1,304 results
Search Results
2. A note of Shimura's paper ?discontinuous groups and abelian varieties?
- Author
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David Mumford
- Subjects
Shimura variety ,Discrete mathematics ,Pure mathematics ,Abelian variety of CM-type ,General Mathematics ,Schottky problem ,Elementary abelian group ,Abelian category ,Hilbert's twelfth problem ,Abelian group ,Mathematics ,Arithmetic of abelian varieties - Published
- 1969
3. Remarks on blowing-up divisorial ideals
- Author
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Lorenzo Robbiano
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Algebraic Geometry ,Number theory ,Mathematics::Commutative Algebra ,General Mathematics ,Prime ideal ,Short paper ,Local ring ,Topological group ,Algebraic geometry ,Blowing up ,Mathematics - Abstract
In this short paper I describe special situations where a local ring is normally flat along a divisorial prime ideal.
- Published
- 1979
4. Redheffer type bounds for Bessel and modified Bessel functions of the first kind
- Author
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Árpád Baricz and Khaled Mehrez
- Subjects
Discrete mathematics ,Pure mathematics ,Hankel transform ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dirichlet eta function ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Bessel polynomials ,Struve function ,symbols ,Discrete Mathematics and Combinatorics ,Bessel's inequality ,0101 mathematics ,Bessel function ,Mathematics - Abstract
In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.
- Published
- 2018
5. The lattices of invariant subspaces of a class of operators on the Hardy space
- Author
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Zeljko Cuckovic and Bhupendra Paudyal
- Subjects
Discrete mathematics ,Pure mathematics ,Volterra operator ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,Hardy space ,Reflexive operator algebra ,01 natural sciences ,Linear subspace ,symbols.namesake ,Operator (computer programming) ,Lattice (order) ,FOS: Mathematics ,symbols ,Complex Variables (math.CV) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**
- Published
- 2018
6. Some weak specification properties and strongly mixing
- Author
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Jiandong Yin, Tao Wang, and Qi Yan
- Subjects
010101 applied mathematics ,Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Equivalence (formal languages) ,01 natural sciences ,Mathematics - Abstract
In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.
- Published
- 2017
7. L p estimates of rough maximal functions along surfaces with applications
- Author
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Abdulla M. Jarrah and Ahmad Al-Salman
- Subjects
Discrete mathematics ,Class (set theory) ,Pure mathematics ,General theorem ,Applied Mathematics ,General Mathematics ,Block (permutation group theory) ,Maximal function ,Singular integral ,Space (mathematics) ,Singular integral operators ,Mathematics - Abstract
In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn−1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.
- Published
- 2016
8. Étale extensions with finitely many subextensions
- Author
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Martine Picavet-L'Hermitte and Gabriel Picavet
- Subjects
Discrete mathematics ,Pure mathematics ,Ring (mathematics) ,Canonical decomposition ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Diagonal ,Support of a module ,Artinian ring ,010103 numerical & computational mathematics ,Extension (predicate logic) ,Type (model theory) ,Characterization (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
We study etale extensions of rings that have FIP., Comment: The paper entitled FIP and FCP products of ring morphisms (arXiv: 1312.1250 [math.AC]) is now split into three papers. The present paper contains the last section of the original paper and many other results on etale FIP extensions
- Published
- 2016
9. Uniqueness of meromorphic functions whose nonlinear differential polynomials share a polynomial
- Author
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Pulak Sahoo and Himadri Karmakar
- Subjects
Discrete mathematics ,Pure mathematics ,Polynomial ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Polynomial matrix ,Nonlinear system ,Uniqueness ,0101 mathematics ,Differential (mathematics) ,Mathematics ,Meromorphic function - Abstract
In this paper, we study some uniqueness problems of meromorphic functions when certain nonlinear differential polynomials generated by them share a nonconstant polynomial. The results of the paper improve the concerning results due to Xu et al. (Mat Vesnik 64:1–16, 2012).
- Published
- 2016
10. Results on uniqueness of entire functions whose certain difference polynomials share a small function
- Author
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Pulak Sahoo and Himadri Karmakar
- Subjects
Discrete mathematics ,Pure mathematics ,Difference polynomials ,General Mathematics ,Entire function ,Function (mathematics) ,Uniqueness ,Type (model theory) ,Mathematics - Abstract
In the paper, using the concept of weakly weighted sharing and relaxed weighted sharing, we investigate the uniqueness problems of certain type of difference polynomials that share a small function. The results of the paper improve and extend some recent results due to C. Meng [Math. Bohem., 139(2014), 89–97] and the present first author [Commun. Math. Stat., 3(2015), 227–238].
