570 results
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2. 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
3. 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
4. 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
5. Bounds for Calderón–Zygmund operators with matrix A2 weights
- Author
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Sandra Pott and Andrei Stoica
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Pure mathematics ,Representation theorem ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Mathematics::Classical Analysis and ODEs ,Haar ,01 natural sciences ,Operator (computer programming) ,0103 physical sciences ,Embedding ,010307 mathematical physics ,0101 mathematics ,Special case ,Martingale (probability theory) ,Singular integral operators ,Mathematics - Abstract
It is well-known that dyadic martingale transforms are a good model for Calderon–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A 2 matrix weight, then the weighted L 2 -norm of a Calderon–Zygmund operator with cancellation has the same dependence on the A 2 characteristic of W as the weighted L 2 -norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderon–Zygmund operators on the A 2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderon–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytonen to extend the result to general Calderon–Zygmund operators.
- Published
- 2017
6. Degenerate abstract Volterra equations in locally convex spaces
- Author
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Marko Kostić
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Degenerate energy levels ,Volterra equations ,Equicontinuity ,01 natural sciences ,Volterra integral equation ,010101 applied mathematics ,symbols.namesake ,Locally convex topological vector space ,Resolvent operator ,symbols ,0101 mathematics ,Well posedness ,Mathematics - Abstract
In the paper under review, we analyze various types of degenerate abstract Volterra integrodifferential equations in sequentially complete locally convex spaces. From the theory of non-degenerate equations, it is well known that the class of (a,k)-regularized C-resolvent families provides an efficient tool for dealing with abstract Volterra integro-differential equations of scalar type. Following the approach of T.-J. Xiao and J. Liang [41]-[43], we introduce the class of degenerate exponentially equicontinuous (a,k)- regularized C-resolvent families and discuss its basic structural properties. In the final section of paper, we will look at generation of degenerate fractional resolvent operator families associated with abstract differential operators.
- Published
- 2017
7. On the Langlands correspondence for symplectic motives
- Author
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Benedict H. Gross
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Langlands dual group ,01 natural sciences ,Cohomology ,Langlands program ,Elliptic curve ,Local Langlands conjectures ,Orthogonal group ,0101 mathematics ,Weil group ,Symplectic geometry ,Mathematics - Abstract
In this paper, we present a refinement of the global Langlands correspondence for discrete symplectic motives of rank 2n over Q. To such a motive Langlands conjecturally associates a generic, automorphic representation of the split orthogonal group SO2n+1 over Q, which appears with multiplicity one in the cuspidal spectrum. Using the local theory of generic representations of odd orthogonal groups, we define a new vector F in this representation, which is the tensor product of local test vectors for the Whittaker functionals [9]. I hope that the defining properties ofF will make it easier to investigate the Langlands correspondence computationally, especially for the cohomology of algebraic curves. Our refinement is similar to the refinement that Weil [24] proposed for the conjecture that elliptic curves over Q are modular. Namely, Weil proposed that such a curve should be associated with a homomorphic newform F = P anq n of weight 2 on 0(N), where N is equal to the conductor of the curve. This paper expands on a letter that I wrote to Serre in 2010. It was motivated by a question Serre posed at my 60th birthday conference, and a suggestion Brumer made of a family of discrete subgroups generalizing 0(N). I would like to thank them, and to thank Deligne for his comments.
- 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. Rigidity theory for $C^*$-dynamical systems and the 'Pedersen Rigidity Problem', II
- Author
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Tron Omland, John Quigg, and Steven Kaliszewski
- Subjects
Discrete mathematics ,Exterior equivalences ,Pure mathematics ,Dynamical systems theory ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Outer conjugacy ,Generalized fixed point algebra ,01 natural sciences ,Rigidity (electromagnetism) ,Crossed product ,Primary 46L55 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,Abelian group ,Rigidity theory ,Operator Algebras (math.OA) ,Crossed products ,Mathematics ,Conjugate - Abstract
This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group $G$, saying that two actions $(A,\alpha)$ and $(B,\beta)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_{\alpha}G,\widehat\alpha)$ and $(B\rtimes_{\beta}G,\widehat\beta)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images, and we have decided to use the term "Pedersen rigid" for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call "fixed-point rigidity". In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known., Comment: Minor revision. To appear in Internat. J. Math
- Published
- 2018
10. Uniqueness of meromorphic functions whose nonlinear differential polynomials share a polynomial
- Author
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Pulak Sahoo and Himadri Karmakar
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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
11. Metrically and topologically projective ideals of Banach algebras
- Author
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N. T. Nemesh
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Identity (mathematics) ,Banach algebra ,Bounded function ,0103 physical sciences ,Metric (mathematics) ,Ideal (order theory) ,010307 mathematical physics ,0101 mathematics ,Projective test ,Commutative property ,Approximate identity ,Mathematics - Abstract
In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary C*-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.
