank zero to a unital C*-algebra is a homomorphism when h(3nuy)=h(3nu)h(y) holds for all , and v is invertible}, all , and all . Furthermore, we prove the Hyers-Ulam-Rassias stability of *-homomorphisms between unital C*-algebras, and -linear *-derivations on unital C*-algebras. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300.
Lie algebra type noncommutative phase spaces are Hopf algebroids
NASA Astrophysics Data System (ADS)
Meljanac, Stjepan; Škoda, Zoran; Stojić, Martina
2016-11-01
For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.
A note on derivations of Murray–von Neumann algebras
Kadison, Richard V.; Liu, Zhe
2014-01-01
A Murray–von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we first present a brief introduction to the theory of derivations of operator algebras from both the physical and mathematical points of view. We then describe our recent work on derivations of Murray–von Neumann algebras. We show that the “extended derivations” of a Murray–von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray–von Neumann algebra associated with a factor of type II1 into that factor is 0. Those results are extensions of Singer’s seminal result answering a question of Kaplansky, as applied to von Neumann algebras: The algebra may be noncommutative and may even contain unbounded elements. PMID:24469831
A double commutant theorem for Murray–von Neumann algebras
Liu, Zhe
2012-01-01
Murray–von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra of the Murray–von Neumann algebra associated with a finite von Neumann algebra is the Murray–von Neumann algebra , where is a maximal abelian self-adjoint subalgebra of and, in addition, is . We also prove that the Murray–von Neumann algebra with the center of is the center of the Murray–von Neumann algebra . Von Neumann’s celebrated double commutant theorem characterizes von Neumann algebras as those for which , where , the commutant of , is the set of bounded operators on the Hilbert space that commute with all operators in . At the end of this article, we present a double commutant theorem for Murray–von Neumann algebras. PMID:22543165
Bai, Xiaomei; Wen, Zhongming; An, Shaoshan; Li, Bicheng
2015-01-01
Evaluating the sustainability of cropland use is essential for guaranteeing a secure food supply and accomplishing agriculture sustainable development. This study was conducted in the ecologically vulnerable Loess Plateau region of China to evaluate the sustainability of cropland use based on an ecological footprint model that integrates emergy analysis. One modified method proposed in 2005 is known as the emergetic ecological footprint (EEF). We enhanced the method by accounting for both the surface soil energy in the carrying capacity calculation and the net topsoil loss for human consumption in the EF calculation. This paper evaluates whether the cropland of the study area was overloaded or sustainably managed during the period from 1981 to 2009. Toward this end, the final results obtained from EEF were compared to conventional EF and previous methods. The results showed that the cropland of Yuanzhou County has not been used sustainably since 1983, and the conventional EF analysis provided similar results. In contrast, a deficit did not appear during this time period when previous calculation methods of others were used. Additionally, the ecological sustainable index (ESI) from three models indicated that the recently used cropland system is unlikely to be unsustainable.
Highest-weight representations of Brocherd`s algebras
Slansky, R.
1997-01-01
General features of highest-weight representations of Borcherd`s algebras are described. to show their typical features, several representations of Borcherd`s extensions of finite-dimensional algebras are analyzed. Then the example of the extension of affine- su(2) to a Borcherd`s algebra is examined. These algebras provide a natural way to extend a Kac-Moody algebra to include the hamiltonian and number-changing operators in a generalized symmetry structure.
BiHom-Associative Algebras, BiHom-Lie Algebras and BiHom-Bialgebras
NASA Astrophysics Data System (ADS)
Graziani, Giacomo; Makhlouf, Abdenacer; Menini, Claudia; Panaite, Florin
2015-10-01
A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α,β\\colon A→ A such that α (a)(bc)=(ab)β (c), for all a, b, cin A. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).
Using Schemas to Develop Algebraic Thinking
ERIC Educational Resources Information Center
Steele, Diana F.
2005-01-01
This article describes ways in which students develop schemas as they generalize and formalize patterns when solving related algebraic problems that involve size, shape, growth, and change. (Contains 7 figures and 3 tables.)
Cohomological invariants of central simple algebras
NASA Astrophysics Data System (ADS)
Merkurjev, A. S.
2016-10-01
We determine the indecomposable degree 3 cohomological invariants of tuples of central simple algebras with linear relations. Equivalently, we determine the degree 3 reductive cohomological invariants of all split semisimple groups of type A.
Cyclic Cocycles on Twisted Convolution Algebras
NASA Astrophysics Data System (ADS)
Angel, Eitan
2013-01-01
We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper étale groupoids, Tu and Xu (Adv Math 207(2):455-483, 2006) provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to the construction of Mathai and Stevenson (Adv Math 200(2):303-335, 2006). When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras.
ALGEBRAIC DEPENDENCE THEOREMS ON COMPLEX PSEUDOCONCAVE SPACES
The notion of pseudoconcave space is introduced and classical theorems on algebraic dependence of meromorphic functions are extended for this new class of spaces and for sections in a coherent sheaf. (Author)
Applications: Using Algebra in an Accounting Practice.
ERIC Educational Resources Information Center
Eisner, Gail A.
1994-01-01
Presents examples of algebra from the field of accounting including proportional ownership of stock, separation of a loan payment into principal and interest portions, depreciation methods, and salary withholdings computations. (MKR)
Lisa's Lemonade Stand: Exploring Algebraic Ideas.
ERIC Educational Resources Information Center
Billings, Esther M. H.; Lakatos, Tracy
2003-01-01
Presents an activity, "Lisa's Lemonade Stand," that actively engages students in algebraic thinking as they analyze change by investigating relationships between variables and gain experience describing and representing these relationships graphically. (YDS)
Algebraic operator approach to gas kinetic models
NASA Astrophysics Data System (ADS)
Il'ichov, L. V.
1997-02-01
Some general properties of the linear Boltzmann kinetic equation are used to present it in the form ∂ tϕ = - Â†Âϕ with the operators ÂandÂ† possessing some nontrivial algebraic properties. When applied to the Keilson-Storer kinetic model, this method gives an example of quantum ( q-deformed) Lie algebra. This approach provides also a natural generalization of the “kangaroo model”.
Dynamical systems and quantum bicrossproduct algebras
NASA Astrophysics Data System (ADS)
Arratia, Oscar; del Olmo, Mariano A.
2002-06-01
We present a unified study of some aspects of quantum bicrossproduct algebras of inhomogeneous Lie algebras, such as Poincaré, Galilei and Euclidean in N dimensions. The action associated with the bicrossproduct structure allows us to obtain a nonlinear action over a new group linked to the translations. This new nonlinear action associates a dynamical system with each generator which is the object of our study.
Algebra and topology for applications to physics
NASA Technical Reports Server (NTRS)
Rozhkov, S. S.
1987-01-01
The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.
Vague Congruences and Quotient Lattice Implication Algebras
Qin, Xiaoyan; Xu, Yang
2014-01-01
The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given. PMID:25133207
Edge covers and independence: Algebraic approach
NASA Astrophysics Data System (ADS)
Kalinina, E. A.; Khitrov, G. M.; Pogozhev, S. V.
2016-06-01
In this paper, linear algebra methods are applied to solve some problems of graph theory. For ordinary connected graphs, edge coverings and independent sets are considered. Some results concerning minimum edge covers and maximum matchings are proved with the help of linear algebraic approach. The problem of finding a maximum matching of a graph is fundamental both practically and theoretically, and has numerous applications, e.g., in computational chemistry and mathematical chemistry.
Algebraic Sub-Structuring for Electromagnetic Applications
Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC
2006-06-30
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebraic sub-structuring for electromagnetic applications
Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.
2004-09-14
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
NASA Astrophysics Data System (ADS)
Sati, Hisham; Schreiber, Urs
2017-03-01
We uncover higher algebraic structures on Noether currents and BPS charges. It is known that equivalence classes of conserved currents form a Lie algebra. We show that at least for target space symmetries of higher parameterized WZW-type sigma-models this naturally lifts to a Lie ( p + 1)-algebra structure on the Noether currents themselves. Applied to the Green-Schwarz-type action functionals for super p-brane sigma-models this yields super Lie ( p+1)-algebra refinements of the traditional BPS brane charge extensions of supersymmetry algebras. We discuss this in the generality of higher differential geometry, where it applies also to branes with (higher) gauge fields on their worldvolume. Applied to the M5-brane sigma-model we recover and properly globalize the M-theory super Lie algebra extension of 11-dimensional superisometries by 2-brane and 5-brane charges. Passing beyond the infinitesimal Lie theory we find cohomological corrections to these charges in higher analogy to the familiar corrections for D-brane charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.
On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions
NASA Astrophysics Data System (ADS)
Ballesteros, A.; Herranz, F. J.; Musso, F.
2014-09-01
Quantum deformations of (anti-)de Sitter (A)dS algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincare algebras and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the K-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the κ-(A)dS deformation that preserves the space isotropy is found.
I CAN Learn[R] Pre-Algebra and Algebra. What Works Clearinghouse Intervention Report
ERIC Educational Resources Information Center
What Works Clearinghouse, 2007
2007-01-01
The "I CAN Learn[R] Pre-Algebra" and "Algebra" computerized curricula are designed to cover mathematics and problem-solving skills for ethnically diverse, inner-city students in grades 6-12. The curricula are designed to equip students with the skills they need to meet district, state, and national math objectives through an…
ERIC Educational Resources Information Center
Okpube, Nnaemeka Michael; Anugwo, M. N.
2016-01-01
This study investigated the Card Games and Algebra tic-Tacmatics on Junior Secondary II Students' Achievement in Algebraic Expressions. Three research questions and three null hypotheses guided the study. The study adopted the pre-test, post-test control group design. A total of two hundred and forty (240) Junior Secondary School II students were…
ERIC Educational Resources Information Center
Ormond, Christine
2012-01-01
Primary teachers play a key role in their students' future mathematical success in the early secondary years. While the word "algebra" may make some primary teachers feel uncomfortable or worried, the basic arithmetic ideas underlying algebra are vitally important for older primary students as they are increasingly required to use "algebraic…
The Application of a Computer Algebra System as a Tool in College Algebra.
ERIC Educational Resources Information Center
Mayes, Robert L.
1995-01-01
Students (n=61) in an experimental course stressing active student involvement and the use of a computer algebra system scored higher than students (n=76) in a traditional college algebra course on final measures of inductive reasoning, visualization, and problem solving while maintaining equivalent manipulation and computation skills. (Author/MLB)
ERIC Educational Resources Information Center
Davies Gomez, Lisa
2012-01-01
Algebra is the gatekeeper of access to higher-level math and science courses, higher education and future earning opportunities. Unequal numbers of African-American males drop out of Algebra and mathematics courses and underperform on tests of mathematical competency and are thus denied both essential skills and a particularly important pathway to…
Ternary Hom-Nambu-Lie algebras induced by Hom-Lie algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Makhlouf, Abdenacer; Silvestrov, Sergei
2010-04-01
The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom-Nambu-Lie algebras obtained using this construction.
Classification of central extensions of Lax operator algebras
Schlichenmaier, Martin
2008-11-18
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
Linearizing W2,4 and WB2 algebras
NASA Astrophysics Data System (ADS)
Bellucci, S.; Krivonos, S.; Sorin, A.
1995-02-01
It has recently been shown that the W3 and W3(2) algebras can be considered as subalgebras in some linear conformal algebras. In this paper we show that the nonlinear algebras W2,4 and WB2 as well as Zamolodchikov's spin {5}/{2} superalgebra also can be embedded as subalgebras into some linear conformal algebras with a finite set of currents. These linear algebras give rise to new realizations of the nonlinear algebras which could be suitable in the construction of W-string theories.
Classification of central extensions of Lax operator algebras
NASA Astrophysics Data System (ADS)
Schlichenmaier, Martin
2008-11-01
Lax operator algebras were introduced by Krichever and Sheinman as further developments of Krichever's theory of Lax operators on algebraic curves. They are infinite dimensional Lie algebras of current type with meromorphic objects on compact Riemann surfaces (resp. algebraic curves) as elements. Here we report on joint work with Oleg Sheinman on the classification of their almost-graded central extensions. It turns out that in case that the finite-dimensional Lie algebra on which the Lax operator algebra is based on is simple there is a unique almost-graded central extension up to equivalence and rescaling of the central element.
TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION
NASA Technical Reports Server (NTRS)
Smith, R. E.
1994-01-01
TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.
Spinor representations of affine Lie algebras
Frenkel, I. B.
1980-01-01
Let [unk] be an infinite-dimensional Kac-Moody Lie algebra of one of the types Dl+1(2), Bl(1), or Dl(1). These algebras are characterized by the property that an elimination of any endpoint of their Dynkin diagrams gives diagrams of types Bl or Dl of classical orthogonal Lie algebras. We construct two representations of a Lie algebra [unk], which we call spinor representations, following the analogy with the classical case. We obtain that every spinor representation is either irreducible or has two irreducible components. This provides us with an explicit construction of fundamental representations of [unk], two for the type Dl+1(2), three for Bl(1), and four for Dl(1). We note the profound connection of our construction with quantum field theory—in particular, with fermion fields. Comparing the character formulas of our representations with another construction of the fundamental representations of Kac-Moody Lie algebras of types Al(1), Dl(1), El(1), we obtain classical Jacobi identities and addition formulas for elliptic θ-functions. PMID:16592912
A process algebra model of QED
NASA Astrophysics Data System (ADS)
Sulis, William
2016-03-01
The process algebra approach to quantum mechanics posits a finite, discrete, determinate ontology of primitive events which are generated by processes (in the sense of Whitehead). In this ontology, primitive events serve as elements of an emergent space-time and of emergent fundamental particles and fields. Each process generates a set of primitive elements, using only local information, causally propagated as a discrete wave, forming a causal space termed a causal tapestry. Each causal tapestry forms a discrete and finite sampling of an emergent causal manifold (space-time) M and emergent wave function. Interactions between processes are described by a process algebra which possesses 8 commutative operations (sums and products) together with a non-commutative concatenation operator (transitions). The process algebra possesses a representation via nondeterministic combinatorial games. The process algebra connects to quantum mechanics through the set valued process and configuration space covering maps, which associate each causal tapestry with sets of wave functions over M. Probabilities emerge from interactions between processes. The process algebra model has been shown to reproduce many features of the theory of non-relativistic scalar particles to a high degree of accuracy, without paradox or divergences. This paper extends the approach to a semi-classical form of quantum electrodynamics.
ADA interpretative system for image algebra
NASA Astrophysics Data System (ADS)
Murillo, Juan J.; Wilson, Joseph N.
1992-06-01
An important research problem in image processing is to find appropriate tools to support algorithm development. There have been efforts to build algorithm development support systems for image algebra in several languages, but these systems still have the disadvantage of the time consuming algorithm development style associated with compilation-oriented programming. This paper starts with a description of the Run-Time Support Library (RTSL), which serves as the base for executing programs on both the Image Algebra Ada Translator (IAAT) and Image Algebra Ada Interpreter (IAAI). A presentation on the current status of IAAT and its capabilities is followed by a brief introduction to the utilization of the Image Display Manager (IDM) for image manipulation and analysis. We then discuss in detail the current development stage of IAAI and its relation with RTSL and IDM. The last section describes the design of a syntax-directed graphical user interface for IAAI. We close with an analysis of the current performance of IAAI, and future trends are discussed. Appendix A gives a brief introduction to Image Algebra (IA), and in Appendix B the reader is presented to the Image Algebra Ada (IAA) grammar.
Permutation centralizer algebras and multimatrix invariants
NASA Astrophysics Data System (ADS)
Mattioli, Paolo; Ramgoolam, Sanjaye
2016-03-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.
ENVIRONMENTAL GOODS AND SERIVCES FROM RESTORATION ALTERNATIVES: EMERGY-BASED METHOD OF BENEFIT VALUE
Although economic benefit-cost analyses of environmental regulations has been conducted since the 1970s, an effective methodology has yet to be developed for the integrated assessment of regulatory impacts on the larger system as a whole, including its social and environmental as...
ERIC Educational Resources Information Center
Walston, Jill; McCarroll, Jill Carlivati
2010-01-01
This Brief is intended to provide descriptive statistics about various aspects of algebra enrollment in the eighth grade. The first analyses describe who is taking algebra in the eighth grade; findings compare eighth-grade course enrollment percentages for students with different demographic, family, geographic, and school characteristics (table…
An algebra of discrete event processes
NASA Technical Reports Server (NTRS)
Heymann, Michael; Meyer, George
1991-01-01
This report deals with an algebraic framework for modeling and control of discrete event processes. The report consists of two parts. The first part is introductory, and consists of a tutorial survey of the theory of concurrency in the spirit of Hoare's CSP, and an examination of the suitability of such an algebraic framework for dealing with various aspects of discrete event control. To this end a new concurrency operator is introduced and it is shown how the resulting framework can be applied. It is further shown that a suitable theory that deals with the new concurrency operator must be developed. In the second part of the report the formal algebra of discrete event control is developed. At the present time the second part of the report is still an incomplete and occasionally tentative working paper.
Operator algebra in logarithmic conformal field theory
Nagi, Jasbir
2005-10-15
For some time now, conformal field theories in two dimensions have been studied as integrable systems. Much of the success of these studies is related to the existence of an operator algebra of the theory. In this paper, some of the extensions of this machinery to the logarithmic case are studied and used. More precisely, from Moebius symmetry constraints, the generic three- and four-point functions of logarithmic quasiprimary fields are calculated in closed form for arbitrary Jordan rank. As an example, c=0 disordered systems with nondegenerate vacua are studied. With the aid of two-, three-, and four-point functions, the operator algebra is obtained and associativity of the algebra studied.
Computational algebraic geometry of epidemic models
NASA Astrophysics Data System (ADS)
Rodríguez Vega, Martín.
2014-06-01
Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
Sound Off! A Dialogue between Calculator and Algebra
ERIC Educational Resources Information Center
Wade, William R.
2006-01-01
This article illustrates the fact that unless tempered by algebraic reasoning, a graphing calculator can lead one to erroneous conclusions. It also demonstrates that some problems can be solved by combining technology with algebra.
A new algebra core for the minimal form' problem
Purtill, M.R. . Center for Communications Research); Oliveira, J.S.; Cook, G.O. Jr. )
1991-12-20
The demands of large-scale algebraic computation have led to the development of many new algorithms for manipulating algebraic objects in computer algebra systems. For instance, parallel versions of many important algorithms have been discovered. Simultaneously, more effective symbolic representations of algebraic objects have been sought. Also, while some clever techniques have been found for improving the speed of the algebraic simplification process, little attention has been given to the issue of restructuring expressions, or transforming them into minimal forms.'' By minimal form,'' we mean that form of an expression that involves a minimum number of operations. In a companion paper, we introduce some new algorithms that are very effective at finding minimal forms of expressions. These algorithms require algebraic and combinatorial machinery that is not readily available in most algebra systems. In this paper we describe a new algebra core that begins to provide the necessary capabilities.
Infinitesimal deformations of naturally graded filiform Leibniz algebras
NASA Astrophysics Data System (ADS)
Khudoyberdiyev, A. Kh.; Omirov, B. A.
2014-12-01
In the present paper we describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any n-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra Fn3(0) . We establish that in the same way any n-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras Fn1,Fn2and Fn3(α) . Moreover, we describe the linear integrable deformations of the above-mentioned algebras with a fixed basis of HL2 in the set of all n-dimensional Leibniz algebras. Among these deformations one new rigid algebra has been found.
Current algebra and the nonlinear σ-model
NASA Astrophysics Data System (ADS)
Ghosh, S.
2007-06-01
We present the current algebra of a particular form in the nonlinear σ-model. The algebra has a non-Abelian form with field-dependent structure functions. We comment on the connection of the model with noncommutative space.
Kac-Moody algebra and nonlinear sigma model
NASA Astrophysics Data System (ADS)
Ogura, Waichi; Hosoya, Akio
1985-12-01
We investigate the nonlinear sigma model over an arbitrary homogeneous space. Then it is shown that the sigma model realizes the Kac-Moody algebra as current algebra only if the homogeneous space is restricted to the group manifold.
Upper bound for the length of commutative algebras
NASA Astrophysics Data System (ADS)
Markova, Ol'ga V.
2009-12-01
By the length of a finite system of generators for a finite-dimensional associative algebra over an arbitrary field one means the least positive integer k such that the words of length not exceeding k span this algebra (as a vector space). The maximum length for the systems of generators of an algebra is referred to as the length of the algebra. In the present paper, an upper bound for the length of a commutative algebra in terms of a function of two invariants of the algebra, the dimension and the maximal degree of the minimal polynomial for the elements of the algebra, is obtained. As a corollary, a formula for the length of the algebra of diagonal matrices over an arbitrary field is obtained. Bibliography: 8 titles.
Rees algebras, Monomial Subrings and Linear Optimization Problems
NASA Astrophysics Data System (ADS)
Dupont, Luis A.
2010-06-01
In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut property or the packing property. We also study algebraic properties of symbolic Rees algebras of edge ideals of graphs, edge ideals of clique clutters of comparability graphs, and Stanley-Reisner rings.
Supersymmetry in physics: an algebraic overview
Ramond, P.
1983-01-01
In 1970, while attempting to generalize the Veneziano model (string model) to include fermions, I introduced a new algebraic structure which turned out to be a graded Lie algebra; it was used as a spectrum-generating algebra. This approach was soon after generalized to include interactions, yielding a complete model of fermions and boson (RNS model). In an unrelated work in the Soviet Union, it was shown how to generalize the Poincare group to include fermionic charges. However it was not until 1974 that an interacting field theory invariant under the Graded Poincare group in 3 + 1 dimensions was built (WZ model). Supersymmetric field theories turned out to have less divergent ultraviolet behavior than non-supersymmetric field theories. Gravity was generalized to include supersymmetry, to a theory called supergravity. By now many interacting local field theories exhibiting supersymmetry have been built and studied from 1 + 1 to 10 + 1 dimensions. Supersymmetric local field theories in less than 9 + 1 dimensions, can be understood as limits of multilocal (string) supersymmetric theories, in 9 + 1 dimensions. On the other hand, graded Lie algebras have been used in non-relativistic physics as approximate symmetries of Hamiltonians. The most striking such use so far helps comparing even and odd nuclei energy levels. It is believed that graded Lie algebras can be used whenever paired and unpaired fermions excitations can coexist. In this overview of a tremendously large field, I will only survey finite graded Lie algebras and their representations. For non-relativistic applications, all of GLA are potentially useful, while for relativistic applications, only these which include the Poincare group are to be considered.
Double conformal space-time algebra
NASA Astrophysics Data System (ADS)
Easter, Robert Benjamin; Hitzer, Eckhard
2017-01-01
The Double Conformal Space-Time Algebra (DCSTA) is a high-dimensional 12D Geometric Algebra G 4,8that extends the concepts introduced with the Double Conformal / Darboux Cyclide Geometric Algebra (DCGA) G 8,2 with entities for Darboux cyclides (incl. parabolic and Dupin cyclides, general quadrics, and ring torus) in spacetime with a new boost operator. The base algebra in which spacetime geometry is modeled is the Space-Time Algebra (STA) G 1,3. Two Conformal Space-Time subalgebras (CSTA) G 2,4 provide spacetime entities for points, flats (incl. worldlines), and hyperbolics, and a complete set of versors for their spacetime transformations that includes rotation, translation, isotropic dilation, hyperbolic rotation (boost), planar reflection, and (pseudo)spherical inversion in rounds or hyperbolics. The DCSTA G 4,8 is a doubling product of two G 2,4 CSTA subalgebras that inherits doubled CSTA entities and versors from CSTA and adds new bivector entities for (pseudo)quadrics and Darboux (pseudo)cyclides in spacetime that are also transformed by the doubled versors. The "pseudo" surface entities are spacetime hyperbolics or other surface entities using the time axis as a pseudospatial dimension. The (pseudo)cyclides are the inversions of (pseudo)quadrics in rounds or hyperbolics. An operation for the directed non-uniform scaling (anisotropic dilation) of the bivector general quadric entities is defined using the boost operator and a spatial projection. DCSTA allows general quadric surfaces to be transformed in spacetime by the same complete set of doubled CSTA versor (i.e., DCSTA versor) operations that are also valid on the doubled CSTA point entity (i.e., DCSTA point) and the other doubled CSTA entities. The new DCSTA bivector entities are formed by extracting values from the DCSTA point entity using specifically defined inner product extraction operators. Quadric surface entities can be boosted into moving surfaces with constant velocities that display the length
Constraint algebra for interacting quantum systems
NASA Astrophysics Data System (ADS)
Fubini, S.; Roncadelli, M.