- Published
- 2015
11. The Fundamental Theorem of Algebra: A Visual Approach
- Author
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Daniel J. Velleman
- Subjects
Discrete mathematics ,Pure mathematics ,Fundamental theorem of algebra ,Color scheme ,History and Philosophy of Science ,Fundamental theorem ,General Mathematics ,Fundamental theorem of linear algebra ,Color wheel ,Mathematical proof ,Complex plane ,Complex number ,Mathematics - Abstract
Some version of the statement of the Fundamental Theorem of Algebra first appeared early in the 17th century in the writings of several mathematicians, including Peter Roth, Albert Girard, and Rene Descartes. The first proof of the Fundamental Theorem was published by Jean Le Rond d’Alembert in 1746 [2], but his proof was not very rigorous. Carl Friedrich Gauss is often credited with producing the first correct proof in his doctoral dissertation of 1799 [15], although this proof also had gaps. (For a comparison of these two proofs, see [26, pp. 195–200].) Today there are many known proofs of the Fundamental Theorem of Algebra, including proofs using methods of algebra, analysis, and topology. (The references include many papers and books containing proofs of the Fundamental Theorem; [14] alone contains 11 proofs.) Our focus in this paper will be on the use of pictures to see why the theorem is true. Of course, if we want to use pictures to display the behavior of polynomials defined on the complex numbers, we are immediately faced with a difficulty: the complex numbers are two-dimensional, so it appears that a graph of a complex-valued function on the complex numbers will require four dimensions. Our solution to this problem will be to use color to represent some dimensions. We begin by assigning a color to every number in the complex plane. Figure 1 is a picture of the complex plane in which every point has been assigned a different color. The origin is colored black. Traveling counterclockwise around a circle centered at the origin, we go through the colors of a standard color wheel: red, yellow, green, cyan, blue, magenta, and back to red. Points near the origin have dark colors, with the color assigned to a complex number z approaching black as z approaches 0. Points far from the origin are light, with the color of z approaching white as |z| approaches infinity. Every complex number has a different color in this picture, so a complex number can be uniquely specified by giving its color. We can now use this color scheme to draw a picture of a function f : C → C as follows: we simply color each point z in the complex plane with the color corresponding to the value of f(z). From such a picture, we can read off the value of f(z), for any complex number z, by determining the color of the point z in the picture, and then consulting Figure 1 to see what complex number is represented by that color.
- Published
- 2015
12. On the Convolution of a Finite Number of Analytic Functions Involving a Generalized Srivastava–Attiya Operator
- Author
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Janusz Sokół, Ravinder Krishna Raina, and Poonam Sharma
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Convolution power ,01 natural sciences ,Convexity ,Circular convolution ,Riemann zeta function ,Convolution ,symbols.namesake ,Operator (computer programming) ,symbols ,0101 mathematics ,Finite set ,Mathematics ,Analytic function - Abstract
The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.
- Published
- 2015
13. The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups
- Author
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Marcin Bownik and Kenneth A. Ross
- Subjects
Discrete mathematics ,Pointwise ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Dimension function ,Second-countable space ,Linear subspace ,Euclidean geometry ,Locally compact space ,Abelian group ,Invariant (mathematics) ,Analysis ,Mathematics - Abstract
Let \(\Gamma \) be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group \(G\). In this paper we study translation-invariant (TI) subspaces of \(L^2(G)\) by elements of \(\Gamma \). We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that \(\Gamma \) be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of \(G=\mathbb {R}^n\). We also extend the notion of the spectral function in \(\mathbb {R}^n\) to the LCA setting. It is shown that spectral functions, initially defined in terms of \(\Gamma \), do not depend on \(\Gamma \). Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of \(G\) with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit.
- Published
- 2015
14. Graded Annihilators and Uniformly F-Compatible Ideals
- Author
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Rodney Y. Sharp
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,13A35, 16S36 ,Semiprime ring ,Excellent ring ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Associated prime ,Cohen–Macaulay ring ,Module ,FOS: Mathematics ,Krull dimension ,Ideal (ring theory) ,Tight closure ,Mathematics - Abstract
Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$ that is $F$-pure. This paper is concerned with comparison of three finite sets of radical ideals of $R$, one of which is only defined in the case when $R$ is $F$-finite (that is, is finitely generated when viewed as a module over itself via the Frobenius homomorphism). Two of the afore-mentioned three sets have links to tight closure, via test ideals. Among the aims of the paper are a proof that two of the sets are equal, and a proposal for a generalization of I. M. Aberbach's and F. Enescu's splitting prime., 17 pages. This paper has been accepted for publication in Acta Mathematica Vietnamica
- Published
- 2015
15. The split common null point problem in Banach spaces
- Author
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Wataru Takahashi
- Subjects
Discrete mathematics ,Pure mathematics ,Fréchet space ,General Mathematics ,Topological tensor product ,Eberlein–Šmulian theorem ,Banach space ,Interpolation space ,Birnbaum–Orlicz space ,Banach manifold ,Lp space ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
In this paper, we consider the split common null point problem in Banach spaces. Then using the metric resolvents of maximal monotone operators and the metric projections, we prove a strong convergence theorem for finding a solution of the split common null point problem in Banach spaces. The result of this paper seems to be the first one to study it outside Hilbert spaces.
- Published
- 2015
16. Appendix: On some Gelfand pairs and commutative association schemes
- Author
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Eiichi Bannai and Hajime Tanaka
- Subjects
Discrete mathematics ,Pure mathematics ,medicine.anatomical_structure ,Association scheme ,General Mathematics ,medicine ,Commutative property ,Gelfand pair ,Appendix ,Mathematics - Abstract
This paper is Appendix of the paper of T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli [8].