- Published
- 2016
12. Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems
- Author
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Raffaella Servadei, Giovanni Molica Bisci, Alessio Fiscella, Fiscella, A, Molica Bisci, G, and Servadei, R
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,variational techniques ,010102 general mathematics ,Multiplicity (mathematics) ,integrodifferential operators ,01 natural sciences ,Dirichlet distribution ,Fractional Laplacian ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,critical nonlinearities ,Operator (computer programming) ,Fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,Bounded function ,best fractional critical Sobolev constant ,fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,symbols ,Exponent ,0101 mathematics ,Bifurcation ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.
- Published
- 2016
13. Degeneration of the Hilbert pairing in formal groups over local fields
- Author
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O. Yu. Podkopaeva, Sergei V. Vostokov, and Regina P. Vostokova
- Subjects
Discrete mathematics ,Ring (mathematics) ,Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Formal group ,Field (mathematics) ,Subring ,01 natural sciences ,010305 fluids & plasmas ,Formal derivative ,Pairing ,0103 physical sciences ,0101 mathematics ,Symbol (formal) ,Mathematics - Abstract
For an arbitrary local field K (a finite extension of the field Qp) and an arbitrary formal group law F over K, we consider an analog cF of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing cF has a certain fundamental symbol property for all Lubin–Tate formal groups, then cF = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring OK of integers of K and have a fixed ring O0 of endomorphisms, where O0 is a subring of OK. We prove that if the symbol cF has the above-mentioned symbol property, then cF = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.
- Published
- 2016
14. 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
15. Effective Twisted Conjugacy Separability of Nilpotent Groups
- Author
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Jonas Deré and Mark Pengitore
- Subjects
Discrete mathematics ,Pure mathematics ,Polynomial ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,01 natural sciences ,Upper and lower bounds ,Nilpotent ,Mathematics::Group Theory ,Conjugacy class ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Finitely-generated abelian group ,0101 mathematics ,Nilpotent group ,Mathematics - Group Theory ,Quotient ,Mathematics - Abstract
This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a precise calculation of conjugacy separability for finitely generated nilpotent groups of nilpotency class 2., V2: removed reference to false result of other paper. Accepted for publication in Math. Z
- Published
- 2017
16. 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
17. Locally piecewise affine functions and their order structure
- Author
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Vladimir G. Troitsky and Samer Adeeb
- Subjects
Discrete mathematics ,Pure mathematics ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,46A40, 46E05 ,01 natural sciences ,Affine plane ,Theoretical Computer Science ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Affine geometry ,Affine coordinate system ,Affine combination ,Affine representation ,Affine geometry of curves ,Affine hull ,Affine group ,FOS: Mathematics ,0101 mathematics ,Analysis ,Mathematics - Abstract
Piecewise affine functions on subsets of $\mathbb R^m$ were studied in \cite{Ovchinnikov:02,Aliprantis:06a,Aliprantis:07a,Aliprantis:07}. In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in $C(\mathbb R^m)$, while piecewise affine functions are sequentially order dense in $C(\mathbb R^m)$. This paper is partially based on \cite{Adeeb:14}., Comment: 11 pages
- Published
- 2016
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18. Algebraic Rational Cells And Equivariant Intersection Theory
- Author
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Richard Gonzales
- Subjects
medicine.medical_specialty ,Pure mathematics ,General Mathematics ,01 natural sciences ,Mathematics - Algebraic Geometry ,03 medical and health sciences ,0302 clinical medicine ,Mathematics::Algebraic Geometry ,medicine ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,030212 general & internal medicine ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics ,Singular point of an algebraic variety ,Discrete mathematics ,Intersection theory ,Function field of an algebraic variety ,14C15, 14L30, 14M27 ,010102 general mathematics ,Toric variety ,Algebraic variety ,Birational geometry ,16. Peace & justice ,Algebraic cycle - Abstract
We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of $\mathbb{Q}$-filtrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any $\mathbb{Q}$-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more generally to spherical varieties. This paper is an extension of arxiv.org/abs/1112.0365 to equivariant Chow groups., Comment: Second version. 24 pages. Substantial changes in the presentation. In particular, the results on Poincar\'e duality (Section 6 of first version) are omitted; they are published in a separate paper (see http://revistas.pucp.edu.pe/index.php/promathematica/article/view/11235)
- Published
- 2016
19. Micro-local analysis in some spaces of ultradistributions
- Author
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Karoline Johansson, Nenad Teofanov, Joachim Toft, and Stevan Pilipović
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Modulation space ,Pure mathematics ,Class (set theory) ,35A18, 35S30, 42B05, 35H10 ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Local analysis ,FOS: Mathematics ,0101 mathematics ,Nonlinear Sciences::Pattern Formation and Solitons ,Mathematics - Abstract
In this paper we extend some results from our earlier papers on wave-front sets, concerning wave-front sets of Fourier-Lebesgue and modulation space types, to a broader class of spaces of ultradistributions, and relate these wave-front sets with the usual wave-front sets of ultradistributions. Furthermore, we use Gabor frames for the description of discrete wave-front sets, and prove that these wave-front sets coincide with corresponding continuous ones., Comment: 28 pages