1988-04-01
We consider relativistic constrained systems interacting with external fields. We provide physical arguments to support the idea that the quantum constraint algebra should be the same as in the free quantum case. For systems with ordering ambiguities this principle is essential to obtain a unique quantization. This is shown explicitly in the case of a relativistic spinning particle, where our assumption about the constraint algebra plus invariance under general coordinate transformations leads to a unique S-matrix. On leave from Dipartimento di Fisica Nucleare e Teorica, Università di Pavia and INFN, I-27100 Pavia, Italy.
Weak Lie symmetry and extended Lie algebra
Goenner, Hubert
2013-04-15
The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found ('extended Lie algebras') which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).
Algebraic surface design and finite element meshes
NASA Technical Reports Server (NTRS)
Bajaj, Chandrajit L.
1992-01-01
Some of the techniques are summarized which are used in constructing C sup 0 and C sup 1 continuous meshes of low degree, implicitly defined, algebraic surface patches in three dimensional space. These meshes of low degree algebraic surface patches are used to construct accurate computer models of physical objects. These meshes are also used in the finite element simulation of physical phenomena (e.g., heat dissipation, stress/strain distributions, fluid flow characteristics) required in the computer prototyping of both the manufacturability and functionality of the geometric design.
Fréchet-algebraic deformation quantizations
NASA Astrophysics Data System (ADS)
Waldmann, S.
2014-09-01
In this review I present some recent results on the convergence properties of formal star products. Based on a general construction of a Fréchet topology for an algebra with countable vector space basis I discuss several examples from deformation quantization: the Wick star product on the flat phase space m2n gives a first example of a Fréchet algebraic framework for the canonical commutation relations. More interesting, the star product on the Poincare disk can be treated along the same lines, leading to a non-trivial example of a convergent star product on a curved Kahler manifold.
Shapes and stability of algebraic nuclear models
NASA Technical Reports Server (NTRS)
Lopez-Moreno, Enrique; Castanos, Octavio
1995-01-01
A generalization of the procedure to study shapes and stability of algebraic nuclear models introduced by Gilmore is presented. One calculates the expectation value of the Hamiltonian with respect to the coherent states of the algebraic structure of the system. Then equilibrium configurations of the resulting energy surface, which depends in general on state variables and a set of parameters, are classified through the Catastrophe theory. For one- and two-body interactions in the Hamiltonian of the interacting Boson model-1, the critical points are organized through the Cusp catastrophe. As an example, we apply this Separatrix to describe the energy surfaces associated to the Rutenium and Samarium isotopes.
Bohr model as an algebraic collective model
Rowe, D. J.; Welsh, T. A.; Caprio, M. A.
2009-05-15
Developments and applications are presented of an algebraic version of Bohr's collective model. Illustrative examples show that fully converged calculations can be performed quickly and easily for a large range of Hamiltonians. As a result, the Bohr model becomes an effective tool in the analysis of experimental data. The examples are chosen both to confirm the reliability of the algebraic collective model and to show the diversity of results that can be obtained by its use. The focus of the paper is to facilitate identification of the limitations of the Bohr model with a view to developing more realistic, computationally tractable models.
Riemannian manifolds as Lie-Rinehart algebras
NASA Astrophysics Data System (ADS)
Pessers, Victor; van der Veken, Joeri
2016-07-01
In this paper, we show how Lie-Rinehart algebras can be applied to unify and generalize the elementary theory of Riemannian geometry. We will first review some necessary theory on a.o. modules, bilinear forms and derivations. We will then translate some classical theory on Riemannian geometry to the setting of Rinehart spaces, a special kind of Lie-Rinehart algebras. Some generalized versions of classical results will be obtained, such as the existence of a unique Levi-Civita connection, inducing a Levi-Civita connection on a submanifold, and the construction of spaces with constant sectional curvature.
Quantum walled Brauer algebra: commuting families, Baxterization, and representations
NASA Astrophysics Data System (ADS)
Semikhatov, A. M.; Tipunin, I. Yu
2017-02-01
For the quantum walled Brauer algebra, we construct its Specht modules and (for generic parameters of the algebra) seminormal modules. The latter construction yields the spectrum of a commuting family of Jucys-Murphy elements. We also propose a Baxterization prescription; it involves representing the quantum walled Brauer algebra in terms of morphisms in a braided monoidal category and introducing parameters into these morphisms, which allows constructing a ‘universal transfer matrix’ that generates commuting elements of the algebra.
Dynamical algebras for Poeschl-Teller Hamiltonian hierarchies
Kuru, S.; Negro, J.
2009-12-15
The dynamical algebras of the trigonometric and hyperbolic symmetric Poeschl-Teller Hamiltonian hierarchies are obtained. A kind of discrete-differential realizations of these algebras are found which are isomorphic to so(3, 2) Lie algebras. In order to get them, first the relation between ladder and factor operators is investigated. In particular, the action of the ladder operators on normalized eigenfunctions is found explicitly. Then, the whole dynamical algebras are generated in a straightforward way.
Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups
NASA Astrophysics Data System (ADS)
Batat, W.; Onda, K.
2017-04-01
We study algebraic Ricci solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are solvsolitons. In particular, we obtain new solitons on G2, G5, and G6, and we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not be algebraic Ricci solitons.
Capability and Schur multiplier of a pair of Lie algebras
NASA Astrophysics Data System (ADS)
Johari, Farangis; Parvizi, Mohsen; Niroomand, Peyman
2017-04-01
The aim of this work is to find some criteria for detecting the capability of a pair of Lie algebras. We characterize the exact structure of all pairs of capable Lie algebras in the class of abelian and Heisenberg ones. Among the other results, we also give some exact sequences on the Schur multiplier and exterior product of Lie algebras.
The Ideas of Algebra, K-12. 1988 Yearbook.
ERIC Educational Resources Information Center
Coxford, Arthur F., Ed.; Shulte, Albert P., Ed.
This volume is organized into six parts. Chapters 1-5, which make up Part 1, first discuss the forces impinging on algebra in the curriculum and suggest possible directions for change. Chapters 6-8, Part 2, concentrate on concepts and teaching possibilities available prior to the formal introduction of algebra. The notion that algebraic ideas are…
The Impact of Early Algebra: Results from a Longitudinal Intervention
ERIC Educational Resources Information Center
Brizuela, Bárbara M.; Martinez, Mara V.; Cayton-Hodges, Gabrielle A.
2013-01-01
In this paper, we provide evidence of the impact of early algebra (EA) over time. We document this impact in the following ways: (a) by showing the performance over time of an experimental group of 15 children on an algebra assessment, from 3rd to 5th grade; and (b) by showing how the performance on an algebra assessment of children from an…
Changing Pre-Service Elementary Teachers' Attitudes to Algebra.
ERIC Educational Resources Information Center
McGowen, Mercedes A.; Davis, Gary E.
This article addresses the question: "What are the implications for the preparation of prospective elementary teachers of 'early algebra' in the elementary grades curriculum?" Part of the answer involves language aspects of algebra: in particular, how a change in pre-service teachers' attitudes to algebra, from instrumental to relational, is…
A Research Base Supporting Long Term Algebra Reform?
ERIC Educational Resources Information Center
Kaput, James J.
This paper discusses three dimensions of algebra reform: breadth, integration, and pedagogy. Breadth of algebra includes algebra as: generalizing and formalizing patterns and constraints; syntactically-guided manipulation of formalisms; study of structures abstracted from computations and relations; study of functions, relations, and joint…
Classical versus Computer Algebra Methods in Elementary Geometry
ERIC Educational Resources Information Center
Pech, Pavel
2005-01-01
Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…
Remarks on Virasoro and Kac-Moody Algebras
NASA Astrophysics Data System (ADS)
Grabowski, J.; Marmo, G.; Perelomov, A.; Simoni, A.
Parametric realizations of Virasoro or Kac-Moody algebras are constructed on a generic manifold carrying an appropriate vector field. It is shown that the centrally extended algebras cannot be realized as algebras of vector fields on finite-dimensional manifolds.
Processes Used by College Students in Understanding Basic Algebra.
ERIC Educational Resources Information Center
Rachlin, Sidney Lee
The purpose of this study was to uncover information about and gain a greater insight into the extent to which students who are successful in a basic algebra course: l) demonstrate a reversibility of reasoning processes when solving algebraic problems; 2) demonstrate a flexibility of reasoning processes when solving algebraic problems; 3)…
Effectiveness of Cognitive Tutor Algebra I at Scale
ERIC Educational Resources Information Center
Pane, John F.; Griffin, Beth Ann; McCaffrey, Daniel F.; Karam, Rita
2014-01-01
This article examines the effectiveness of a technology-based algebra curriculum in a wide variety of middle schools and high schools in seven states. Participating schools were matched into similar pairs and randomly assigned to either continue with the current algebra curriculum for 2 years or to adopt Cognitive Tutor Algebra I (CTAI), which…
Abstract Numeric Relations and the Visual Structure of Algebra
ERIC Educational Resources Information Center
Landy, David; Brookes, David; Smout, Ryan
2014-01-01
Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition,…
Static friction, differential algebraic systems and numerical stability
NASA Astrophysics Data System (ADS)
Chen, Jian; Schinner, Alexander; Matuttis, Hans-Georg
We show how Differential Algebraic Systems (Ordinary Differential Equations with algebraic constraints) in mechanics are affected by stability issues and we implement Lubich's projection method to reduce the error to practically zero. Then, we explain how the "numerically exact" implementation for static friction by Differential Algebraic Systems can be stabilized. We conclude by comparing the corresponding steps in the "Contact mechanics" introduced by Moreau.
Pilot Study on Algebra Learning among Junior Secondary Students
ERIC Educational Resources Information Center
Poon, Kin-Keung; Leung, Chi-Keung
2010-01-01
The purpose of the study reported herein was to identify the common mistakes made by junior secondary students in Hong Kong when learning algebra and to compare teachers' perceptions of students' ability with the results of an algebra test. An algebra test was developed and administered to a sample of students (aged between 13 and 14 years). From…
Evolution of a Teaching Approach for Beginning Algebra
ERIC Educational Resources Information Center
Banerjee, Rakhi; Subramaniam, K.
2012-01-01
The article reports aspects of the evolution of a teaching approach over repeated trials for beginning symbolic algebra. The teaching approach emphasized the structural similarity between arithmetic and algebraic expressions and aimed at supporting students in making a transition from arithmetic to beginning algebra. The study was conducted with…
Should College Algebra be a Prerequisite for Taking Psychology Statistics?
ERIC Educational Resources Information Center
Sibulkin, Amy E.; Butler, J. S.
2008-01-01
In order to consider whether a course in college algebra should be a prerequisite for taking psychology statistics, we recorded students' grades in elementary psychology statistics and in college algebra at a 4-year university. Students who earned credit in algebra prior to enrolling in statistics for the first time had a significantly higher mean…
How Middle Grade Teachers Think about Algebraic Reasoning
ERIC Educational Resources Information Center
Glassmeyer, David; Edwards, Belinda
2016-01-01
Algebraic reasoning is an essential habit of mind for building conceptual knowledge in K-12 mathematics, yet little is known about how middle school mathematics teachers think about algebraic reasoning. In this article we describe a research project examining how algebraic reasoning was considered by grades 6, 7, or 8 mathematics teachers in a…
Solving Our Algebra Problem: Getting All Students through Algebra I to Improve Graduation Rates
ERIC Educational Resources Information Center
Schachter, Ron
2013-01-01
graduation as well as admission to most colleges. But taking algebra also can turn into a pathway for failure, from which some students never recover. In 2010, a national U.S. Department of Education study…
ERIC Educational Resources Information Center
Burrill, Gail
This paper is a reaction to a plenary address, "A Research Base Supporting Long Term Algebra Reform?" by James Kaput (SE 057 182). Three dimensions of algebra reform identified by Kaput (breadth, integration, and pedagogy) are discussed and contrasted with the draft version of the Algebra Document from the National Council of Teachers of…
ERIC Educational Resources Information Center
Nomi, Takako; Raudenbush, Stephen W.