- Published
- 2014
17. Krein-Space Representations of Arithmetic Functions Determined by Primes
- Author
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Palle E. T. Jorgensen and Ilwoo Cho
- Subjects
Discrete mathematics ,Pure mathematics ,Operator algebra ,General Mathematics ,Prime number ,Arithmetic function ,Primes in arithmetic progression ,Algebra over a field ,Space (mathematics) ,Free probability ,Mathematics - Abstract
In this paper, we study representations of the algebra \(\mathcal {A}\) generated by all arithmetic functions, determined by fixed primes (or prime numbers). The main purposes of this paper are (i) to establish nice representational models of \(\mathcal {A}\) under primes, (ii) to study fundamental properties of such representations, (iii) to investigate how \( \mathcal {A}\) is acting as operators in representations, and (iv) to consider new free probability models on Krein-space operator algebras.
- Published
- 2014
18. Torsion Abelian RAI-Groups
- Author
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Pham Thi Thu Thuy
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Torsion subgroup ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,Elementary abelian group ,Rank of an abelian group ,Divisible group ,Non-abelian group ,Free abelian group ,Torsion (algebra) ,Condensed Matter::Strongly Correlated Electrons ,Abelian group ,Mathematics - Abstract
This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.
- Published
- 2014
19. Some Integral Type Fixed-Point Theorems and an Application to Systems of Functional Equations
- Author
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Sunny Chauhan, Hassen Aydi, Wasfi Shatanawi, Calogero Vetro, Chauhan, S, Aydi, H, Shatanawi, W, and Vetro, C
- Subjects
Discrete mathematics ,Subsequential limit ,Subcompatible mapping ,Pure mathematics ,Compatible mapping ,General Mathematics ,Reciprocal continuity ,Fixed-point theorem ,Fixed point ,Metric space ,Settore MAT/05 - Analisi Matematica ,Subsequential continuity ,Coincidence point ,Common fixed point theorem ,Reciprocal ,Mathematics - Abstract
In this paper, we prove a new common fixed point theorem for four self mappings by using the notions of compatibility and subsequential continuity (alternate subcompatibility and reciprocal continuity) in metric spaces satisfying a general contractive condition of integral type. We give some examples to support the useability of our main result. Also, we obtain some fixed point theorems of Gregus type for four mappings satisfying a strict general contractive condition of integral type in metric spaces. We conclude the paper with an application of our main result to solvability of systems of functional equations.
- Published
- 2013
20. On isotopies, parastrophies, and orthogonality of quasigroups
- Author
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K. K. Shchukin
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Orthogonality ,Applied Mathematics ,General Mathematics ,Order (group theory) ,Quasigroup ,Mathematics - Abstract
In V. D. Belousov’s papers, some properties of parastrophies were studied and some relations between parastrophies of a given quasigroup were obtained. Also some invariants of a parastrophy were found. This article continues our paper with V. V. Gushan, in which minimal sets of parastrophy systems for quasigroups of order 6 were obtained and some questions about orthogonality of parastrophies of a given quasigroup were studied as well.
- Published
- 2013
21. On λ-compact operators
- Author
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Antara Bhar and Manjul Gupta
- Subjects
Discrete mathematics ,Unbounded operator ,Pure mathematics ,Nuclear operator ,Approximation property ,Applied Mathematics ,General Mathematics ,Finite-rank operator ,Spectral theorem ,Operator theory ,Operator norm ,Compact operator on Hilbert space ,Mathematics - Abstract
Using the duality theory of sequence spaces, we study in this paper λ-compact operators defined on Banach spaces, corresponding to a sequence space λ. We show that these operators form a quasi-normed operator ideal under suitable restrictions on λ. We also study the relationships of these operators with λ-summing, λ-nuclear and quasi-λ-nuclear operators. The results of this paper generalize the earlier results proved by Sinha and Karn; and also Delgado, Pineiro and Serrano.
- Published
- 2013
22. On the conjugacy of nilpotent injectors in finite groups
- Author
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Anni Neumann
- Subjects
Discrete mathematics ,Mathematics::Group Theory ,Nilpotent ,Finite group ,Pure mathematics ,Conjugacy class ,General Mathematics ,Physics::Accelerator Physics ,Nilpotent group ,Type (model theory) ,Mathematics - Abstract
If a finite group G is \({\mathcal{N}}\)-constrained, then the nilpotent injectors of G form a single conjugacy class of subgroups. In this paper we shall generalize this result. This paper is part of a larger program investigating a special type of nilpotent injectors in arbitrary finite groups.
- Published
- 2013
23. On locally analytic Beilinson–Bernstein localization and the canonical dimension
- Author
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Tobias Schmidt
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Localization theorem ,Dimension (graph theory) ,Congruence (manifolds) ,Field (mathematics) ,Reductive group ,Type (model theory) ,Prime (order theory) ,Mathematics - Abstract
Let $$\mathbf{G}$$ be a connected split reductive group over a $$p$$ -adic field. In the first part of the paper we prove, under certain assumptions on $$\mathbf{G}$$ and the prime $$p$$ , a localization theorem of Beilinson–Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of $$\mathbf{G}$$ . In doing so we take up and extend some recent methods and results of Ardakov–Wadsley on completed universal enveloping algebras (Ardakov and Wadsley, Ann. Math., 2013) to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith’s theorem on the canonical dimension.