- Published
- 2012
20. Nilpotence and descent in equivariant stable homotopy theory
- Author
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Justin Noel, Akhil Mathew, and Niko Naumann
- Subjects
Discrete mathematics ,Pure mathematics ,Finite group ,General Mathematics ,010102 general mathematics ,Mathematics - Category Theory ,01 natural sciences ,Mathematics::Algebraic Topology ,Stable homotopy theory ,Nilpotent ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Spectral sequence ,FOS: Mathematics ,Torsion (algebra) ,Algebraic Topology (math.AT) ,Equivariant cohomology ,Equivariant map ,Category Theory (math.CT) ,010307 mathematical physics ,Mathematics - Algebraic Topology ,0101 mathematics ,Abelian group ,Mathematics - Abstract
Let $G$ be a finite group and let $\mathscr{F}$ be a family of subgroups of $G$. We introduce a class of $G$-equivariant spectra that we call $\mathscr{F}$-nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable $\infty$-category, with which we begin. We then develop some of the basic properties of $\mathscr{F}$-nilpotent $G$-spectra, which are explored further in the sequel to this paper. In the rest of the paper, we prove several general structure theorems for $\infty$-categories of module spectra over objects such as equivariant real and complex $K$-theory and Borel-equivariant $MU$. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex $K$-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property., 63 pages. Revised version, to appear in Advances in Mathematics
- Published
- 2015
21. Characterising subspaces of Banach spaces with a Schauder basis having the shift property
- Author
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Christian Rosendal
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Pure mathematics ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,Linear subspace ,Tsirelson space ,Sequence space ,Functional Analysis (math.FA) ,Separable space ,Schauder basis ,Mathematics - Functional Analysis ,Distortion problem ,Computer Science::Logic in Computer Science ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,46B03 ,Mathematics - Abstract
We give an intrinsic characterisation of the separable reflexive Banach spaces that embed into separable reflexive spaces with an unconditional basis all of whose normalised block sequences with the same growth rate are equivalent. This uses methods of E. Odell and T. Schlumprecht. 1. THE SHIFT PROPERTY We consider in this paper a property of Schauder bases that has come up on sev- eral occasions since the first construction of a truly non-classical Banach space by B. S. Tsirelson in 1974 (11). It is a weakening of the property of perfect homogeneity, which replaces the condition all normalised block bases are equivalent with the weaker all normalised block bases with the same growth rate are equivalent, and is satisfied by bases constructed along the lines of the Tsirelson basis, including the standard bases for the Tsirelson space and its dual. To motivate our study and in order to fix ideas, in the following result we sum up a number of conditions that have been studied at various occasions in the literature and that can all be seen to be reformulations of the aforementioned property. Though I know of no single reference for the proof of the equivalence, parts of it are implicit in J. Lindenstrauss and L. Tzafriri's paper (7) and the paper by P. G. Casazza, W. B. Johnson and L. Tzafriri (2). Moreover, any idea needed for the proof can be found in
- Published
- 2011
22. Polynomial functors from algebras over a set-operad and nonlinear Mackey functors
- Author
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Christine Vespa, Teimuraz Pirashvili, Manfred Hartl, Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 (LAMAV), Centre National de la Recherche Scientifique (CNRS)-Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France), Department of Mathematics [Leicester], University of Leicester, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Calculus of functors ,polynomial functors ,Derived functor ,General Mathematics ,Functor category ,010103 numerical & computational mathematics ,01 natural sciences ,Mathematics::Algebraic Topology ,non-linear Mackey functors ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Adjoint functors ,Mathematics ,Discrete mathematics ,010102 general mathematics ,set-operads ,[MATH.MATH-AT]Mathematics [math]/Algebraic Topology [math.AT] ,Natural transformation ,Ext functor ,Tor functor ,Abelian category ,18D ,18A25 ,55U - Abstract
In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (fi nitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-e ffects and maps relating them which satisfy a list of relations., Comment: 58 pages
- Published
- 2015
23. Some Groups of Type E7
- Author
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T. A. Springer
- Subjects
Discrete mathematics ,Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Alternating group ,Outer automorphism group ,Type (model theory) ,01 natural sciences ,Inner automorphism ,Irreducible representation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Homogeneous space ,Identity component ,0101 mathematics ,Group theory ,Mathematics - Abstract
An algebraic group of type E7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F. Certain F-forms of E7 appear in the automorphism groups of quartic forms over F, as well as forms of E6. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.
- Published
- 2006
24. Infinitesimal operations on complexes of graphs
- Author
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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
25. Convergence of the zeta functions of prehomogeneous vector spaces
- Author
-
Hiroshi Saito
- Subjects
Discrete mathematics ,Pure mathematics ,Prehomogeneous vector space ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Algebraic number field ,01 natural sciences ,Riemann zeta function ,Arithmetic zeta function ,symbols.namesake ,Hypersurface ,Hasse principle ,0103 physical sciences ,symbols ,11S90 ,0101 mathematics ,Abelian group ,11S40 ,Mathematics ,Vector space - Abstract
Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.