2014-01-01
Algebra is often considered as a gateway for later achievement. A recent report by the Mathematics Advisory Panel (2008) underscores the importance of improving algebra learning in secondary school. Today, a growing number of states and districts require algebra for all students in ninth grade or earlier. Chicago is at the forefront of this…
Regular Gleason Measures and Generalized Effect Algebras
NASA Astrophysics Data System (ADS)
Dvurečenskij, Anatolij; Janda, Jiří
2015-12-01
We study measures, finitely additive measures, regular measures, and σ-additive measures that can attain even infinite values on the quantum logic of a Hilbert space. We show when particular classes of non-negative measures can be studied in the frame of generalized effect algebras.
A Photographic Assignment for Abstract Algebra
ERIC Educational Resources Information Center
Warrington, Gregory S.
2009-01-01
We describe a simple photographic assignment appropriate for an abstract algebra (or other) course. Students take digital pictures around campus of various examples of symmetry. They then classify these pictures according to which of the 17 plane symmetry groups they belong. (Contains 2 figures.)
Using Group Explorer in Teaching Abstract Algebra
ERIC Educational Resources Information Center
Schubert, Claus; Gfeller, Mary; Donohue, Christopher
2013-01-01
This study explores the use of Group Explorer in an undergraduate mathematics course in abstract algebra. The visual nature of Group Explorer in representing concepts in group theory is an attractive incentive to use this software in the classroom. However, little is known about students' perceptions on this technology in learning concepts in…
I Teach Economics, Not Algebra and Calculus
ERIC Educational Resources Information Center
Hey, John D.
2005-01-01
Most people learn to drive without knowing how the engine works. In a similar vein, the author believes that students can learn economics without knowing the algebra and calculus underlying the results. If instructors follow the philosophy of other economics courses in using graphs to illustrate the results, and draw the graphs accurately, then…
Generalizing: The Core of Algebraic Thinking
ERIC Educational Resources Information Center
Kinach, Barbara M.
2014-01-01
Generalizing--along with conjecturing, representing, justifying, and refuting--are forms of mathematical reasoning important in all branches of mathematics (Lannin, Ellis, and Elliott 2011). Increasingly, however, generalizing is recognized as the essence of thinking in algebra (Mason, Graham, and Johnston-Wilder 2010; Kaput, Carraher, and Blanton…
Hungry for Early Spatial and Algebraic Reasoning
ERIC Educational Resources Information Center
Cross, Dionne I.; Adefope, Olufunke; Lee, Mi Yeon; Perez, Arnulfo
2012-01-01
Tasks that develop spatial and algebraic reasoning are crucial for learning and applying advanced mathematical ideas. In this article, the authors describe how two early childhood teachers used stories as the basis for a unit that supports spatial reasoning in kindergartners and first graders. Having mathematical experiences that go beyond…
Journal Writing: Enlivening Elementary Linear Algebra.
ERIC Educational Resources Information Center
Meel, David E.
1999-01-01
Examines the various issues surrounding the implementation of journal writing in an undergraduate linear algebra course. Identifies the benefits of incorporating journal writing into an undergraduate mathematics course, which are supported with students' comments from their journals and their reflections on the process. Contains 14 references.…
The geometric semantics of algebraic quantum mechanics.
Cruz Morales, John Alexander; Zilber, Boris
2015-08-06
In this paper, we will present an ongoing project that aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We argue that this approach provides a geometric semantics for such a formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects.
An Evaluation of Saxon's Algebra Test.
ERIC Educational Resources Information Center
Johnson, Dale M.; Smith, Blaine
1987-01-01
John Saxon's incremental development model has been proclaimed as a superior teaching strategy for mathematics. This study evaluated the Saxon approach and textbook using 276 Algebra I students in experimental and control groups. The groups were compared in cognitive and affective areas. Results are presented. (Author/MT)
Programed Instruction in Elementary Algebra: An Experiment
ERIC Educational Resources Information Center
Lial, Margaret L.
1970-01-01
Report of an experiment which investigated the use of a programed elementary algebra text as a teaching method. The method was evaluated on the basis of student evaluation of the course and the percentage of students achieving a grade of C or better. Results indicated that the use of programed texts was superior to the traditional approach using…
Private quantum subsystems and quasiorthogonal operator algebras
NASA Astrophysics Data System (ADS)
Levick, Jeremy; Jochym-O'Connor, Tomas; Kribs, David W.; Laflamme, Raymond; Pereira, Rajesh
2016-03-01
We generalize a recently discovered example of a private quantum subsystem to find private subsystems for Abelian subgroups of the n-qubit Pauli group, which exist in the absence of private subspaces. In doing so, we also connect these quantum privacy investigations with the theory of quasiorthogonal operator algebras through the use of tools from group theory and operator theory.
Pre-Algebra Groups. Concepts & Applications.
ERIC Educational Resources Information Center
Montgomery County Public Schools, Rockville, MD.
Discussion material and exercises related to pre-algebra groups are provided in this five chapter manual. Chapter 1 (mappings) focuses on restricted domains, order of operations (parentheses and exponents), rules of assignment, and computer extensions. Chapter 2 considers finite number systems, including binary operations, clock arithmetic,…
Parallel Algebraic Multigrids for Structural mechanics
Brezina, M; Tong, C; Becker, R
2004-05-11
This paper presents the results of a comparison of three parallel algebraic multigrid (AMG) preconditioners for structural mechanics applications. In particular, they are interested in investigating both the scalability and robustness of the preconditioners. Numerical results are given for a range of structural mechanics problems with various degrees of difficulty.
Thinking Algebraically across the Elementary School Curriculum
ERIC Educational Resources Information Center
Soares, June; Blanton, Maria L.; Kaput, James J.
2006-01-01
With testing and accountability on everyone's mind, teachers are looking for creative ways to teach "all" subjects. Literacy is on the top of the list for testing, so it seems to get top priority. But how can teachers make sure that mathematics, especially a crucial area such as algebraic thinking, is a priority as well? Integrating subject matter…
A Concurrent Support Course for Intermediate Algebra
ERIC Educational Resources Information Center
Cooper, Cameron I.
2011-01-01
This article summarizes the creation and implementation of a concurrent support class for TRS 92--Intermediate Algebra, a developmental mathematics course at Fort Lewis College in Durango, Colorado. The concurrent course outlined in this article demonstrates a statistically significant increase in student success rates since its inception.…
Using Technology to Balance Algebraic Explorations
ERIC Educational Resources Information Center
Kurz, Terri L.
2013-01-01
In 2000, the "National Council of Teachers of Mathematics" recommended that Algebra Standards, "instructional programs from prekindergarten through grade 12 should enable all students to use mathematical models to represent and understand quantitative relationships." In this article, the authors suggest the "Balance"…
Noise limitations in optical linear algebra processors.
Batsell, S G; Jong, T L; Walkup, J F; Krile, T F
1990-05-10
A general statistical noise model is presented for optical linear algebra processors. A statistical analysis which includes device noise, the multiplication process, and the addition operation is undertaken. We focus on those processes which are architecturally independent. Finally, experimental results which verify the analytical predictions are also presented.
Lie algebras and linear differential equations.
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
Constructive Learning in Undergraduate Linear Algebra
ERIC Educational Resources Information Center
Chandler, Farrah Jackson; Taylor, Dewey T.
2008-01-01
In this article we describe a project that we used in our undergraduate linear algebra courses to help our students successfully master fundamental concepts and definitions and generate interest in the course. We describe our philosophy and discuss the projects overall success.
Stability of Linear Equations--Algebraic Approach
ERIC Educational Resources Information Center
Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.
2012-01-01
This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
On a Equation in Finite Algebraically Structures
ERIC Educational Resources Information Center
Valcan, Dumitru
2013-01-01
Solving equations in finite algebraically structures (semigroups with identity, groups, rings or fields) many times is not easy. Even the professionals can have trouble in such cases. Therefore, in this paper we proposed to solve in the various finite groups or fields, a binomial equation of the form (1). We specify that this equation has been…
Fundamental Theorems of Algebra for the Perplexes
ERIC Educational Resources Information Center
Poodiak, Robert; LeClair, Kevin
2009-01-01
The fundamental theorem of algebra for the complex numbers states that a polynomial of degree n has n roots, counting multiplicity. This paper explores the "perplex number system" (also called the "hyperbolic number system" and the "spacetime number system") In this system (which has extra roots of +1 besides the usual [plus or minus]1 of the…
Using geometric algebra to study optical aberrations
Hanlon, J.; Ziock, H.
1997-05-01
This paper uses Geometric Algebra (GA) to study vector aberrations in optical systems with square and round pupils. GA is a new way to produce the classical optical aberration spot diagrams on the Gaussian image plane and surfaces near the Gaussian image plane. Spot diagrams of the third, fifth and seventh order aberrations for square and round pupils are developed to illustrate the theory.
A Microcomputer Lab for Algebra & Calculus.
ERIC Educational Resources Information Center
Avery, Chris; And Others
An overview is provided of De Anza College's use of computerized instruction in its mathematics courses. After reviewing the ways in which computer technology is changing math instruction, the paper looks at the use of computers in several course sequences. The instructional model for the algebra sequence is based on a large group format of…
A Linear Algebra Measure of Cluster Quality.
ERIC Educational Resources Information Center
Mather, Laura A.
2000-01-01
Discussion of models for information retrieval focuses on an application of linear algebra to text clustering, namely, a metric for measuring cluster quality based on the theory that cluster quality is proportional to the number of terms that are disjoint across the clusters. Explains term-document matrices and clustering algorithms. (Author/LRW)
Learning Activity Package, Algebra-Trigonometry.
ERIC Educational Resources Information Center
Holland, Bill
A series of ten teacher-prepared Learning Activity Packages (LAPs) in advanced algebra and trigonometry, the units cover logic; absolute value, inequalities, exponents, and complex numbers; functions; higher degree equations and the derivative; the trigonometric function; graphs and applications of the trigonometric functions; sequences and…
Baxter Operator and Archimedean Hecke Algebra
NASA Astrophysics Data System (ADS)
Gerasimov, A.; Lebedev, D.; Oblezin, S.
2008-12-01
In this paper we introduce Baxter integral {mathcal{Q}} -operators for finite-dimensional Lie algebras {mathfrak{gl}_{ell+1}} and {mathfrak{so}_{2ell+1}} . Whittaker functions corresponding to these algebras are eigenfunctions of the {mathcal{Q}}-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = GL( ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter {mathcal{Q}}-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter {mathcal{Q}}-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra {mathcal {H}(G(mathbb{R}), K)} , K being a maximal compact subgroup of G. Finally we stress an analogy between {mathcal{Q}}-operators and certain elements of the non-Archimedean Hecke algebra {mathcal {H}(G(mathbb{Q}_p),G(mathbb{Z}_p))}.
Modern Geometric Algebra: A (Very Incomplete!) Survey
ERIC Educational Resources Information Center
Suzuki, Jeff
2009-01-01
Geometric algebra is based on two simple ideas. First, the area of a rectangle is equal to the product of the lengths of its sides. Second, if a figure is broken apart into several pieces, the sum of the areas of the pieces equals the area of the original figure. Remarkably, these two ideas provide an elegant way to introduce, connect, and…
Titration Calculations with Computer Algebra Software
ERIC Educational Resources Information Center
Lachance, Russ; Biaglow, Andrew
2012-01-01
This article examines the symbolic algebraic solution of the titration equations for a diprotic acid, as obtained using "Mathematica," "Maple," and "Mathcad." The equilibrium and conservation equations are solved symbolically by the programs to eliminate the approximations that normally would be performed by the student. Of the three programs,…
Some Unexpected Results Using Computer Algebra Systems.
ERIC Educational Resources Information Center
Alonso, Felix; Garcia, Alfonsa; Garcia, Francisco; Hoya, Sara; Rodriguez, Gerardo; de la Villa, Agustin
2001-01-01
Shows how teachers can often use unexpected outputs from Computer Algebra Systems (CAS) to reinforce concepts and to show students the importance of thinking about how they use the software and reflecting on their results. Presents different examples where DERIVE, MAPLE, or Mathematica does not work as expected and suggests how to use them as a…
Connecting Functions in Geometry and Algebra
ERIC Educational Resources Information Center
Steketee, Scott; Scher, Daniel
2016-01-01
One goal of a mathematics education is that students make significant connections among different branches of mathematics. Connections--such as those between arithmetic and algebra, between two-dimensional and three-dimensional geometry, between compass-and-straight-edge constructions and transformations, and between calculus and analytic…
Computer Algebra, Instrumentation and the Anthropological Approach
ERIC Educational Resources Information Center
Monaghan, John
2007-01-01
This article considers research and scholarship on the use of computer algebra in mathematics education following the instrumentation and the anthropological approaches. It outlines what these approaches are, positions them with regard to other approaches, examines tensions between the two approaches and makes suggestions for how work in this…
A Linear Algebraic Approach to Teaching Interpolation
ERIC Educational Resources Information Center
Tassa, Tamir
2007-01-01
A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…
Digital Maps, Matrices and Computer Algebra
ERIC Educational Resources Information Center
Knight, D. G.