- Published
- 2013
24. Weak Characterizations of Stochastic Integrability and Dudley’s Theorem in Infinite Dimensions
- Author
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Mark Veraar and Martin Ondreját
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Representation theorem ,General Mathematics ,Representation theory ,Dudley's theorem ,Doob decomposition theorem ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Random variable ,Brownian motion ,Counterexample ,Mathematics - Abstract
In this paper we consider stochastic integration with respect to cylindrical Brownian motion in infinite-dimensional spaces. We study weak characterizations of stochastic integrability and present a natural continuation of results of van Neerven, Weis and the second named author. The limitation of weak characterizations will be demonstrated with a nontrivial counterexample. The second subject treated in the paper addresses representation theory for random variables in terms of stochastic integrals. In particular, we provide an infinite-dimensional version of Dudley’s representation theorem for random variables and an extension of Doob’s representation for martingales.
- Published
- 2013
25. On Oka’s extra-zero problem and examples
- Author
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Junjiro Noguchi, Makoto Abe, and Sachiko Hamano
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Dimension (graph theory) ,Cousin ,Zero (complex analysis) ,Disjoint sets ,Special case ,Mathematics - Abstract
After the solution of Cousin II problem by Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and some new questions.
- Published
- 2012
26. Generalized Derivations of Hom–Lie Triple Systems
- Author
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Jia Zhou, Yao Ma, and Liangyun Chen
- Subjects
Discrete mathematics ,Pure mathematics ,Triple system ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Algebra over a field ,01 natural sciences ,Mathematics - Abstract
In this paper, we give some properties of the generalized derivation algebra \(\mathrm{GDer}(T)\) of a Hom–Lie triple systems T. In particular, we prove that \(\mathrm{GDer}(T) = \mathrm{QDer}(T) + \mathrm{QC}(T)\), the sum of the quasiderivation algebra and the quasicentroid. We also prove that \(\mathrm{QDer}(T)\) can be embedded as derivations in a larger Hom–Lie triple system. General results on centroids of Hom–Lie triple systems are also developed in this paper.
- Published
- 2016
27. Univalence criterion and quasiconformal extension of holomorphic mappings
- Author
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Gabriela Kohr and Hidetaka Hamada
- Subjects
Unit sphere ,Discrete mathematics ,Pure mathematics ,Number theory ,Mathematics::Complex Variables ,General Mathematics ,Euclidean geometry ,Holomorphic function ,Extension (predicate logic) ,Algebraic geometry ,Mathematics ,Loewner differential equation - Abstract
In this paper we are concerned with solutions, in particular with the univalent solutions, of the Loewner differential equation associated with non-normalized subordination chains on the Euclidean unit ball B in \({\mathbb{C}^n}\). We also give applications to univalence conditions and quasiconformal extensions to \({\mathbb{C}^n}\) of holomorphic mappings on B. Finally we consider the asymptotical case of these results. The results in this paper are complete generalizations to higher dimensions of well known results due to Becker. They improve and extend previous sufficient conditions for univalence and quasiconformal extension to \({\mathbb{C}^n}\) of holomorphic mappings on B.
- Published
- 2012
28. CLT for linear random fields with martingale increments
- Author
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Vygantas Paulauskas and Povilas Banys
- Subjects
Discrete mathematics ,Continuation ,Multivariate statistics ,Pure mathematics ,Random field ,Number theory ,General Mathematics ,Ordinary differential equation ,Local martingale ,Martingale (probability theory) ,Mathematics ,Central limit theorem - Abstract
In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, which can be regarded as a continuation of the above-mentioned paper, CLT for sums of linear random field are proved in the case where innovations form martingale differences on the plane (that can be defined in several ways). In both papers, the so-called Beveridge–Nelson decomposition is used.
- Published
- 2012
29. Projections on weak*-closed subspace of dual Banach algebras
- Author
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Ali Ghaffari
- Subjects
Discrete mathematics ,Cancellative semigroup ,Pure mathematics ,Semigroup ,Applied Mathematics ,General Mathematics ,Invariant subspace ,Nest algebra ,Locally compact space ,Locally compact group ,Invariant subspace problem ,Subspace topology ,Mathematics - Abstract
Our first purpose in this paper is to provide necessary conditions for a weak*-closed translation invariant subspace in the semigroup algebra of a locally compact topological foundation semigroup to be completely complemented. We give conditions when a weak*-closed left translation invariant subspace in Ma(S)* of a compact cancellative foundation semigroup S is the range of a weak*-weak* continuous projection on Ma(S)* commuting with translations. Let G be a locally compact group and A be a Banach G-module. Our second purpose in this paper is to study some projections on A* and B(A*) which commutes with translations and convolution.
