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This is the first of two volumes of a stateoftheart survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and nonNoetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: nonnoetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Fløystad on BoijSöderburg theory, of Geramita, Harbourne and Migliore, and of Cooper on Hilbert functions, of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also utilize algebraic invariants to understand combinatorial structures like graphs, hypergraphs, and simplicial complexes such as in the paper of Morey and Villarreal on edge ideals. Homological techniques have become indispensable tools for the study of noetherian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of Dmodules), and combinatorics (as described in the previous paragraph). The homological art
Commutative Algebra  Combinatorics  Homology
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This book reviews current research, including applications of matrices, spaces, and other characteristics. It discusses the application of matrices, which has become an area of great importance in many scientific fields. The theory of row/column determinants of a partial solution to the system of twosided quaternion matrix equations is analyzed. It introduces a matrix that has the exponential function as one of its eigenvectors and realizes that this matrix represents finite difference derivation of vectors on a partition. Mixing problems and the corresponding associated matrices have different structures that deserve to be studied in depth. Special compound magic squares will be considered. Finally, a new type of regular matrix generated by Fibonacci numbers is introduced and we shall investigate its various topological properties.
Physical Sciences, Engineering and Technology  Mathematics  Algebra  Linear Algebra
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SteenrodAlgebra  Homologische Algebra  Stabile Homotopietheorie  Adams Spektralreihe  Homotopiegruppen
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This is the second of two volumes of a stateoftheart survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and nonNoetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: nonnoetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and one by Enescu which discusses the action of the Frobenius on finite dimensional vector spaces both of which are related to tight closure. Finiteness properties of rings and modules or the lack of them come up in all aspects of commutative algebra. However, in the study of nonnoetherian rings it is much easier to find a ring having a finite number of prime ideals. The editors have included papers by Boynton and SatherWagstaff and by Watkins that discuss the relationship of rings with finite Krull dimension and their finite extensions. Finiteness properties in commutative group rings are discussed in Glaz and Schwarz's paper. And Olberding's selection presents us with constructions that produce rings whose integral closure in their field of fractions is not finitely
Commutative Algebra  Closure  Decomposition  Factorization
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This book studies the foundations of quantum theory through its relationship to classical physics. This idea goes back to the Copenhagen Interpretation (in the original version due to Bohr and Heisenberg), which the author relates to the mathematical formalism of operator algebras originally created by von Neumann. The book therefore includes comprehensive appendices on functional analysis and C*algebras, as well as a briefer one on logic, category theory, and topos theory. Matters of foundational as well as mathematical interest that are covered in detail include symmetry (and its "spontaneous" breaking), the measurement problem, the KochenSpecker, Free Will, and Bell Theorems, the KadisonSinger conjecture, quantization, indistinguishable particles, the quantum theory of large systems, and quantum logic, the latter in connection with the topos approach to quantum theory.
Quantum physics  Mathematical physics  Matrix theory  Algebra
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Norway is a leading country in the world when it comes to not prioritizing algebra in school mathematics, the mathematical language many students will need for further education and professions. Analysis of data from TIMSS, TIMSS Advanced and other international comparative studies have concluded that this result is consistent across 20 years of research. Norwegian students perform well in domains such as statistics/data, and alarmingly low in algebra. This conclusion is based on data from all levels in school: Primary school, lower secondary school, upper secondary school, and teacher education. The international studies indicate that East Asian, Eastern European and Latin European countries prioritize algebra more strongly than Norway or other Nordic and Englishspeaking countries. Several of the countries performing well in algebra, also seem to have some sort of balance between teaching algebra and teaching statistics, while there is a huge difference between the student scores on these domains in Norway. It almost seems as though “students learn statistics instead of learning algebra”. The importance of taking into account the type of mathematical content which is tested in different studies before drawing conclusions, is also addressed in the book. For instance, PISA mathematics is compared to TIMSS grade 8 mathematics when it comes to mathematical theory involvement. On the other hand, it is emphasized that it is necessary to have information from different studies to make valid conclusions concerning mathematics in schools. The importance of cooperation between countries all over the world in developing school mathematics, with countries learning from each other, is also discussed. The book is a result of cooperation between researchers at the Department of teacher education and school research (ILS) and the Department of mathematics, both at the University of Oslo, and school teachers in an upper secondary school. The results are discussed from a teacher education perspective, a school teacher perspective and from the perspective of the national curriculum for schools in Norway. Some of the chapters in the book are devoted to going through all the test items in TIMSS Advanced 2015 which can be published, and it is described how these items can be used both by teachers in school and in teacher education at universities.