2005-01-01
The way in which computer algebra systems, such as Maple, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in…
Modules as Learning Tools in Linear Algebra
ERIC Educational Resources Information Center
Cooley, Laurel; Vidakovic, Draga; Martin, William O.; Dexter, Scott; Suzuki, Jeff; Loch, Sergio
2014-01-01
This paper reports on the experience of STEM and mathematics faculty at four different institutions working collaboratively to integrate learning theory with curriculum development in a core undergraduate linear algebra context. The faculty formed a Professional Learning Community (PLC) with a focus on learning theories in mathematics and…
NASA Astrophysics Data System (ADS)
Dobrev, V. K.
2013-02-01
In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(-25) , the parabolic subalgebras including E 6(6) and E 6(-26). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so( n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so( n - 1, 1) and its analogs so( p - 1, q - 1). We consider also E6(6) and E6(2) which are parabolically related to the hermitian symmetric case E6(-14) , the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so( p, q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.
NASA Astrophysics Data System (ADS)
Masood, Syed; Faizal, Mir; Zaz, Zaid; Ali, Ahmed Farag; Raza, Jamil; Shah, Mushtaq B.
2016-12-01
In this paper, we will propose the most general form of the deformation of Heisenberg algebra motivated by the generalized uncertainty principle. This deformation of the Heisenberg algebra will deform all quantum mechanical systems. The form of the generalized uncertainty principle used to motivate these results will be motivated by the space fractional quantum mechanics, and non-locality in quantum mechanical systems. We also analyse a specific limit of this generalized deformation for one dimensional system, and in that limit, a nonlocal deformation of the momentum operator generates a local deformation of all one dimensional quantum mechanical systems. We analyse the low energy effects of this deformation on a harmonic oscillator, Landau levels, Lamb shift, and potential barrier. We also demonstrate that this deformation leads to a discretization of space.
Algebraic direct methods for few-atoms structure models.
Hauptman, Herbert A; Guo, D Y; Xu, Hongliang; Blessing, Robert H
2002-07-01
As a basis for direct-methods phasing at very low resolution for macromolecular crystal structures, normalized structure-factor algebra is presented for few-atoms structure models with N = 1, 2, 3, em leader equal atoms or polyatomic globs per unit cell. Main results include: [see text]. Triplet discriminant Delta(hk) and triplet weight W(hk) parameters, a approximately 4.0 and b approximately 3.0, respectively, were determined empirically in numerical error analyses. Tests with phases calculated for few-atoms 'super-glob' models of the protein apo-D-glyceraldehyde-3-phosphate dehydrogenase (approximately 10000 non-H atoms) showed that low-resolution phases from the new few-atoms tangent formula were much better than conventional tangent formula phases for N = 2 and 3; phases from the two formulae were essentially the same for N > or = 4.
Unified algebraic method to non-Hermitian systems with Lie algebraic linear structure
NASA Astrophysics Data System (ADS)
Zhang, Hong-Biao; Jiang, Guang-Yuan; Wang, Gang-Cheng
2015-07-01
We suggest a generic algebraic method to solve non-Hermitian Hamiltonian systems with Lie algebraic linear structure. Such method can not only unify the non-Hermitian Hamiltonian and the Hermitian Hamiltonian with the same structure but also keep self-consistent between similarity transformation and unitary transformation. To clearly reveal the correctness and physical meaning of such algebraic method, we illustrate our method with two different types of non-Hermitian Hamiltonians: the non-Hermitian Hamiltonian with Heisenberg algebraic linear structure and the non-Hermitian Hamiltonian with su(1, 1) algebraic linear structure. We obtain energy eigenvalues and the corresponding eigenstates of non-Hermitian forced harmonic oscillator with Heisenberg algebra structure via a proper similarity transformation. We also calculate the eigen-problems of general non-Hermitian Hamiltonian with su(1, 1) structure in terms of the similarity transformation. As an application, we focus on studying the non-Hermitian single-mode squeezed and coherent harmonic oscillator system and find that such similarity transformation associated with this model is in fact gauge-like transformation for simple harmonic oscillator.
The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra
NASA Astrophysics Data System (ADS)
Hallowell, Karl; Waldron, Andrew
2007-09-01
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra! was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.
Current algebra formulation of M-theory based on E11 Kac-Moody algebra
NASA Astrophysics Data System (ADS)
Sugawara, Hirotaka
2017-02-01
Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac-Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the E11 Kac-Moody algebra that was shown recently1‑5 to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside E11, so that, by constructing the current algebra of E11, I obtain both internal gauge theory and gravity theory. The energy-momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein-Hilbert action.
Relation of deformed nonlinear algebras with linear ones
NASA Astrophysics Data System (ADS)
Nowicki, A.; Tkachuk, V. M.
2014-01-01
The relation between nonlinear algebras and linear ones is established. For a one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between the Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated in an example of a harmonic oscillator.
Differential geometry on Hopf algebras and quantum groups
Watts, Paul
1994-12-15
The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined.
Weak Hom-Hopf algebras and their (co)representations
NASA Astrophysics Data System (ADS)
Zhang, Xiaohui; Wang, Shuanhong
2015-08-01
In this paper we will introduce the notion of a weak Hom-Hopf algebra, generalizing both weak Hopf algebras and Hom-Hopf algebras. Then we study the category Rep(H) of Hom-modules with bijective Hom-structure maps over a weak Hom-Hopf algebra H and show that the tensor functor of a (weak) Hom-bialgebra is actually a (weak) bimonad on a Hom-type category. At last, we prove that if H is a quasitriangular weak Hom-bialgebra (resp. ribbon weak Hom-Hopf algebra), then Rep(H) is a braided monoidal category (resp. ribbon category).
A Topos for Algebraic Quantum Theory
NASA Astrophysics Data System (ADS)
Heunen, Chris; Landsman, Nicolaas P.; Spitters, Bas
2009-10-01
The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr’s idea that the empirical content of quantum physics is accessible only through classical physics, we show how a noncommutative C*-algebra of observables A induces a topos {mathcal{T}(A)} in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra {A} . According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum {\\underline{Σ}(A)} in {mathcal{T}(A)} , which in our approach plays the role of the quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on {\\underline{Σ}} , and self-adjoint elements of A define continuous functions (more precisely, locale maps) from {\\underline{Σ}} to Scott’s interval domain. Noting that open subsets of {\\underline{Σ}(A)} correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos {mathcal{T}(A)}. These results were inspired by the topos-theoretic approach to quantum physics proposed by Butterfield and Isham, as recently generalized by Döring and Isham.
The automorphisms of Novikov algebras in low dimensions
NASA Astrophysics Data System (ADS)
Bai, Chengming; Meng, Daoji
2003-07-01
Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. They also correspond to a class of vertex algebras. An automorphism of a Novikov algebra is a linear isomorphism varphi satisfying varphi(xy) = varphi(x)varphi(y) which keeps the algebraic structure. The set of automorphisms of a Novikov algebra is a Lie group whose Lie algebra is just the Novikov algebra's derivation algebra. The theory of automorphisms plays an important role in the study of Novikov algebras. In this paper, we study the automorphisms of Novikov algebras. We get some results on their properties and classification in low dimensions. These results are fundamental in a certain sense, and they will serve as a guide for further development. Moreover, we apply these results to classify Gel'fand-Dorfman bialgebras and Novikov-Poisson algebras. These results also can be used to study certain phase spaces and geometric classical r-matrices.
Metric Lie 3-algebras in Bagger-Lambert theory
NASA Astrophysics Data System (ADS)
de Medeiros, Paul; Figueroa-O'Farrill, José; Méndez-Escobar, Elena
2008-08-01
We recast physical properties of the Bagger-Lambert theory, such as shift-symmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3-algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We classify metric Lie 3-algebras of signature (2, p) and study their Lie algebras of derivations, including those which preserve the conformal class of the inner product. We revisit the 3-algebraic criteria spelt out at the start of the paper and select those algebras with signature (2, p) which satisfy them, as well as indicate the construction of more general metric Lie 3-algebras satisfying the ghost-decoupling criterion.
Diagnosing students' misconceptions in algebra: results from an experimental pilot study.
Russell, Michael; O'Dwyer, Laura M; Miranda, Helena
2009-05-01
Computer-based diagnostic assessment systems hold potential to help teachers identify sources of poor performance and to connect teachers and students to learning activities designed to help advance students' conceptual understandings. The present article presents findings from a study that examined how students' performance in algebra and their overcoming of common algebraic misconceptions were affected by the use of a diagnostic assessment system that focused on important algebra concepts. This study used a four-group randomized cluster trial design in which teachers were assigned randomly to one of four groups: a "business as usual" control group, a partial intervention group that was provided with access to diagnostic tests results, a partial intervention group that was provided with access to the learning activities, and a full intervention group that was given access to the test results and learning activities. Data were collected from 905 students (6th-12th grade) nested within 44 teachers. We used hierarchical linear modeling techniques to compare the effects of full, partial, and no (control) intervention on students' algebraic ability and misconceptions. The analyses indicate that full intervention had a net positive effect on ability and misconception measures.
Superintegrability in Two Dimensions and the Racah-Wilson Algebra
NASA Astrophysics Data System (ADS)
Genest, Vincent X.; Vinet, Luc; Zhedanov, Alexei
2014-08-01
The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere is cast in the framework of the Racah problem for the algebra. The Hamiltonian of the 3-parameter system and the generators of its quadratic symmetry algebra are seen to correspond to the total and intermediate Casimir operators of the combination of three algebras, respectively. The construction makes explicit the isomorphism between the Racah-Wilson algebra, which is the fundamental algebraic structure behind the Racah problem for , and the invariance algebra of the generic 3-parameter system. It also provides an explanation for the occurrence of the Racah polynomials as overlap coefficients in this context. The irreducible representations of the Racah-Wilson algebra are reviewed as well as their connection with the Askey scheme of classical orthogonal polynomials.
Generalized conformal realizations of Kac-Moody algebras
NASA Astrophysics Data System (ADS)
Palmkvist, Jakob
2009-01-01
We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n =1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of Hermitian 3×3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n =2. Moreover, we obtain their infinite-dimensional extensions for n ≥3. In the case of 2×2 matrices, the resulting Lie algebras are of the form so(p +n,q+n) and the concomitant nonlinear realization generalizes the conformal transformations in a spacetime of signature (p,q).
Maximizing algebraic connectivity in air transportation networks
NASA Astrophysics Data System (ADS)
Wei, Peng
In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the
A double commutant theorem for Murray-von Neumann algebras.
Liu, Zhe
2012-05-15
Murray-von Neumann algebras are algebras of operators affiliated with finite von Neumann algebras. In this article, we study commutativity and affiliation of self-adjoint operators (possibly unbounded). We show that a maximal abelian self-adjoint subalgebra A of the Murray-von Neumann algebra A(f)(R) associated with a finite von Neumann algebra R is the Murray-von Neumann algebra A(f)(A(0)), where A(0) is a maximal abelian self-adjoint subalgebra of R and, in addition, A(0) is A Π R. We also prove that the Murray-von Neumann algebra A(f)(C) with C the center of R is the center of the Murray-von Neumann algebra A(f)(R). Von Neumann's celebrated double commutant theorem characterizes von Neumann algebras R as those for which R'' = R, where R', the commutant of R, is the set of bounded operators on the Hilbert space that commute with all operators in R. At the end of this article, we present a double commutant theorem for Murray-von Neumann algebras.
Dynamics of gelling liquids: algebraic relaxation.