- Published
- 2011
30. The distribution of normalized zero-sets of random meromorphic functions
- Author
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WeiHong Yao
- Subjects
Normalization (statistics) ,Discrete mathematics ,Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,Holomorphic function ,Dirac delta function ,Hermitian matrix ,Nevanlinna theory ,symbols.namesake ,symbols ,Special case ,Mathematics::Symplectic Geometry ,Mathematics ,Meromorphic function - Abstract
This paper is concerned with the distribution of normalized zero-sets of random meromorphic functions. The normalization of the zero-set plays the same role as the counting function for a meromorphic function in Nevanlinna theory. The results generalize the theory of Shiffman and Zelditch on the distribution of the zeroes of random holomorphic sections of powers of positive Hermitian holomorphic line bundles. As in a very special case, our paper resembles a form of First Main Theorem in classical Nevanlinna Theory.
- Published
- 2011
31. Affine structures on a ringed space and schemes
- Author
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Feng-Wen An
- Subjects
Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,14A15, 14A25, 57R55 ,Affine plane ,Affine coordinate system ,Affine geometry ,Mathematics - Algebraic Geometry ,Complex space ,Ringed space ,Affine hull ,Affine group ,FOS: Mathematics ,Affine space ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we will first introduce the notion of affine structures on a ringed space and then obtain several properties. Affine structures on a ringed space, arising mainly from complex analytical spaces of algebraic schemes over number fields, behave like differential structures on a smooth manifold. As one does for differential manifolds, we will use pseudogroups of affine transformations to define affine atlases on a ringed space. An atlas on a space is said to be an affine structure if it is maximal. An affine structure is admissible if there is a sheaf on the underlying space such that they are coincide on all affine charts, which are in deed affine open sets of a scheme. In a rigour manner, a scheme is defined to be a ringed space with a specified affine structure if the affine structures are in action in some special cases such as analytical spaces of algebraic schemes. Particularly, by the whole of affine structures on a space, we will obtain respectively necessary and sufficient conditions that two spaces are homeomorphic and that two schemes are isomorphic, which are the two main theorems of the paper. It follows that the whole of affine structures on a space and a scheme, as local data, encode and reflect the global properties of the space and the scheme, respectively., Final version. 22 pages. to appear in Chinese Ann of Math, Series B
- Published
- 2010
32. Torsion-free rings
- Author
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E. I. Kompantseva
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Torsion subgroup ,Noncommutative ring ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Elementary abelian group ,Rank of an abelian group ,Divisible group ,Non-abelian group ,Abelian group ,Mathematics ,Group ring - Abstract
The aim of this paper is to study the correlation between the properties of rings and the structure of their additive groups. In this paper, all multiplications on reduced algebraically compact groups and on divisible torsion-free Abelian groups are described. Absolute Jacobson radicals and absolute nil-radicals in some classes of Abelian groups are found. Using these results, we will give a description of semisimple groups and also of groups on which every ring is a nilpotent ring (a nil-ring, a radical ring) in some concerned classes of Abelian groups.
- Published
- 2010
33. Isomorphism Classes of Certain Artinian Gorenstein Algebras
- Author
-
Giuseppe Valla and Juan Elias
- Subjects
Discrete mathematics ,Hilbert series and Hilbert polynomial ,Pure mathematics ,Mathematics::Commutative Algebra ,Degree (graph theory) ,13H10 ,General Mathematics ,Mathematics::Rings and Algebras ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Square (algebra) ,Mathematics - Algebraic Geometry ,symbols.namesake ,FOS: Mathematics ,symbols ,13H15 ,Maximal ideal ,Isomorphism ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables
- Published
- 2009
34. Homomorphisms in quasi-Banach algebras associated with a Pexiderized Cauchy-Jensen functional equation
- Author
-
Abbas Najati
- Subjects
Discrete mathematics ,Linear map ,Mathematics::Functional Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Functional equation ,Banach space ,Cauchy distribution ,Homomorphism ,Stability (probability) ,Stability theorem ,Mathematics - Abstract
In this paper, we prove the generalized Hyers-Ulam stability of homomorphisms in quasi-Banach algebras associated with the following Pexiderized Jensen functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemOzay2aaeWaaeaadaWcaaqaaiabdIha4fXafv3y % SLgzGmvETj2BSbacgaGae83kaSIaemyEaKhabaGae8Nmaidaaiab-T % caRiabdQha6bGaayjkaiaawMcaaiab-jHiTiabdEgaNnaabmaabaWa % aSaaaeaacqWG4baEcqWFsislcqWG5bqEaeaacqWFYaGmaaGae83kaS % IaemOEaOhacaGLOaGaayzkaaGae8xpa0JaemiAaGMae8hkaGIaemyE % aKNae8xkaKIae8Nla4caaa!5ABC! $$ f\left( {\frac{{x + y}} {2} + z} \right) - g\left( {\frac{{x - y}} {2} + z} \right) = h(y). $$ This is applied to investigating homomorphisms between quasi-Banach algebras. The concept of the generalized Hyers-Ulam stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297–300 (1978).
