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Differential difference algebras were introduced by Mansfield and Szanto, which arose naturally from differential difference equations. In this paper, we investigate the Gelfand-Kirillov dimension of differential difference algebras. We give a lower bound of the Gelfand-Kirillov dimension of a differential difference algebra and a sufficient condition under which the lower bound is reached; we also find an upper bound of this Gelfand-Kirillov dimension under some specific conditions and construct an example to show that this upper bound can not be sharpened any more., Comment: 12 pages

In this paper we consider a general class of external cusps defined by linking appropriate collections of John domains. For that class, weighted Korn inequalities are proved by means of rather elementary arguments. Fil: Acosta, Gabriel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina Fil: Ojea, Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina

Frobenius observed that the number of times an element of a finite group is obtained as a commutator is given by a specific combination of the irreducible characters of the group. More generally, for any word w the number of times an element is obtained by substitution in w is a class function. Thus, it has a presentation as a combination of irreducible characters, called its Fourier expansion. In this paper we present formulas regarding the Fourier expansion of words in which some letters appear twice. These formulas give simple proofs for classical results, as well as new ones.

In this paper, we obtain the existence of at least two nontrivial solutions for a Robin-type differential inclusion problem involving p(x)-Laplacian type operator and nonsmooth potentials. Our approach is variational, and it is based on the nonsmooth critical point theory for locally Lipschitz functions. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem CDαu(t)+f(t,u(t))=0,0

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated, and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors., This is a preprint of a paper whose final and definite form will be published in Mathematical Methods in the Applied Sciences, ISSN 0170-4214. Submitted 05/March/2014; revised 24/Feb/2015; accepted 25/Feb/2015. arXiv admin note: text overlap with arXiv:1405.2813

The graded Specht module Sfor a cyclotomic Hecke al- gebra comes with a distinguished generating vector z � 2 S � , which can be thought of as a "highest weight vector of weight �". This paper describes the defining relations for the Specht module Sas a graded module generated by z � . The first three relations say precisely what it means for zto be a highest weight vector of weight �. The remaining relations are homogeneous analogues of the classical Garnir relations. The homogeneous Garnir relations, which are simpler than the classical ones, are associated with a remarkable family of homogeneous operators on the Specht module which satisfy the braid relations.

A relative Schottky set in a planar domain D is a subset of D obtained by removing from D open geometric discs whose closures are in D and are pairwise disjoint. In this paper we study quasisymmetric and related maps between relative Schottky sets of measure zero. We prove, in particular, that quasisymmetric maps between such sets in Jordan domains are conformal, locally bi-Lipschitz, and that their first derivatives are locally Lipschitz. We also provide a locally bi-Lipschitz uniformization result for relative Schottky sets in Jordan domains and establish rigidity with respect to local quasisymmetric maps for relative Schottky sets in the unit disc., Comment: 43 pages

In this paper we consider the singularities of the varieties parameterizing stable vector bundles of fixed rank and degree with sections on a smooth curve of genus at least two. In particular, we extend results of Y. Laszlo, and of the second author, regarding the singularities of generalized theta divisors. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Vojta's Conjectures are well known to imply a wide range of results, known and unknown, in arithmetic geometry. In this paper, we add to the list by proving that they imply that rational points tend to repel each other on algebraic varieties with nonnegative Kodaira dimension. We use this to deduce, from Vojta's Conjectures, conjectures of Batyrev-Manin and Manin on the distribution of rational points on algebraic varieties. In particular, we show that Vojta's Main Conjecture implies the Batyrev-Manin Conjecture for $K3$ surfaces., 10 pages

Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to Bringmann and Mahlburg that refined a conjecture of Garvan. Garvan's conjecture states that the moments of the crank function are always larger than the moments of the rank function, even though the moments have the same main asymptotic term. The proof uses the Hardy-Ramanujan method to provide precise asymptotic estimates for rank and crank moments and their differences., 11 pages

In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations that are formally similar to distribution theory. Further, we give several characterizations of equality in the sense of generalized distributions in these algebras., 20 pages; follow-up of arXiv:math/0611377

This paper studies the problems of embedding and isomorphism for countably generated Hilbert C*-modules over commutative C*-algebras. When the fibre dimensions differ sufficiently, relative to the dimension of the spectrum, we show that there is an embedding between the modules. This result continues to hold over recursive subhomogeneous C*-algebras. For certain modules, including all modules over $C_0(X)$ when $dim X \leq 3$, isomorphism and embedding are determined by the restrictions to the sets where the fibre dimensions are constant. These considerations yield results for the Cuntz semigroup, including a computation of the Cuntz semigroup for $C_0(X)$ when $dim X \leq 3$, in terms of cohomological data about $X$., To appear in Proc. London Math. Soc. The published version differs