Srivastava, Sunita; Kumar, C N; Tankeshwar, K
2009-08-19
The sol-gel system which is known, experimentally, to exhibit a power law decay of stress autocorrelation function has been studied theoretically. A second-order nonlinear differential equation obtained from Mori's integro-differential equation is derived which provides the algebraic decay of a time correlation function. Involved parameters in the expression obtained are related to exact properties of the corresponding correlation function. The algebraic model has been applied to Lennard-Jones and sol-gel systems. The model shows the behaviour of viscosity as has been observed in computer simulation and theoretical studies. The expression obtained for the viscosity predicts a logarithmic divergence at a critical value of the parameter in agreement with the prediction of other theories.
Field Theoretic Investigations in Current Algebra
NASA Astrophysics Data System (ADS)
Jackiw, Roman
The following sections are included: * Introduction * Canonical and Space-Time Constraints in Current Algebra * Canonical Theory of Currents * Space-Time Constraints on Commutators * Space-Time Constraints on Green's Functions * Space-Time Constraints on Ward Identities * Schwinger Terms * Discussion * The Bjorken-Johnson-Low Limit * The π 0 → 2γ Problem * Preliminaries * Sutherland-Veltman Theorem * Model Calculation * Anomalous Ward Identity * Anomalous Commutators * Anomalous Divergence of Axial Current * Discussion * Electroproduction Sum Rules * Preliminaries * Derivation of Sum Rules, Naive Method * Derivation of Sum Rules, Dispersive Method * Model Calculation * Anomalous Commutators * Discussion * Discussion of Anomalies in Current Algebra * Miscellaneous Anomalies * Non-Perturbative Arguments for Anomalies * Models without Anomalies * Discussion * Approximate Scale Symmetry * Introduction * Canonical Theory of Scale and Conformal Transformations * Ward Identities and Trace Identities * False Theorems * True Theorems * EXERCISES * SOLUTIONS
On an algebra of pseudodifferential operators
NASA Astrophysics Data System (ADS)
Zuĭ Bang, Kha
1995-08-01
In this paper we introduce an algebra of pseudodifferential operators with symbols of finite smoothness, acting invariantly and continuously in an Orlicz space of functions of exponential type. The concept of point spectral radius \\displaystyle \\lim_{m\\to\\infty}\\Vert A^m(D)f\\Vert _{\\Phi }^{1/m}is introduced and its existence is proved. Here f is an arbitrary function in this space, A(D) is an arbitrary element of the algebra, and \\Vert\\cdot\\Vert _{\\Phi } is the Luxemburg norm. This point spectral radius is evaluated as the supremum of the modulus of A(D) on the support of the Fourier transform of f. We evaluate the spectral radius of a pseudodifferential operator. As applications, certain non-convex and convex cases of the well-known Paley-Wiener theorem are obtained. We also consider the solvability of pseudodifferential equations.
An algebraic approach to the Hubbard model
NASA Astrophysics Data System (ADS)
de Leeuw, Marius; Regelskis, Vidas
2016-02-01
We study the algebraic structure of an integrable Hubbard-Shastry type lattice model associated with the centrally extended su (2 | 2) superalgebra. This superalgebra underlies Beisert's AdS/CFT worldsheet R-matrix and Shastry's R-matrix. The considered model specializes to the one-dimensional Hubbard model in a certain limit. We demonstrate that Yangian symmetries of the R-matrix specialize to the Yangian symmetry of the Hubbard model found by Korepin and Uglov. Moreover, we show that the Hubbard model Hamiltonian has an algebraic interpretation as the so-called secret symmetry. We also discuss Yangian symmetries of the A and B models introduced by Frolov and Quinn.
Using Group Explorer in teaching abstract algebra
NASA Astrophysics Data System (ADS)
Schubert, Claus; Gfeller, Mary; Donohue, Christopher
2013-04-01
This study explores the use of Group Explorer in an undergraduate mathematics course in abstract algebra. The visual nature of Group Explorer in representing concepts in group theory is an attractive incentive to use this software in the classroom. However, little is known about students' perceptions on this technology in learning concepts in abstract algebra. A total of 26 participants in an undergraduate course studying group theory were surveyed regarding their experiences using Group Explorer. Findings indicate that all participants believed that the software was beneficial to their learning and described their attitudes regarding the software in terms of using the technology and its helpfulness in learning concepts. A multiple regression analysis reveals that representational fluency of concepts with the software correlated significantly with participants' understanding of group concepts yet, participants' attitudes about Group Explorer and technology in general were not significant factors.
Selecting reusable components using algebraic specifications
NASA Technical Reports Server (NTRS)
Eichmann, David A.
1992-01-01
A significant hurdle confronts the software reuser attempting to select candidate components from a software repository - discriminating between those components without resorting to inspection of the implementation(s). We outline a mixed classification/axiomatic approach to this problem based upon our lattice-based faceted classification technique and Guttag and Horning's algebraic specification techniques. This approach selects candidates by natural language-derived classification, by their interfaces, using signatures, and by their behavior, using axioms. We briefly outline our problem domain and related work. Lattice-based faceted classifications are described; the reader is referred to surveys of the extensive literature for algebraic specification techniques. Behavioral support for reuse queries is presented, followed by the conclusions.
Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.
Sialaros, Michalis; Christianidis, Jean
2016-06-01
Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.
Projective Connections and the Algebra of Densities
George, Jacob
2008-11-18
Projective connections first appeared in Cartan's papers in the 1920's. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant quantisation. We recall the notion of projective connection and describe its relation with the algebra of densities on a manifold. In particular, we construct a Laplace-type operator on functions using a Thomas projective connection and a symmetric contravariant tensor of rank 2 ('upper metric')
Charge transfer in algebraic quantum field theory
NASA Astrophysics Data System (ADS)
Wright, Jill Dianne
We discuss aspects of the algebraic structure of quantum field theory. We take the view that the superselection structure of a theory should be determinable from the vacuum representation of the observable algebra, and physical properties of the charge. Hence one determines the nature of the charge transfer operations: the automorphisms of the observable algebra corresponding to the movement of charge along space-time paths. New superselection sectors are obtained from the vacuum sector by an automorphism which is a limit of charge transfer operations along paths with an endpoint tending to spacelike infinity. Roberts has shown that for a gauge theory of the first kind, the charge transfer operations for a given charge form a certain kind of 1-cocycle over Minkowski space. The local 1-cohomology group of their equivalence classes corresponds to the superselection structure. The exact definition of the cohomology group depends on the properties of the charge. Using displaced Fock representations of free fields, we develop model field theories which illustrate this structure. The cohomological classification of displaced Fock representations has been elucidated by Araki. For more general representations, explicit determination of the cohomology group is a hard problem. Using our models, we can illustrate ways in which fields with reasonable physical properties depart fromthe abovementioned structure. In 1+1 dimensions, we use the Streater-Wilde model to illustrate explicitly the representation-dependence of the cohomology structure, and the direction-dependence of the limiting charge transfer operation. The cohomology structure may also be representation-dependent in higher-dimensional theories without strict localization of charge, for example the electromagnetic field. The algebraic structure of the electromagnetic field has many other special features, which we discuss in relation to the concept of charge transfer. We also give some indication of the modifications
On Ternary Quotients of Cubic Hecke Algebras
NASA Astrophysics Data System (ADS)
Cabanes, Marc; Marin, Ivan
2012-08-01
We prove that the quotient of the group algebra of the braid group introduced by Funar (Commun Math Phys 173:513-558, 1995) collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in addition a symmetry of order 3. We also establish conditions on the possible Markov traces factorizing through it.
Aspects of coherent states of nonlinear algebras
NASA Astrophysics Data System (ADS)
Shreecharan, T.; Chaitanya, K. V. S. Shiv
2010-12-01
Various aspects of coherent states of nonlinear su(2) and su(1, 1) algebras are studied. It is shown that the nonlinear su(1, 1) Barut-Girardello and Perelomov coherent states are related by a Laplace transform. We then concentrate on the derivation and analysis of the statistical and geometrical properties of these states. The Berry's phase for the nonlinear coherent states is also derived.
Boolean Operations with Prism Algebraic Patches
Bajaj, Chandrajit; Paoluzzi, Alberto; Portuesi, Simone; Lei, Na; Zhao, Wenqi
2009-01-01
In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C1-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation. PMID:21516262
Using computer algebra and SMT-solvers to analyze a mathematical model of cholera propagation
NASA Astrophysics Data System (ADS)
Trujillo Arredondo, Mariana
2014-06-01
We analyze a mathematical model for the transmission of cholera. The model is already defined and involves variables such as the pathogen agent, which in this case is the bacterium Vibrio cholera, and the human population. The human population is divided into three classes: susceptible, infectious and removed. Using Computer Algebra, specifically Maple we obtain two equilibrium states: the disease free state and the endemic state. Using Maple it is possible to prove that the disease free state is locally asymptotically stable if and only if R0 < 1. Using Maple it is possible to prove that the endemic equilibrium state is locally stable when it exists, it is to say when R0 > 1. Using the package Red-Log of the Computer algebra system Reduce and the SMT-Solver Z3Py it is possible to obtain numerical conditions for the model. The formula for the basic reproductive number makes a synthesis with all epidemic parameters in the model. Also it is possible to make numerical simulations which are very illustrative about the epidemic patters that are expected to be observed in real situations. We claim that these kinds of software are very useful in the analysis of epidemic models given that the symbolic computation provides algebraic formulas for the basic reproductive number and such algebraic formulas are very useful to derive control measures. For other side, computer algebra software is a powerful tool to make the stability analysis for epidemic models given that the all steps in the stability analysis can be made automatically: finding the equilibrium points, computing the jacobian, computing the characteristic polynomial for the jacobian, and applying the Routh-Hurwitz theorem to the characteristic polynomial. Finally, using SMT-Solvers is possible to make automatically checks of satisfiability, validity and quantifiers elimination being these computations very useful to analyse complicated epidemic models.
CULA: hybrid GPU accelerated linear algebra routines
NASA Astrophysics Data System (ADS)
Humphrey, John R.; Price, Daniel K.; Spagnoli, Kyle E.; Paolini, Aaron L.; Kelmelis, Eric J.
2010-04-01
The modern graphics processing unit (GPU) found in many standard personal computers is a highly parallel math processor capable of nearly 1 TFLOPS peak throughput at a cost similar to a high-end CPU and an excellent FLOPS/watt ratio. High-level linear algebra operations are computationally intense, often requiring O(N3) operations and would seem a natural fit for the processing power of the GPU. Our work is on CULA, a GPU accelerated implementation of linear algebra routines. We present results from factorizations such as LU decomposition, singular value decomposition and QR decomposition along with applications like system solution and least squares. The GPU execution model featured by NVIDIA GPUs based on CUDA demands very strong parallelism, requiring between hundreds and thousands of simultaneous operations to achieve high performance. Some constructs from linear algebra map extremely well to the GPU and others map poorly. CPUs, on the other hand, do well at smaller order parallelism and perform acceptably during low-parallelism code segments. Our work addresses this via hybrid a processing model, in which the CPU and GPU work simultaneously to produce results. In many cases, this is accomplished by allowing each platform to do the work it performs most naturally.
Translating cosmological special relativity into geometric algebra
NASA Astrophysics Data System (ADS)
Horn, Martin Erik
2012-11-01
Geometric algebra and Clifford algebra are important tools to describe and analyze the physics of the world we live in. Although there is enormous empirical evidence that we are living in four dimensional spacetime, mathematical worlds of higher dimensions can be used to present the physical laws of our world in an aesthetical and didactical more appealing way. In physics and mathematics education we are therefore confronted with the question how these high dimensional spaces should be taught. But as an immediate confrontation of students with high dimensional compactified spacetimes would expect too much from them at the beginning of their university studies, it seems reasonable to approach the mathematics and physics of higher dimensions step by step. The first step naturally is the step from four dimensional spacetime of special relativity to a five dimensional spacetime world. As a toy model for this artificial world cosmological special relativity, invented by Moshe Carmeli, can be used. This five dimensional non-compactified approach describes a spacetime which consists not only of one time dimension and three space dimensions. In addition velocity is regarded as a fifth dimension. This model very probably will not represent physics correctly. But it can be used to discuss and analyze the consequences of an additional dimension in a clear and simple way. Unfortunately Carmeli has formulated cosmological special relativity in standard vector notation. Therefore a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of cosmological special relativity is discussed.
Multifractal vector fields and stochastic Clifford algebra.