- Published
- 2009
35. On the Multiplicity of Zeroes of Polynomials with Quaternionic Coefficients
- Author
-
Daniele C. Struppa and Graziano Gentili
- Subjects
Discrete mathematics ,Pure mathematics ,symbols.namesake ,Polynomial ,General Mathematics ,Weierstrass factorization theorem ,symbols ,Division ring ,Multiplicity (mathematics) ,Degree of a polynomial ,Mathematical proof ,Mathematics - Abstract
Regular polynomials with quaternionic coefficients admit only isolated zeroes and spherical zeroes. In this paper we prove a factorization theorem for such polynomials. Specifically, we show that every regular polynomial can be written as a product of degree one binomials and special second degree polynomials with real coefficients. The degree one binomials are determined (but not uniquely) by the knowledge of the isolated zeroes of the original polynomial, while the second degree factors are uniquely determined by the spherical zeroes. We also show that the number of zeroes of a polynomial, counted with their multiplicity as defined in this paper, equals the degree of the polynomial. While some of these results are known in the general setting of an arbitrary division ring, our proofs are based on the theory of regular functions of a quaternionic variable, and as such they are elementary in nature and offer explicit constructions in the quaternionic setting.
- Published
- 2008
36. The Kurosh problem, height theorem, nilpotency of the radical, and algebraicity identity
- Author
-
A. Ya. Belov
- Subjects
Statistics and Probability ,Discrete mathematics ,Noetherian ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Commutative ring ,Identity (mathematics) ,Kurosh problem ,Bounded function ,Finitely-generated abelian group ,Algebraic number ,Associative property ,Mathematics - Abstract
This paper is devoted to relations between the Kurosh problem and the Shirshov height theorem. The central point and main technical tool is the identity of algebraicity. The main result of this paper is the following. Let A be a finitely generated PI-algebra and Y be a finite subset of A. For any Noetherian associative and commutative ring {ie125-01}, let any factor of R ⊗ A such that all projections of elements from Y are algebraic over π(R) be a Noetherian R-module. Then A has bounded essential height over Y. If, furthermore, Y generates A as an algebra, then A has bounded height over Y in the Shirshov sense.
- Published
- 2008
37. Notes on sectionally complemented lattices. IV How far does the Atom Lemma go?
- Author
-
M. Roddy and George Grätzer
- Subjects
Discrete mathematics ,Pure mathematics ,High Energy Physics::Lattice ,General Mathematics ,Lattice (order) ,Atom ,Mathematics - Abstract
There are two results in the literature that prove that the ideal lattice of a finite, sectionally complemented, chopped lattice is again sectionally complemented. The first is in the 1962 paper of G. Gratzer and E. T. Schmidt, where the ideal lattice is viewed as a closure space to prove that it is sectionally complemented; we call the sectional complement constructed then the 1960 sectional complement. The second is the Atom Lemma from a 1999 paper of the same authors that states that if a finite, sectionally complemented, chopped lattice is made up of two lattices overlapping in an atom and a zero, then the ideal lattice is sectionally complemented.
- Published
- 2007
38. Explicit relation between the solutions of the heat and the Hermite heat equation
- Author
-
Bang-He Li
- Subjects
Discrete mathematics ,Pure mathematics ,Hermite polynomials ,Uniqueness theorem for Poisson's equation ,Applied Mathematics ,General Mathematics ,Bounded function ,Mathematics::Classical Analysis and ODEs ,General Physics and Astronomy ,Heat equation ,Constant (mathematics) ,Mathematics - Abstract
There are lots of results on the solutions of the heat equation $$\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u,$$ but much less on those of the Hermite heat equation $$\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U$$ due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).
- Published
- 2007
39. Invariants of the stable equivalence of symmetric special biserial algebras
- Author
-
Mikhail Antipov
- Subjects
Statistics and Probability ,Discrete mathematics ,Diagrammatic reasoning ,Pure mathematics ,Triangulated category ,Mathematics::Category Theory ,Applied Mathematics ,General Mathematics ,Cartan matrix ,Bibliography ,Equivalence (formal languages) ,Invariant (mathematics) ,Mathematics - Abstract
The present paper is one in a series of papers devoted to the classification of some classes of tame algebras up to stable category equivalence. In this paper, we study symmetric algebras (their stable categories have a structure of triangulated categories) and the simplest class of tame algebras-the class of special biserial algebras (SB-algebras). In the paper, we give a relevant version of the “diagrammatic method” and study the structure of the triangulated category “in a neighborhood” of the periodic part (with respect to Ω) of the stable category. Thus we prove the invariance of the collection of lengths of G-cycles under equivalence of stable categories (see Theorem 2.12). Then we use the invariance stated above, together with some properties of the Cartan matrix of a symmetric SB-algebra, to prove that the number of A-cycles (but not their lengths!) is also an invariant of stable equivalence. Bibliography: 8 titles.
- Published
- 2007
40. Projective resolutions and Yoneda algebras for algebras of dihedral type: The family $$D(3\mathcal{Q})$$
- Author
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N. V. Kosmatov and A. I. Generalov
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Computation ,Dihedral angle ,Type (model theory) ,Notation ,Diagrammatic reasoning ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Projective test ,Simple module ,Mathematics - Abstract
This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family \(D(3\mathcal{Q})\)) in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in the C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson.