- Published
- 2003
26. On the dimension and multiplicity of local cohomology modules
- Author
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Rodney Y. Sharp, Markus Brodmann, and University of Zurich
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multiplicity of local cohomology module ,Pure mathematics ,13D45 ,General Mathematics ,Group cohomology ,Local cohomology ,01 natural sciences ,Cohen ,510 Mathematics ,Grothendieck topology ,Cup product ,0103 physical sciences ,Equivariant cohomology ,0101 mathematics ,2600 General Mathematics ,Mathematics ,Discrete mathematics ,Macaulay fibers ,Zariski topology ,Noetherian local ring ,Mathematics::Commutative Algebra ,13C15 ,13H10 ,010308 nuclear & particles physics ,Computer Science::Information Retrieval ,010102 general mathematics ,Local ring ,universally catenary module ,Matlis dual ,10123 Institute of Mathematics ,Artinian module ,Maximal ideal - Abstract
This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.
- Published
- 2002
27. Homological dimensions of crossed products
- Author
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Liping Li
- Subjects
Discrete mathematics ,Finite group ,Noetherian ring ,Pure mathematics ,Functor ,General Mathematics ,010102 general mathematics ,Sylow theorems ,Dimension (graph theory) ,Group Theory (math.GR) ,16E10 ,01 natural sciences ,Separable space ,Global dimension ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Group ring ,Mathematics - Abstract
In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions, extending the main results of the author in some previous papers., Comment: Proof simplified, typos and mistakes corrected. A big revision for induction and restriction by using theory of separable extensions
- Published
- 2014
28. Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains
- Author
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Oleksandr Misiats, Oleksandr Stanzhytskyi, and Nung Kwan Yip
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,35, 60 ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Elliptic operator ,Compact space ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Ergodic theory ,Invariant measure ,Uniqueness ,Nabla symbol ,0101 mathematics ,Statistics, Probability and Uncertainty ,Invariant (mathematics) ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type $du = (Au + f(u))dt + \sigma(u) dW(t)$, where $A$ is an elliptic operator, $f$ and $\sigma$ are nonlinear maps and $W$ is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function $f$ possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions $f$ and $\sigma$ and elliptic operators $A$ for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if $A$ is the Shr\"{o}dinger-type operator $A = \frac{1}{\rho}(\text{div} \rho \nabla u)$ where $\rho = e^{-|x|^2}$ is the Gaussian weight., Comment: 25 pages
- Published
- 2014
- Full Text
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29. REMARKS ON THE SIMILARITY DEGREE OF AN OPERATOR ALGEBRA
- Author
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Gilles Pisier, Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Arxiv, Import
- Subjects
Pure mathematics ,General Mathematics ,[MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA] ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,01 natural sciences ,Filtered algebra ,0103 physical sciences ,FOS: Mathematics ,46 L 07, 46 K 99 ,0101 mathematics ,Operator Algebras (math.OA) ,Mathematics ,Symmetric algebra ,Discrete mathematics ,010102 general mathematics ,[MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA] ,Mathematics - Operator Algebras ,Reflexive operator algebra ,Compact operator ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Operator algebra ,Algebra representation ,Cellular algebra ,010307 mathematical physics ,Unitary operator ,[MATH.MATH-OA] Mathematics [math]/Operator Algebras [math.OA] - Abstract
The "similarity degree" of a unital operator algebra A was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type II1 factor with property Γ is at most 5, improving on a previous bound (≤ 44) due to E. Christensen.
- Published
- 2001
30. Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
- Author
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H. E. A. Campbell, R. J. Shank, David L. Wehlau, Anthony V. Geramita, and I.P. Hughes
- Subjects
Principal ideal ring ,Discrete mathematics ,Reduced ring ,Ring (mathematics) ,Pure mathematics ,General Mathematics ,Gorenstein ring ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Primitive ring ,Simple ring ,0101 mathematics ,Quotient ring ,Mathematics ,Group ring - Abstract
This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants ofmcopies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.