Schertzer, Daniel; Tchiguirinskaia, Ioulia
2015-12-01
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Multifractal vector fields and stochastic Clifford algebra
Schertzer, Daniel Tchiguirinskaia, Ioulia
2015-12-15
In the mid 1980s, the development of multifractal concepts and techniques was an important breakthrough for complex system analysis and simulation, in particular, in turbulence and hydrology. Multifractals indeed aimed to track and simulate the scaling singularities of the underlying equations instead of relying on numerical, scale truncated simulations or on simplified conceptual models. However, this development has been rather limited to deal with scalar fields, whereas most of the fields of interest are vector-valued or even manifold-valued. We show in this paper that the combination of stable Lévy processes with Clifford algebra is a good candidate to bridge up the present gap between theory and applications. We show that it indeed defines a convenient framework to generate multifractal vector fields, possibly multifractal manifold-valued fields, based on a few fundamental and complementary properties of Lévy processes and Clifford algebra. In particular, the vector structure of these algebra is much more tractable than the manifold structure of symmetry groups while the Lévy stability grants a given statistical universality.
Algebraic and geometric spread in finite frames
NASA Astrophysics Data System (ADS)
King, Emily J.
2015-08-01
When searching for finite unit norm tight frames (FUNTFs) of M vectors in FN which yield robust representations, one is concerned with finding frames consisting of frame vectors which are in some sense as spread apart as possible. Algebraic spread and geometric spread are the two most commonly used measures of spread. A frame with optimal algebraic spread is called full spark and is such that any subcollection of N frame vectors is a basis for FN. A Grassmannian frame is a FUNTF which satisfies the Grassmannian packing problem; that is, the frame vectors are optimally geometrically spread given fixed M and N. A particular example of a Grassmannian frame is an equiangular frame, which is such that the absolute value of all inner products of distinct vectors is equal. The relationship between these two types of optimal spread is complicated. The folk knowledge for many years was that equiangular frames were full spark; however, this is now known not to hold for an infinite class of equiangular frames. The exact relationship between these types of spread will be further explored in this talk, as well as Plücker coordinates and coherence, which are measures of how much a frame misses being optimally algebraically or geometrically spread.
Structured adaptive grid generation using algebraic methods
NASA Technical Reports Server (NTRS)
Yang, Jiann-Cherng; Soni, Bharat K.; Roger, R. P.; Chan, Stephen C.
1993-01-01
The accuracy of the numerical algorithm depends not only on the formal order of approximation but also on the distribution of grid points in the computational domain. Grid adaptation is a procedure which allows optimal grid redistribution as the solution progresses. It offers the prospect of accurate flow field simulations without the use of an excessively timely, computationally expensive, grid. Grid adaptive schemes are divided into two basic categories: differential and algebraic. The differential method is based on a variational approach where a function which contains a measure of grid smoothness, orthogonality and volume variation is minimized by using a variational principle. This approach provided a solid mathematical basis for the adaptive method, but the Euler-Lagrange equations must be solved in addition to the original governing equations. On the other hand, the algebraic method requires much less computational effort, but the grid may not be smooth. The algebraic techniques are based on devising an algorithm where the grid movement is governed by estimates of the local error in the numerical solution. This is achieved by requiring the points in the large error regions to attract other points and points in the low error region to repel other points. The development of a fast, efficient, and robust algebraic adaptive algorithm for structured flow simulation applications is presented. This development is accomplished in a three step process. The first step is to define an adaptive weighting mesh (distribution mesh) on the basis of the equidistribution law applied to the flow field solution. The second, and probably the most crucial step, is to redistribute grid points in the computational domain according to the aforementioned weighting mesh. The third and the last step is to reevaluate the flow property by an appropriate search/interpolate scheme at the new grid locations. The adaptive weighting mesh provides the information on the desired concentration
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
NASA Astrophysics Data System (ADS)
Anguelova, Iana I.
2013-12-01
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras.
Clifford Algebras, Random Graphs, and Quantum Random Variables
NASA Astrophysics Data System (ADS)
Schott, René; Staples, G. Stacey
2008-08-01
For fixed n > 0, the space of finite graphs on n vertices is canonically associated with an abelian, nilpotent-generated subalgebra of the Clifford algebra {C}l2n,2n which is canonically isomorphic to the 2n-particle fermion algebra. Using the generators of the subalgebra, an algebraic probability space of "Clifford adjacency matrices" associated with finite graphs is defined. Each Clifford adjacency matrix is a quantum random variable whose mth moment corresponds to the number of m-cycles in the graph G. Each matrix admits a canonical "quantum decomposition" into a sum of three algebraic random variables: a = aΔ + aΥ + aΛ, where aΔ is classical while aΥ and aΛ are quantum. Moreover, within the Clifford algebra context the NP problem of cycle enumeration is reduced to matrix multiplication, requiring no more than n4 Clifford (geo-metric) multiplications within the algebra.
Lie-algebraic solutions of the type IIB matrix model
NASA Astrophysics Data System (ADS)
Chatzistavrakidis, Athanasios
2011-11-01
A systematic search for Lie-algebra solutions of the type IIB matrix model is performed. Our survey is based on the classification of all Lie algebras for dimensions up to five and of all nilpotent Lie algebras of dimension six. It is shown that Lie-type solutions of the equations of motion of the type IIB matrix model exist and they correspond to certain nilpotent and solvable Lie algebras. Their representation in terms of Hermitian matrices is discussed in detail. These algebras give rise to certain noncommutative spaces for which the corresponding star products are provided. Finally the issue of constructing quantized compact nilmanifolds and solvmanifolds based on the above algebras is addressed.
Minimax Techniques For Optimizing Non-Linear Image Algebra Transforms
NASA Astrophysics Data System (ADS)
Davidson, Jennifer L.
1989-08-01
It has been well established that the Air Force Armament Technical Laboratory (AFATL) image algebra is capable of expressing all linear transformations [7]. The embedding of the linear algebra in the image algebra makes this possible. In this paper we show a relation of the image algebra to another algebraic system called the minimax algebra. This system is used extensively in economics and operations research, but until now has not been investigated for applications to image processing. The relationship is exploited to develop new optimization methods for a class of non-linear image processing transforms. In particular, a general decomposition technique for templates in this non-linear domain is presented. Template decomposition techniques are an important tool in mapping algorithms efficiently to both sequential and massively parallel architectures.
On left Hopf algebras within the framework of inhomogeneous quantum groups for particle algebras
Rodriguez-Romo, Suemi
2012-10-15
We deal with some matters needed to construct concrete left Hopf algebras for inhomogeneous quantum groups produced as noncommutative symmetries of fermionic and bosonic creation/annihilation operators. We find a map for the bidimensional fermionic case, produced as in Manin's [Quantum Groups and Non-commutative Hopf Geometry (CRM Univ. de Montreal, 1988)] seminal work, named preantipode that fulfills all the necessary requirements to be left but not right on the generators of the algebra. Due to the complexity and importance of the full task, we consider our result as an important step that will be extended in the near future.
Aspects of infinite dimensional ℓ-super Galilean conformal algebra
NASA Astrophysics Data System (ADS)
Aizawa, N.; Segar, J.
2016-12-01
In this work, we construct an infinite dimensional ℓ-super Galilean conformal algebra, which is a generalization of the ℓ = 1 algebra found in the literature. We give a classification of central extensions, the vector field representation, the coadjoint representation, and the operator product expansion of the infinite dimensional ℓ-super Galilean conformal algebra, keeping possible applications in physics and mathematics in mind.
Spectral Metric Spaces on Extensions of C*-Algebras
NASA Astrophysics Data System (ADS)
Hawkins, Andrew; Zacharias, Joachim
2017-03-01
We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.
Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations
2014-07-01
non- linear hybrid systems by linear algebraic methods. In Radhia Cousot and Matthieu Martel, editors, SAS, volume 6337 of LNCS, pages 373–389. Springer...Tarski. A decision method for elementary algebra and geometry. Bulletin of the American Mathematical Society, 59, 1951. [36] Wolfgang Walter. Ordinary...Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 July 2014
Poincare invariant algebra from instant to light-front quantization
Ji, Chueng-Ryong; Mitchell, Chad
2001-10-15
We present the Poincare algebra interpolating between instant and light-front time quantizations. The angular momentum operators satisfying SU(2) algebra are constructed in an arbitrary interpolation angle and shown to be identical to the ordinary angular momentum and Leutwyler-Stern angular momentum in the instant and light-front quantization limits, respectively. The exchange of the dynamical role between the transverse angular mometum and the boost operators is manifest in our newly constructed algebra.
Quantum walks, deformed relativity and Hopf algebra symmetries.
Bisio, Alessandro; D'Ariano, Giacomo Mauro; Perinotti, Paolo
2016-05-28
We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras.
Clifford algebra approach to the coincidence problem for planar lattices.
Rodríguez, M A; Aragón, J L; Verde-Star, L
2005-03-01
The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions.
Twisted conformal algebra related to κ -Minkowski space
NASA Astrophysics Data System (ADS)
Meljanac, Stjepan; Pachoł, Anna; Pikutić, Danijel
2015-11-01
Twisted deformations of the conformal symmetry in the Hopf algebraic framework are constructed. The first one is obtained by a Jordanian twist built up from dilatation and momenta generators. The second is the lightlike κ -deformation of the Poincaré algebra extended to the conformal algebra, obtained by a twist corresponding to the extended Jordanian r -matrix. The κ -Minkowski spacetime is covariant quantum space under both of these deformations. The extension of the conformal algebra by the noncommutative coordinates is presented in two cases. The differential realizations for κ -Minkowski coordinates, as well as their left-right dual counterparts, are also included.
A spatial operator algebra for manipulator modeling and control
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.
1991-01-01
A recently developed spatial operator algebra for manipulator modeling, control, and trajectory design is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high-level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics.
Spatial-Operator Algebra For Flexible-Link Manipulators
NASA Technical Reports Server (NTRS)
Jain, Abhinandan; Rodriguez, Guillermo
1994-01-01
Method of computing dynamics of multiple-flexible-link robotic manipulators based on spatial-operator algebra, which originally applied to rigid-link manipulators. Aspects of spatial-operator-algebra approach described in several previous articles in NASA Tech Briefs-most recently "Robot Control Based on Spatial-Operator Algebra" (NPO-17918). In extension of spatial-operator algebra to manipulators with flexible links, each link represented by finite-element model: mass of flexible link apportioned among smaller, lumped-mass rigid bodies, coupling of motions expressed in terms of vibrational modes. This leads to operator expression for modal-mass matrix of link.
The structure of split regular BiHom-Lie algebras
NASA Astrophysics Data System (ADS)
Calderón, Antonio J.; Sánchez, José M.
2016-12-01
We introduce the class of split regular BiHom-Lie algebras as the natural extension of the one of split Hom-Lie algebras and so of split Lie algebras. We show that an arbitrary split regular BiHom-Lie algebra L is of the form L = U +∑jIj with U a linear subspace of a fixed maximal abelian subalgebra H and any Ij a well described (split) ideal of L, satisfying [Ij ,Ik ] = 0 if j ≠ k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its simple ideals.
Conformal manifolds in four dimensions and chiral algebras
NASA Astrophysics Data System (ADS)
Buican, Matthew; Nishinaka, Takahiro
2016-11-01
Any { N }=2 superconformal field theory (SCFT) in four dimensions has a sector of operators related to a two-dimensional chiral algebra containing a Virasoro sub-algebra. Moreover, there are well-known examples of isolated SCFTs whose chiral algebra is a Virasoro algebra. In this note, we consider the chiral algebras associated with interacting { N }=2 SCFTs possessing an exactly marginal deformation that can be interpreted as a gauge coupling (i.e., at special points on the resulting conformal manifolds, free gauge fields appear that decouple from isolated SCFT building blocks). At any point on these conformal manifolds, we argue that the associated chiral algebras possess at least three generators. In addition, we show that there are examples of SCFTs realizing such a minimal chiral algebra: they are certain points on the conformal manifold obtained by considering the low-energy limit of type IIB string theory on the three complex-dimensional hypersurface singularity {x}13+{x}23+{x}33+α {x}1{x}2{x}3+{w}2=0. The associated chiral algebra is the { A }(6) theory of Feigin, Feigin, and Tipunin. As byproducts of our work, we argue that (i) a collection of isolated theories can be conformally gauged only if there is a SUSY moduli space associated with the corresponding symmetry current moment maps in each sector, and (ii) { N }=2 SCFTs with a≥slant c have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators).