- Published
- 2007
41. Group-cograded Multiplier Hopf ${\left( { * {\text{ - }}} \right)}$ algebras
- Author
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Lydia Delvaux, A. T. Abd El-hafez, and A. Van Daele
- Subjects
Discrete mathematics ,Pure mathematics ,Quantum group ,Direct sum ,General Mathematics ,Mathematics::Rings and Algebras ,Quantum algebra ,Representation theory of Hopf algebras ,Quasitriangular Hopf algebra ,Hopf algebra ,Multiplier (Fourier analysis) ,Mathematics::Quantum Algebra ,Hopf lemma ,Mathematics - Abstract
Let G be a group and assume that (Ap)p∈G is a family of algebras with identity. We have a Hopf G-coalgebra (in the sense of Turaev) if, for each pair p,q ∈ G, there is given a unital homomorphism Δp,q : Apq → Ap ⊗ Aq satisfying certain properties. Consider now the direct sum A of these algebras. It is an algebra, without identity, except when G is a finite group, but the product is non-degenerate. The maps Δp,q can be used to define a coproduct Δ on A and the conditions imposed on these maps give that (A,Δ) is a multiplier Hopf algebra. It is G-cograded as explained in this paper. We study these so-called group-cograded multiplier Hopf algebras. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras.
- Published
- 2006
42. The Auslander–Reiten Quiver of a Poincaré Duality Space
- Author
-
Peter Jørgensen
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,Quiver ,Algebraic topology ,Topological space ,Space (mathematics) ,symbols.namesake ,Mathematics::Category Theory ,Differential graded algebra ,symbols ,Component (group theory) ,Mathematics::Representation Theory ,Poincaré duality ,Mathematics - Abstract
In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincare duality space, each component of the Auslander–Reiten quiver is isomorphic to \(\mathbb{Z}A_{\infty }\).
- Published
- 2006
43. On some local cohomology invariants of local rings
- Author
-
Gennady Lyubeznik
- Subjects
Discrete mathematics ,Noetherian ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Local ring ,Field (mathematics) ,Local cohomology ,Commutative Algebra (math.AC) ,13D45 (Primary) ,14B15 (Secondary) ,Mathematics - Commutative Algebra ,Mathematics - Algebraic Geometry ,Integer ,FOS: Mathematics ,Algebraic Geometry (math.AG) ,Commutative property ,Topology (chemistry) ,DIMA ,Mathematics - Abstract
Let A be a commutative Noetherian local ring containing a field of characteristic p>0. The integer invariants $\lambda_{i,j}(A)$ have been introduced in an old paper of ours. In this paper we completely describe $\lambda_{d,d}(A)$, where d=dimA, in terms of the topology of SpecA., Comment: 14 pages
- Published
- 2006
44. The functor A min on p-local spaces
- Author
-
Jie Wu and Paul Selick
- Subjects
Condensed Matter::Soft Condensed Matter ,Loop (topology) ,Discrete mathematics ,Connected space ,Pure mathematics ,Functor ,General Mathematics ,Suspension (topology) ,Mathematics - Abstract
In a previous paper, the authors gave the finest functorial decomposition of the loop suspension of a p-torsion suspension. The purpose of this paper is to generalize this theorem to the loop suspension of arbitrary p-local path connected spaces.
- Published
- 2006
45. Existence Results for Functional Differential Inclusions with Infinite Delay
- Author
-
Shi Huang Hong
- Subjects
Discrete mathematics ,Set (abstract data type) ,Pure mathematics ,Differential inclusion ,Applied Mathematics ,General Mathematics ,Phase space ,Banach space ,Fixed-point theorem ,C0-semigroup ,Axiom ,Mathematics - Abstract
The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.
- Published
- 2005
46. A new proof of the Gerritzen-Grauert theorem
- Author
-
Michael Temkin
- Subjects
Discrete mathematics ,Pure mathematics ,Compact space ,Analytic geometry ,Proofs of Fermat's little theorem ,General Mathematics ,Structure (category theory) ,Variety (universal algebra) ,Bruck–Ryser–Chowla theorem ,Ground field ,Mathematics ,Valuation (algebra) - Abstract
The Gerritzen-Grauert theorem ([GG], [BGR, 7.3.5/1]) is one of the most important foundational results of rigid analytic geometry. It describes so called locally closed immersions between affinoid varieties, and this description implies the fact that any affinoid subdomain of an affinoid variety is a finite union of rational domains. In its turn, the latter fact allowed one to extend Tate’s theorem (see [Tate], [BGR, 8.2.1/1]) on acyclicity of the Cech complex associated to a finite rational covering of an affinoid variety to finite covering by arbitrary affinoid domains. The same fact also plays an important role in foundations of non-Archimedean analytic geometry developed by V. Berkovich in [Ber1] and [Ber2]. Recall that building blocks of the latter are affinoid spaces associated to a class of affinoid algebras broader than that considered in rigid analytic geometry (the latter were called in [Ber1] strictly affinoid) and, besides, the valuation on the ground field is not assumed to be nontrivial. In the recent papers by A. Ducros [Duc, 2.4] and the author [Tem, 3.5], the above fact on the structure of affinoid domains was extended to arbitrary affinoid spaces, but its proof was based on the case of strictly affinoid ones (i.e., affinoid varieties). The original proof of the Gerritzen-Grauert theorem is not easy, and since then the only different proof was found by M. Raynaud in the framework of his approach to rigid analytic geometry (see [Ray], [BL]). Although that proof is more conceptual, it is based on a complicated algebraic technics. The purpose of this paper is to give a new proof of the Gerritzen-Grauert theorem which uses basic properties of affinoid algebras in a standard way. The only novelty is in using the whole spectrum M(A) of an affinoid algebra A, introduced in [Ber1], instead of the maximal spectrum Max(A), considered in rigid analytic geometry. The use of the whole spectrum allows one to apply additional but standard compactness arguments. In §§1-2, we work in the setting of rigid analytic geometry, i.e., the valuation on the ground field is assumed to be nontrivial and only the class of strictly affinoid algebras is considered. In §1, we recall basic definitions of an affinoid algebra, an affinoid domain, all notions necessary for the formulation of the Gerritzen-Grauert theorem, and formulate it (Theorem 1.1). The only new fact is Proposition 1.2 which establishes the simple fact that a morphism of affinoid varieties is a locally closed immersion if and only if it is a