- Published
- 1999
31. Integral Points on Subvarieties of Semiabelian Varieties, II
- Author
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Paul Vojta
- Subjects
Discrete mathematics ,11G10 (Primary) 11J25, 14G05, 14K15 (Secondary) ,Pure mathematics ,Mathematics - Number Theory ,Subvariety ,Divisor ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Context (language use) ,Algebraic geometry ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Logic ,Mathematics::Algebraic Geometry ,Number theory ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Variety (universal algebra) ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
This paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties. Combined with the main theorem of the first part of this paper, this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety, minus a divisor. In addition, the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors., Comment: 31 pages, amstex with rsfs fonts
- Published
- 1999
32. Localization in equivariant intersection theory and the Bott residue formula
- Author
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William Graham and Dan Edidin
- Subjects
medicine.medical_specialty ,Pure mathematics ,General Mathematics ,Cyclic group ,Homology (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Localization theorem ,FOS: Mathematics ,medicine ,Equivariant cohomology ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics ,Discrete mathematics ,Intersection theory ,010308 nuclear & particles physics ,010102 general mathematics ,Equivariant map ,Isomorphism ,Character group - Abstract
The purpose of this paper is to prove the localization theorem for torus actions in equivariant intersection theory. Using the theorem we give another proof of the Bott residue formula for Chern numbers of bundles on smooth complete varieties. In addition, our techniques allow us to obtain residue formulas for bundles on a certain class of singular schemes which admit torus actions. This class is rather special, but it includes some interesting examples such as complete intersections and Schubert varieties., This paper is a substantially revised version of our preprint "Equivariant Chow groups and the Bott residue formula". Current email address for Dan Edidin is "edidin@math.missouri.edu" and current email for William Graham is "wag@math.ias.edu" Amslatex
- Published
- 1998
33. On Homogeneous Images of Compact Ordered Spaces
- Author
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Jacek Nikiel and E. D. Tymchatyn
- Subjects
Discrete mathematics ,Pure mathematics ,Continuum (topology) ,General Mathematics ,First-countable space ,010102 general mathematics ,Hausdorff space ,Mathematics::General Topology ,Disjoint sets ,01 natural sciences ,Jordan curve theorem ,symbols.namesake ,Metrization theorem ,0103 physical sciences ,Homogeneous space ,symbols ,010307 mathematical physics ,0101 mathematics ,Indecomposable module ,Mathematics - Abstract
We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.
- Published
- 1993
34. Homotopy Theory of Diagrams and CW-Complexes Over a Category
- Author
-
Robert J. Piacenza
- Subjects
Discrete mathematics ,Pure mathematics ,Homotopy category ,Brown's representability theorem ,Model category ,Computer Science::Information Retrieval ,General Mathematics ,Homotopy ,010102 general mathematics ,Whitehead theorem ,Mathematics::Algebraic Topology ,01 natural sciences ,Weak equivalence ,n-connected ,Mathematics::K-Theory and Homology ,Mathematics::Category Theory ,Homotopy hypothesis ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to introduce the notion of a CW complex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category.A brief description of the paper goes as follows: in Section 1 we introduce the homotopy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type homotopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method.
- Published
- 1991
35. Pointwise Trichotomy for Skew-Evolution Semiflows on Banach Spaces
- Author
-
Codruta Stoica, Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Mathematics::Analysis of PDEs ,Banach space ,pointwise exponential trichotomy ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,Mathematics Subject Classification: 34D09 ,State evolution ,exponential trichotomy ,Exponential stability ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Skew-evolution semiflow ,0101 mathematics ,Computer Science::Databases ,Mathematics ,Pointwise ,Discrete mathematics ,Applied Mathematics ,Exponential dichotomy ,010102 general mathematics ,Skew ,010101 applied mathematics ,Mathematics - Classical Analysis and ODEs ,34D09 ,Trichotomy (mathematics) - Abstract
The paper introduces the notion of skew-evolution semiflows and presents the concept of pointwise trichotomy in the case of skew-evolution semiflows on a Banach space. The connection with the classic notion of trichotomy presented by us in a previous paper in 2006 for evolution operators, is also emphasized, as well as some characterizations. The approach of the theory is from uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows.
- Published
- 2008
36. On the Conley decomposition of Mather sets
- Author
-
Patrick Bernard, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), IUF, École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,chain transitivity ,Context (language use) ,Dynamical Systems (math.DS) ,01 natural sciences ,37J50 ,Set (abstract data type) ,symbols.namesake ,0103 physical sciences ,Decomposition (computer science) ,FOS: Mathematics ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Dynamical Systems ,Mathematics::Symplectic Geometry ,Mathematics ,Discrete mathematics ,49L25 ,37B20 ,010102 general mathematics ,Function (mathematics) ,semi-continuity of the Aubry set ,37J50, 37B20, 49L25 ,symbols ,010307 mathematical physics ,minimizing measures ,chain transitivity 116 P Bernard ,Lagrangian - Abstract
International audience; In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian. In the study of Lagrangian systems, John Mather introduced several invariant sets composed of globally minimizing extremals. He developed methods to construct several orbits undergoing interesting behaviors in phase space under some assumptions on these invariant sets, see [14]. In order to pursue this theory and to apply it on examples, it is necessary to have tools to describe precisely the invariant sets. At least two points of view can be adopted. One can study the invariant set from a purely topological point of view in the style of Conley as compact metric spaces with flows, and study their transitive components. One can also study these set from the point of view of action minimization, and decompose them in invariant subsets that have been called static classes. These points of view are very closely related, but each of them has specific features. For example, understanding the decomposition in static classes is necessary for the variational construction of interesting orbits, while the topological decomposition behaves well under perturbations. Our goal in the present paper is to explicit the links between these two decompositions. We explain that the topological decomposition is finer than the variational one, and that they coincide for most (but not all) systems. As an application, we prove a result of semi-continuity of the so-called Aubry set as a function of the Lagrangian, under certain non-degeneracy hypotheses. The semi-continuity of the Aubry set is a subtle problem, which has remained open for several years, until John Mather gave a counter example, see §18 in [16]. In the same paper, he also states without proof that semi-continuity holds under appropriate hypotheses. Our result extends the one of Mather. The methods we use are inspired from the recent work of Fathi, Figalli and Rifford, [9].