timms  mathematics  algebra  undervisning  skole  matematikk  education
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Polynomials are well known for their ability to improve their properties and for their applicability in the interdisciplinary fields of engineering and science. Many problems arising in engineering and physics are mathematically constructed by differential equations. Most of these problems can only be solved using special polynomials. Special polynomials and orthonormal polynomials provide a new way to analyze solutions of various equations often encountered in engineering and physical problems. In particular, special polynomials play a fundamental and important role in mathematics and applied mathematics. Until now, research on polynomials has been done in mathematics and applied mathematics only. This book is based on recent results in all areas related to polynomials. Divided into sections on theory and application, this book provides an overview of the current research in the field of polynomials. Topics include cyclotomic and Littlewood polynomials; Descartes' rule of signs; obtaining explicit formulas and identities for polynomials defined by generating functions; polynomials with symmetric zeros; numerical investigation on the structure of the zeros of the qtangent polynomials; investigation and synthesis of robust polynomials in uncertainty on the basis of the root locus theory; pricing basket options by polynomial approximations; and orthogonal expansion in time domain method for solving Maxwell's equations using parallelinginorder scheme.
Physical Sciences, Engineering and Technology  Mathematics  Algebra
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Despite the importance of mathematics in our educational systems little is known about how abstract mathematical thinking emerges. Under the uniting thread of mathematical development, we hope to connect researchers from various backgrounds to provide an integrated view of abstract mathematical cognition. Much progress has been made in the last 20 years on how numeracy is acquired. Experimental psychology has brought to light the fact that numerical cognition stems from spatial cognition. The findings from neuroimaging and single cell recording experiments converge to show that numerical representations take place in the intraparietal sulcus. Further research has demonstrated that supplementary neural networks might be recruited to carry out subtasks; for example, the retrieval of arithmetic facts is done by the angular gyrus. Now that the neural networks in charge of basic mathematical cognition are identified, we can move onto the stage where we seek to understand how these basics skills are used to support the acquisition and use of abstract mathematical concepts.
Mathematical Cognition  abstract  algebra  Arithmetic  Expertise  development  Neuroimaging  gifted  numerosity
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Since the end of the 19th century when the prominent Norwegian mathematician Sophus Lie created the theory of Lie algebras and Lie groups and developed the method of their applications for solving differential equations, his theory and method have continuously been the research focus of many wellknown mathematicians and physicists. This book is devoted to recent development in Lie theory and its applications for solving physically and biologically motivated equations and models. The book contains the articles published in two Special Issue of the journal Symmetry, which are devoted to analysis and classification of Lie algebras, which are invariance algebras of realword models; Lie and conditional symmetry classification problems of nonlinear PDEs; the application of symmetrybased methods for finding new exact solutions of nonlinear PDEs (especially reactiondiffusion equations) arising in applications; the application of the Lie method for solving nonlinear initial and boundaryvalue problems (especially those for modelling processes with diffusion, heat transfer, and chemotaxis).
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The YangBaxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C.N. Yang and in the work of R.J. Baxter in the field of Statistical Mechanics. At the 1990 International Mathematics Congress, Vladimir Drinfeld, Vaughan F. R. Jones, and Edward Witten were awarded Fields Medals for their work related to the YangBaxter equation. It turned out that this equation is one of the basic equations in mathematical physics; more precisely, it is used for introducing the theory of quantum groups. It also plays a crucial role in: knot theory, braided categories, the analysis of integrable systems, noncommutative descent theory, quantum computing, noncommutative geometry, etc. Many scientists have used the axioms of various algebraic structures (quasitriangular Hopf algebras, YetterDrinfeld categories, quandles, group actions, Lie (super)algebras, brace structures, (co)algebra structures, Jordan triples, Boolean algebras, relations on sets, etc.) or computer calculations (and Grobner bases) in order to produce solutions for the YangBaxter equation. However, the full classification of its solutions remains an open problem. At present, the study of solutions of the YangBaxter equation attracts the attention of a broad circle of scientists. The current volume highlights various aspects of the YangBaxter equation, related algebraic structures, and applications.
Quantum Group  YangBaxter equation  Hopf algebra  Rmatrix  Lie algebra  braided category  duality  sixvertex model  startriangle relation  quantum integrability  braid group  quasitriangular structure  quantum projective space  bundle
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