NASA Astrophysics Data System (ADS)
Mukhin, Evgeny; Tarasov, Vitaly; Varchenko, Alexander
2011-10-01
Consider a tensor product of finite-dimensional irreducible ??;N+1-modules and its decomposition into irreducible modules. The ??;N+1 Gaudin model assigns to each multiplicity space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra. In this paper we consider the algebra of functions on the critical set of the associated master function and show that the action of this algebra on itself is isomorphic to the action of the Bethe algebra on a suitable subspace of the multiplicity space. As a byproduct we prove that the Bethe vectors corresponding to different critical points of the master function are linearly independent and, in particular, nonzero.
Remarks on the differential algebraic approach to particle beam optics by M. Berz
Garczynski, V.
1992-12-31
The underlying mathematical structure of the differential algebraic approach of M. Berz to particle beam optics is isomorphic to the familiar truncated polynomial algebra. Concrete examples of derivations in this algebra, consistent with the truncation operation, are given.
Capable n-Lie algebras and the classification of nilpotent n-Lie algebras with s(A) = 3
NASA Astrophysics Data System (ADS)
Darabi, Hamid; Saeedi, Farshid; Eshrati, Mehdi
2016-12-01
Darabi et al. (2016) associate to each d-dimensional nilpotent n-Lie algebra A, a number s(A) where s(A) =(d-1/n) + n - 1 - dim M(A) and M(A) denotes the multiplier of A. They prove that s(A) is non-negative and classify all nilpotent n-Lie algebras A for which s(A) = 0 , 1 , 2. In this paper, we will classify all capable n-Lie algebras with 1-dimensional derived subalgebra and apply it to obtain a classification of those nilpotent n-Lie algebras satisfying s(A) = 3.
Cartan-Weyl 3-algebras and the BLG theory. I: classification of Cartan-Weyl 3-algebras
NASA Astrophysics Data System (ADS)
Chu, Chong-Sun
2010-10-01
As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H I and a number of step generators E α that are characterized by a root space of non-degenerate one-forms α. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.
The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra.
ERIC Educational Resources Information Center
Carlson, David; And Others
1993-01-01
Presents five recommendations of the Linear Algebra Curriculum Study Group: (1) The syllabus must respond to the client disciplines; (2) The first course should be matrix oriented; (3) Faculty should consider the needs and interests of students; (4) Faculty should use technology; and (5) At least one follow-up course should be required. Provides a…
Access to Algebra I, Gateway to Success: The Impact of Eighth-Grade Algebra I
ERIC Educational Resources Information Center
Darling, Emily Jean Skelton
2010-01-01
An understanding of Algebra I and the role that this foundational course plays as an entry to the college preparatory pathway in secondary education and its influence on mathematical achievement is an integral component for the education of American youth in the global world of science and technology. Achievements in high school curricula are…
Combinatorial Hopf Algebras in Quantum Field Theory I
NASA Astrophysics Data System (ADS)
Figueroa, Héctor; Gracia-Bondía, José M.
This paper stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Sec. 1.1 is the introduction, and contains an elementary invitation to the subject as well. The rest of Sec. 1 is devoted to the basics of Hopf algebra theory and examples in ascending level of complexity. Section 2 turns around the all-important Faà di Bruno Hopf algebra. Section 2.1 contains a first, direct approach to it. Section 2.2 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 2.3 rederives the related Connes-Moscovici algebras. In Sec. 3, we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Sec. 3.1, we describe the first. Then in Sec. 3.2, we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Sec. 3.3, general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Sec. 3.4, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained. The structure results for commutative Hopf algebras are found in Sec. 4. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.
PREFACE: Algebra, Geometry, and Mathematical Physics 2010
NASA Astrophysics Data System (ADS)
Stolin, A.; Abramov, V.; Fuchs, J.; Paal, E.; Shestopalov, Y.; Silvestrov, S.
2012-02-01
This proceedings volume presents results obtained by the participants of the 6th Baltic-Nordic workshop 'Algebra, Geometry, and Mathematical Physics (AGMP-6)' held at the Sven Lovén Centre for Marine Sciences in Tjärnö, Sweden on October 25-30, 2010. The Baltic-Nordic Network AGMP 'Algebra, Geometry, and Mathematical Physics' http://www.agmp.eu was created in 2005 on the initiative of two Estonian universities and two Swedish universities: Tallinn University of Technology represented by Eugen Paal (coordinator of the network), Tartu University represented by Viktor Abramov, Lund University represented by Sergei Silvestrov, and Chalmers University of Technology and the University of Gothenburg represented by Alexander Stolin. The goal was to promote international and interdisciplinary cooperation between scientists and research groups in the countries of the Baltic-Nordic region in mathematics and mathematical physics, with special emphasis on the important role played by algebra and geometry in modern physics, engineering and technologies. The main activities of the AGMP network consist of a series of regular annual international workshops, conferences and research schools. The AGMP network also constitutes an important educational forum for scientific exchange and dissimilation of research results for PhD students and Postdocs. The network has expanded since its creation, and nowadays its activities extend beyond countries in the Baltic-Nordic region to universities in other European countries and participants from elsewhere in the world. As one of the important research-dissimilation outcomes of its activities, the network has a tradition of producing high-quality research proceedings volumes after network events, publishing them with various international publishers. The PDF also contains the following: List of AGMP workshops and other AGMP activities Main topics discussed at AGMP-6 Review of AGMP-6 proceedings Acknowledgments List of Conference Participants
Basic linear algebra subprograms for FORTRAN usage
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Hanson, R. J.; Kincaid, D. R.; Krogh, F. T.
1977-01-01
A package of 38 low level subprograms for many of the basic operations of numerical linear algebra is presented. The package is intended to be used with FORTRAN. The operations in the package are dot products, elementary vector operations, Givens transformations, vector copy and swap, vector norms, vector scaling, and the indices of components of largest magnitude. The subprograms and a test driver are available in portable FORTRAN. Versions of the subprograms are also provided in assembly language for the IBM 360/67, the CDC 6600 and CDC 7600, and the Univac 1108.
Optical linear algebra processors - Architectures and algorithms
NASA Technical Reports Server (NTRS)
Casasent, David
1986-01-01
Attention is given to the component design and optical configuration features of a generic optical linear algebra processor (OLAP) architecture, as well as the large number of OLAP architectures, number representations, algorithms and applications encountered in current literature. Number-representation issues associated with bipolar and complex-valued data representations, high-accuracy (including floating point) performance, and the base or radix to be employed, are discussed, together with case studies on a space-integrating frequency-multiplexed architecture and a hybrid space-integrating and time-integrating multichannel architecture.
Algebraic approach to electronic spectroscopy and dynamics.
Toutounji, Mohamad
2008-04-28
Lie algebra, Zassenhaus, and parameter differentiation techniques are utilized to break up the exponential of a bilinear Hamiltonian operator into a product of noncommuting exponential operators by the virtue of the theory of Wei and Norman [J. Math. Phys. 4, 575 (1963); Proc. Am. Math. Soc., 15, 327 (1964)]. There are about three different ways to find the Zassenhaus exponents, namely, binomial expansion, Suzuki formula, and q-exponential transformation. A fourth, and most reliable method, is provided. Since linearly displaced and distorted (curvature change upon excitation/emission) Hamiltonian and spin-boson Hamiltonian may be classified as bilinear Hamiltonians, the presented algebraic algorithm (exponential operator disentanglement exploiting six-dimensional Lie algebra case) should be useful in spin-boson problems. The linearly displaced and distorted Hamiltonian exponential is only treated here. While the spin-boson model is used here only as a demonstration of the idea, the herein approach is more general and powerful than the specific example treated. The optical linear dipole moment correlation function is algebraically derived using the above mentioned methods and coherent states. Coherent states are eigenvectors of the bosonic lowering operator a and not of the raising operator a(+). While exp(a(+)) translates coherent states, exp(a(+)a(+)) operation on coherent states has always been a challenge, as a(+) has no eigenvectors. Three approaches, and the results, of that operation are provided. Linear absorption spectra are derived, calculated, and discussed. The linear dipole moment correlation function for the pure quadratic coupling case is expressed in terms of Legendre polynomials to better show the even vibronic transitions in the absorption spectrum. Comparison of the present line shapes to those calculated by other methods is provided. Franck-Condon factors for both linear and quadratic couplings are exactly accounted for by the herein calculated
Vector fields and nilpotent Lie algebras
NASA Technical Reports Server (NTRS)
Grayson, Matthew; Grossman, Robert
1987-01-01
An infinite-dimensional family of flows E is described with the property that the associated dynamical system: x(t) = E(x(t)), where x(0) is a member of the set R to the Nth power, is explicitly integrable in closed form. These flows E are of the form E = E1 + E2, where E1 and E2 are the generators of a nilpotent Lie algebra, which is either free, or satisfies some relations at a point. These flows can then be used to approximate the flows of more general types of dynamical systems.
Spatial Operator Algebra for multibody system dynamics
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Jain, A.; Kreutz-Delgado, K.
1992-01-01
The Spatial Operator Algebra framework for the dynamics of general multibody systems is described. The use of a spatial operator-based methodology permits the formulation of the dynamical equations of motion of multibody systems in a concise and systematic way. The dynamical equations of progressively more complex grid multibody systems are developed in an evolutionary manner beginning with a serial chain system, followed by a tree topology system and finally, systems with arbitrary closed loops. Operator factorizations and identities are used to develop novel recursive algorithms for the forward dynamics of systems with closed loops. Extensions required to deal with flexible elements are also discussed.
Noncommutative Pfaffians associated with the orthogonal algebra
Artamonov, Dmitrii V; Golubeva, Valentina A
2012-12-31
Commutators of Pfaffians associated with the orthogonal algebra are found in skew-symmetric and root realizations of o{sub N}. A generating function of Pfaffians is proved to satisfy the reflection equation. A relation between Pfaffians in skew-symmetric and root realizations of o{sub N} is established. Using these results we construct an integrable equation of Knizhnik-Zamolodchikov type using the Capelli central elements in U(o{sub N}), which are sums of squares of the considered Pfaffians. A classical limit of the obtained Knizhnik-Zamolodchikov type equation turns out to be a very specific system of equations of isomonodromic deformations. Bibliography: 18 titles.
Bialgebra deformations and algebras of trees
NASA Technical Reports Server (NTRS)
Grossman, Robert; Radford, David
1991-01-01
Let A denote a bialgebra over a field k and let A sub t = A((t)) denote the ring of formal power series with coefficients in A. Assume that A is also isomorphic to a free, associative algebra over k. A simple construction is given which makes A sub t a bialgebra deformation of A. In typical applications, A sub t is neither commutative nor cocommutative. In the terminology of Drinfeld, (1987), A sub t is a quantum group. This construction yields quantum groups associated with families of trees.
Spurious Roots in the Algebraic Dirac Equation
NASA Astrophysics Data System (ADS)
Pestka, Grzegorz
The nature of spurious roots discovered by Drake and Goldman [G. W. F. Drake and S. P. Goldman, Phys. Rev. A 23, 2093 (1981)] among solutions of the algebraic Dirac Hamiltonian eigenvalue problem is discussed. It is shown that the spurious roots represent the positive spectrum states of the Dirac Hamiltonian and that each of them has its variational non-relativistic counterpart. Sufficient conditions to avoid the appearance of the spuriouses in the forbidden gap of Dirac energies are formulated. Numerical examples for κ = 1 ( P1/2) states of an electron in a spherical Coulomb potential (in Slater-type bases) are presented.
Reasoning algebraically with IT: A cognitive perspective
NASA Astrophysics Data System (ADS)
Mok, Ida; Johnson, David
2000-12-01
The focus of this paper is on the implications of key findings and theoretical positions from social psychology and cognitive developmental psychology (Piagetian/neo-Piagetian) for the use of IT tools to support learning in algebra. Particular reference is made to the research of the UK Cognitive Acceleration through Mathematics Education (CAME) project. The feasibility of the CAME model in the exploration of mathematical relationships supported by graphics calculators was addressed in a small-scale study in Hong Kong. The research provides evidence that, with appropriate mediation, cognitive conflict can be utilised to provide valuable appropriate for students to engage in increasingly higher levels of mathematical thinking.