- Published
- 2008
- Full Text
- View/download PDF
37. On the cohomology of Young modules for the symmetric group
- Author
-
Daniel K. Nakano, David J. Hemmer, and Frederick R. Cohen
- Subjects
Pure mathematics ,Mathematics(all) ,Galois cohomology ,General Mathematics ,Group cohomology ,Young modules ,01 natural sciences ,Cohomology ,Symmetric group ,0103 physical sciences ,De Rham cohomology ,FOS: Mathematics ,Equivariant cohomology ,Algebraic Topology (math.AT) ,55S12 ,Mathematics - Algebraic Topology ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics ,Discrete mathematics ,20C30 ,55P47 ,010102 general mathematics ,Representation theory of the symmetric group ,Mayer–Vietoris sequence ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $\Sigma_d$, with `twisted' coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces $Q(X)$ with mod-$p$ coefficients to describe the homology $\HH_{\bullet}(\Sigma_d, V^{\otimes d})$ as a module for the general linear group $GL(V)$ over an algebraically closed field $k$ of characteristic $p$. As a direct application, these results provide a method of reducing the computation of $\text{Ext}^{\bullet}_{\Sigma_{d}}(Y^{\lambda},Y^{\mu})$ (where $Y^{\lambda}$, $Y^{\mu}$ are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many $d$, a complete determination of $\Hs Y^\lambda)$ can be found. This is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given. For arbitrary $d$ the authors determine $\HH^i(\Sigma_d,Y^\lambda)$ for $i=0,1,2$. An interesting phenomenon is uncovered--namely a stability result reminiscent of generic cohomology for algebraic groups. For each $i$ the cohomology $\HH^i(\Sigma_{p^ad}, Y^{p^a\lambda})$ stabilizes as $a$ increases. The methods in this paper are also powerful enough to determine, for any $p$ and $\lambda$, precisely when $\HH^{\bullet}(\sd,Y^\lambda)=0$. Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus., Comment: Substantially revised, original stability conjecture proven for all primes. To appear, Advances in Mathematics
- Published
- 2008
- Full Text
- View/download PDF
38. Alcove path and Nichols-Woronowicz model of the equivariant K-theory of generalized flag varieties
- Author
-
Cristian Lenart and Toshiaki Maeno
- Subjects
Pure mathematics ,General Mathematics ,01 natural sciences ,symbols.namesake ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Quantum Algebra (math.QA) ,Generalized flag variety ,Braided Hopf algebra ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics ,Discrete mathematics ,Weyl group ,Quantum group ,Flag (linear algebra) ,010102 general mathematics ,16. Peace & justice ,Noncommutative geometry ,Cohomology ,symbols ,Equivariant map ,Combinatorics (math.CO) ,010307 mathematical physics - Abstract
Fomin and Kirillov initiated a line of research into the realization of the cohomology and K-theory of generalized flag varieties G/B as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the T -equivariant K-theory of a generalized flag variety KT (G/B) in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for KT (G/B) due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach. Dedicated to Professor Kenji Ueno on the occasion of his sixtieth birthday
- Published
- 2006
39. An uncountably infinite number of indecomposable totally reflexive modules
- Author
-
Ryo Takahashi
- Subjects
Pure mathematics ,General Mathematics ,Prime ideal ,Commutative Algebra (math.AC) ,01 natural sciences ,Residue field ,0103 physical sciences ,FOS: Mathematics ,Countable set ,0101 mathematics ,Mathematics ,Discrete mathematics ,Countable Cohen-Macaulay type ,Conjecture ,Semidualizing ,Cohen-Macaulay ring ,13C14 ,Mathematics::Commutative Algebra ,010308 nuclear & particles physics ,010102 general mathematics ,13C14, 16G60 ,Local ring ,16G60 ,Totally reflexive ,Mathematics - Commutative Algebra ,Uncountable set ,Isomorphism ,Indecomposable module - Abstract
A few years ago, Huneke and Leuschke proved a theorem which solved a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is considered over an arbitrary local ring. The main purpose of this paper is to prove that the existence of a certain prime ideal and a certain totally reflexive module implies the existence of an uncountably infinite number of isomorphism classes of indecomposable totally reflexive modules., Comment: 10 pages, minor changes, to appear in Nagoya Math. J
- Published
- 2006
- Full Text
- View/download PDF
40. Concepts of curvatures in normed planes
- Author
-
Horst Martini, Emad Shonoda, and Vitor Balestro
- Subjects
Discrete mathematics ,Pure mathematics ,Fundamental theorem ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Four-vertex theorem ,Curvature ,01 natural sciences ,Constant curvature ,Fundamental theorem of curves ,Euclidean geometry ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.