- Published
- 2005
47. Operator K-Theory and Functor N0. Some Applications
- Author
-
A. A. Pavlov
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,Fiber functor ,Functor ,Brown's representability theorem ,Mathematics::Operator Algebras ,Direct image functor ,Applied Mathematics ,General Mathematics ,Quiver ,Cone (category theory) ,Mathematics::Algebraic Topology ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Natural transformation ,Exact functor ,Mathematics - Abstract
In this paper, we present a general introduction to the K-theory of C *-algebras and survey of our previous papers, where the functor N 0 from the category of von Neumann algebras to the category of Abelian groups was defined. We investigate the properties of this functor (in particular, its interrelation with the functor K 0) and point out some applications of the functor N 0 in noncommutative geometry. In addition, we recall facts of theory of C *-algebras, von Neumann algebras, and Hilbert C *-modules.
- Published
- 2004
48. Automorphisms of Green Orders and Their Derived Categories
- Author
-
Alexander Zimmermann
- Subjects
Discrete mathematics ,Pure mathematics ,Finite group ,Brauer tree ,Brauer's theorem on induced characters ,General Mathematics ,Braid group ,Homomorphism ,Group homomorphism ,Group representation ,Brauer group ,Mathematics - Abstract
In an earlier paper, Raphael Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.
- Published
- 2004
49. Infinitesimal operations on complexes of graphs
- Author
-
Karen Vogtmann and James Conant
- Subjects
Discrete mathematics ,Pure mathematics ,Functor ,Lie bialgebra ,General Mathematics ,010102 general mathematics ,Outer automorphism group ,Homology (mathematics) ,Automorphism ,Mathematics::Algebraic Topology ,Mathematics::Geometric Topology ,01 natural sciences ,Mapping class group ,Moduli space ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In two seminal papers Kontsevich used a construction called graph homology as a bridge between certain infinite dimensional Lie algebras and various topological objects, includ- ing moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we show that Kontsevich's graph complexes, which include graph complexes studied earlier by Culler and Vogtmann and by Penner, have a rich algebraic structure. We define a Lie bracket and cobracket on graph complexes, and in fact show that they are Batalin-Vilkovisky algebras, and therefore Gerstenhaber algebras. We also find natural subcomplexes on which the bracket and cobracket are compatible as a Lie bialgebra. Kontsevich's graph complex construction was generalized to the context of operads by Ginzburg and Kapranov, with later generalizations by Getzler-Kapranov and Markl. In (CoV), we show that Kontsevich's results in fact extend to general cyclic operads. For some operads, including the examples associated to moduli space and outer automorphism groups of free groups, the subcomplex on which we have a Lie bi-algebra structure is quasi-isomorphic to the entire con- nected graph complex. In the present paper we show that all of the new algebraic operations canonically vanish when the homology functor is applied, and we expect that the resulting con- straints will be useful in studying the homology of the mapping class group, finite type manifold invariants and the homology of Out(F n).
- Published
- 2003
50. On the structure of inner ideals of nest algebras
- Author
-
Lina Oliveira
- Subjects
Discrete mathematics ,Set (abstract data type) ,Class (set theory) ,Pure mathematics ,Rank (linear algebra) ,Direct sum ,General Mathematics ,Structure (category theory) ,Nest algebra ,Characterization (mathematics) ,Linear subspace ,Mathematics - Abstract
IfA is a nest algebra andA s=A ∩ A* , whereA* is the set of the adjoints of the operators lying inA, then the pair (A, A s) forms a partial Jordan *-triple. Important tools when investigating the structure of a partial Jordan *-triple are its tripotents. In particular, given an orthogonal family of tripotents of the partial Jordan *-triple (A, A s), the nest algebraA splits into a direct sum of subspaces known as the Peirce decomposition relative to that family. In this paper, the Peirce decomposition relative to an orthogonal family of minimal tripotents is used to investigate the structure of the inner ideals of (A, A s), whereA is a nest algebra associated with an atomic nest. A property enjoyed by inner ideals of the partial Jordan *-triple (A, A s) is presented as the main theorem. This result is then applied in the final part of the paper to provide examples of inner ideals. A characterization of the minimal tripotents as a certain class of rank one operators is also obtained as a means to deduce the principal theorem.
- Published
- 2003
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