- Published
- 2019
41. Toroidal varieties and the weak Factorization Theorem
- Author
-
Jarosław Włodarczyk
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,14E05 ,14L30 ,14M25 ,58E05 ,57R90 ,General Mathematics ,01 natural sciences ,law.invention ,symbols.namesake ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Factorization ,law ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Algebraically closed field ,Complex Variables (math.CV) ,Algebraic Geometry (math.AG) ,Mathematics ,Discrete mathematics ,Toroid ,Conjecture ,Mathematics - Complex Variables ,010102 general mathematics ,Zero (complex analysis) ,Birational geometry ,Invertible matrix ,Differential Geometry (math.DG) ,Weierstrass factorization theorem ,symbols ,010307 mathematical physics - Abstract
The main goal of the present paper is two-fold. First we extend the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to the class of toroidal varieties with stratifications (which is the main body of the paper). Second we give a proof of the following weak factorization theorem as an application and illustration of the theory: A birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow ups and blow downs with smooth centers. Another proof of the weak factorization theorem appeared in a joint paper with Abramovich, Karu and Matsuki (math.AG/9904135) In that paper the theorem is stated and proven in general for proper algebraic and analytic spaces., Comment: 69 pages, a revised version with conceptual simplifications. The present version is self-contained and it does not rely on weak factorization theorem for toric varieties
- Published
- 1999
- Full Text
- View/download PDF
42. Effective topological complexity of spaces with symmetries
- Author
-
Marek Kaluba and Zbigniew Błaszczyk
- Subjects
Pure mathematics ,motion planning problem ,Topological algebra ,General Mathematics ,equivariant topological complexity ,01 natural sciences ,Mathematics::Algebraic Topology ,Equivariant topological complexity ,55M30 ,Motion planning problem ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Discrete mathematics ,Topological complexity ,68T40 ,Topological tensor product ,010102 general mathematics ,010101 applied mathematics ,Bounded function ,Homogeneous space ,55M30, 68T40 - Abstract
We introduce a version of Farber's topological complexity suitable for investigating mechanical systems whose configuration spaces exhibit symmetries. Our invariant has vastly different properties to the previous approaches of Colman-Grant, Dranishnikov and Lubawski-Marzantowicz. In particular, it is bounded from above by Farber's topological complexity., Comment: New title; a short section with open problems included at the end of the paper. Numerous minor improvements throughout the text. Final version, to appear in Publ. Mat. 19 pages, 2 figures
- Published
- 2021
43. Uniqueness of Coxeter structures on Kac–Moody algebras
- Author
-
Valerio Toledano Laredo and Andrea Appel
- Subjects
Pure mathematics ,Lie bialgebra ,General Mathematics ,Braid group ,Category O ,01 natural sciences ,symbols.namesake ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics ,Discrete mathematics ,Weyl group ,Functor ,Quantum group ,010102 general mathematics ,Coxeter group ,Mathematics - Category Theory ,Monodromy ,symbols ,010307 mathematical physics ,Mathematics - Representation Theory - Abstract
Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages
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- 2019
44. On the Hodge Structure of Projective Hypersurfaces in Toric Varieties
- Author
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David A. Cox and Victor V. Batyrev
- Subjects
14F10 ,Pure mathematics ,General Mathematics ,Homogeneous coordinate ring ,01 natural sciences ,14C30 ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Projective space ,0101 mathematics ,14M25 ,Algebraic Geometry (math.AG) ,Mathematics ,Discrete mathematics ,Mathematics::Commutative Algebra ,010308 nuclear & particles physics ,Complex projective space ,010102 general mathematics ,14D07 ,Toric variety ,Cohomology ,Hypersurface ,Affine space ,Hodge structure - Abstract
This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive cohomology of $X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$ and $T$ is an affine hypersurface in $T$, and we show how recent results of the second author can be stated using various ideals in the ring $S$. To prove our results, we must give a careful description (in terms of $S$) of $d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties. Other topics considered in the paper include quasi-smooth hypersurfaces and $V$-submanifolds, the structure of the complement of $U$ when $P$ is represented as the quotient of an open subset $U$ of affine space, a generalization of the Euler exact sequence on projective space, and the relation between graded pieces of $R/J(f)$ and the moduli of ample hypersurfaces in $P$., Comment: 43 pages, LaTeX Version 2.09
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- 1993
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45. Bounding Harish-Chandra series
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Olivier Dudas and Gunter Malle
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Discrete mathematics ,Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,Unipotent ,Reductive group ,01 natural sciences ,Matrix (mathematics) ,Character (mathematics) ,Bounding overwatch ,0103 physical sciences ,FOS: Mathematics ,Irreducibility ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,20C33, 20C08 ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Simple module ,Mathematics - Representation Theory ,Mathematics - Abstract
We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees., Comment: arXiv admin note: text overlap with arXiv:1611.07373
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- 2018
46. Lifting non-ordinary cohomology classes for $\mathrm{SL}_3$
- Author
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Chris Williams
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,Group cohomology ,010102 general mathematics ,Factor system ,Extension (predicate logic) ,11F75 ,11F85 ,01 natural sciences ,Cohomology ,$\mathrm{SL}_3$ ,control theorem ,Operator (computer programming) ,Congruence (geometry) ,modular symbols ,0103 physical sciences ,Equivariant cohomology ,010307 mathematical physics ,Overconvergent ,0101 mathematics ,Mathematics - Abstract
In this paper, we present a generalisation of a theorem of David and Rob Pollack. In [PP], they give a very general argument for lifting ordinary eigenclasses (with respect to a suitable operator) in the group cohomology of certain arithmetic groups. With slightly tighter conditions, we prove the same result for non-ordinary classes. Pollack and Pollack apply their results to the case of $p$-ordinary classes in the group cohomology of congruence subgroups for $\mathrm{SL}_3$, constructing explicit overconvergent classes in this setting. As an application of our results, we give an extension of their results to the case of non-critical slope classes in the same setting.
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- 2018
47. A Lemma for Microlocal Sheaf Theory in the $\infty$-Categorical Setting
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Marco Robalo and Pierre Schapira
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Discrete mathematics ,Lemma (mathematics) ,Derived category ,Pure mathematics ,Functor ,General Mathematics ,Global section functor ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,010201 computation theory & mathematics ,Mathematics::Category Theory ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Sheaf ,Five lemma ,Mathematics - Algebraic Topology ,0101 mathematics ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Categorical variable ,Mathematics - Abstract
Microlocal sheaf theory of \cite{KS90} makes an essential use of an extension lemma for sheaves due to Kashiwara, and this lemma is based on a criterion of the same author giving conditions in order that a functor defined in $\mathbb{R}$ with values in the category $Sets$ of sets be constant. In a first part of this paper, using classical tools, we show how to generalize the extension lemma to the case of the unbounded derived category. In a second part, we extend Kashiwara's result on constant functors by replacing the category $Sets$ with the $\infty$-category of spaces and apply it to generalize the extension lemma to $\infty$-sheaves, the $\infty$-categorical version of sheaves. Finally, we define the micro-support of sheaves with values in a stable $(\infty,1)$-category.
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- 2018
48. Livsic-type determinantal representations and hyperbolicity
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Victor Vinnikov and Eli Shamovich
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Discrete mathematics ,Pure mathematics ,Subvariety ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,010103 numerical & computational mathematics ,Algebraic geometry ,Codimension ,01 natural sciences ,Hermitian matrix ,Linear subspace ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,Compact Riemann surface ,0101 mathematics ,Algebraic Geometry (math.AG) ,Meromorphic function ,Mathematics - Abstract
Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety X ⊂ P d of an arbitrary codimension l with respect to a real l − 1 -dimensional linear subspace V ⊂ P d and study its basic properties. We also consider a class of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces ( l = 1 ), the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in P d hyperbolic with respect to some real d − 2 -dimensional linear subspace admits a definite Hermitian, or even definite real symmetric, very reasonable Livsic-type determinantal representation.
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- 2018
49. Zero product determined Lie algebras
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Kaiming Zhao, Rencai Lu, Genqiang Liu, Matej Brešar, and Xiangqian Guo
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Discrete mathematics ,Pure mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Non-associative algebra ,010103 numerical & computational mathematics ,Killing form ,01 natural sciences ,Affine Lie algebra ,Lie conformal algebra ,Graded Lie algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,0101 mathematics ,Mathematics - Abstract
A Lie algebra L over a field $$\mathbb {F}$$ is said to be zero product determined (zpd) if every bilinear map with the property that $$f(x,y)=0$$ , whenever x and y commute, is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra , where V is a simple $$\mathfrak {sl}_2$$ -module, is zpd if and only if $$\dim V =2$$ or $$\dim V$$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $$\mathscr {L}_q$$ and $$\mathscr {L}^+_q$$ , the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most 3, while the class of non-zpd Lie algebras includes the (4-dimensional) aging Lie algebra and all Lie algebras of dimension more than 3 in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of zpd Lie algebra by using it in the study of commutativity preserving linear maps.
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- 2018
50. Modular characteristic classes for representations over finite fields
- Author
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Anssi Lahtinen and David Sprehn
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Group cohomology ,010102 general mathematics ,Étale cohomology ,Mathematics::Algebraic Topology ,01 natural sciences ,Cohomology ,Characteristic class ,Group of Lie type ,Mayer–Vietoris sequence ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,De Rham cohomology ,Algebraic Topology (math.AT) ,Equivariant cohomology ,Mathematics - Algebraic Topology ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,20J06, 20C20 ,Mathematics - Representation Theory ,Mathematics - Abstract
The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood., Accepted to the Advances in Mathematics
- Published
- 2018
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