Let A1 and A2 be expansive dilations, respectively, on ℝn and ℝm. Let A ≡ (A1, A2) and p(A) be the class of product Muckenhoupt weights on ℝn × ℝm for p ∈ (1, ∞]. When p ∈ (1, ∞) and w ∈ p(A), the authors characterize the weighted Lebesgue space Lpw(ℝn × ℝm) via the anisotropic Lusin-area function associated with A. When p ∈ (0, 1], w ∈ ∞(A), the authors introduce the weighted anisotropic product Hardy space Hp w (ℝn × ℝm; A) via the anisotropic Lusin-area function and establish its atomic decomposition. Moreover, the authors prove that finite atomic norm on a dense subspace of Hpw (ℝn×ℝm; A) is equivalent with the standard infinite atomic decomposition norm. As an application, the authors prove that if T is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space ℬ, then T uniquely extends to a bounded sublinear operator from Hp w (ℝn × ℝm; A) to ℬ. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

This paper, devoted to the study of spectral pollution, contains both abstract results and applications to some self-adjoint operators with a gap in their essential spectrum occuring in Quantum Mechanics. First we consider Galerkin basis which respect the decomposition of the ambient Hilbert space into a direct sum $H=PH\oplus(1-P)H$, given by a fixed orthogonal projector $P$, and we localize the polluted spectrum exactly. This is followed by applications to periodic Schrodinger operators (pollution is absent in a Wannier-type basis), and to Dirac operator (several natural decompositions are considered). In the second part, we add the constraint that within the Galerkin basis there is a certain relation between vectors in $PH$ and vectors in $(1-P)H$. Abstract results are proved and applied to several practical methods like the famous "kinetic balance" of relativistic Quantum Mechanics.

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements., Comment: 42 pages, 23 figures. Slightly expanded exposition and references

This is a companion paper to math.AT/0609762. For a filtered colimit of commutative rings k=colim k_i, we prove that the homotopy theory of smooth and proper dg-algebras over k is the colimit of the homotopy theories of smooth and proper dg-algebras over k_i. As a consequence, we deduce that any smooth and proper dg-algebra can be defined over a commutative Z-algebra of finite type.

Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, Ts_Q) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of {\em every} singularity with local algebra Q can be recovered from Ts_Q.

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

Recently, J. Roe proved by a direct computation that the Baum–Connes assembly map can be expressed as a boundary map associated to a certain short exact sequence of C*-algebras. In this paper we deduce the same result as a corollary of the author's characterisation of the Baum–Connes assembly map.

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type., Comment: 20 pages. In version 3, some new material is added in the end of section 2, discussing the classification and characterizations of positive quasi-Cartan matrices. In final version 4, Proposition 2.9 is corrected and its proof expanded. To appear in J. London Math. Soc. In version 5 only typos in the arXiv data fixed

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations. In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that $f_o$ is a homeomorphism of the circle with $f_{o}^{-1} \in {\cal W}^{1/2, 2}$. Then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non-vanishing Jacobian determinant. The condition on $f_o$ is sharp. Classically the mapping $f_o$ is assumed to be quasisymmetric. Then there is an extremal quasiconformal mapping with boundary values $f_o$, but it is not always unique and it is seldom smooth. Indeed, even when $f_o$ is quasisymmetric, the ${\cal L}^1$-minimiser for the distortion function will almost never be quasiconformal. We further find that there are many new and unexpected phenomena concerning existence, uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex. These seem to differ markedly from phenomena observed when studying multi-well type functionals.

Let X be a Banach space, let K be a non–empty closed subset of X and let T : K X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ∂K (∂K the boundary of K) into K then T has a unique fixed point in K.

Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for descendent integrals on the moduli space of curves, is a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one ‘$\lambda$-class’), and show that, after a non-trivial change of variables, G = H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber. We can produce recursions and explicit formulas for Hurwitz numbers; the algorithm presented proves all such recursions. As examples we present surprisingly simple new recursions in genus 0 to 3. Similar recursions should exist for all genera. As we expect this paper also to be of interest to combinatorialists, we have tried to make it as self-contained as possible, including reviewing some results and definitions well known in algebraic and symplectic geometry, and mathematical physics. 2000 Mathematical Subject Classification: primary 14H10, 81T40; secondary 05C30, 58D29.

The purpose of this paper is to develop a geometric approach to Wall's finiteness obstruction. We will do this for equivariant CW-complexes. The main advantage will be that we can derive all the formal properties of the equivariant finiteness obstruction easily from this geometric description. Namely, the obstruction property, homotopy invariance, the sum and product formulas, and the restriction formula can be stated and proved in a simple manner. Also a characterization of the finiteness obstruction by a universal property is quickly available. This geometric approach is similar to the treatment of Whitehead torsion by Cohen in [3]. In the first section we define a functor Wa from the category of G-spaces to the category of abelian groups. We assign to a finitely dominated G-CW-complex X an element w(X) e Wa(X) called its finiteness obstruction. The finiteness obstruction vanishes if and only if X is G-homotopic to a finite G-CW-complex and satisfies a sum formula and is homotopy invariant. The notion of a universal functorial additive invariant is introduced in the second section where its existence and uniqueness are proved. Product and restriction formulas for the universal additive invariant are obtained by abstract nonsense. We define equivariant Euler characteristics in the third section generalizing the notion of the Euler characteristic of a finite CW-complex. The goal of the fourth section is to prove that the equivariant Euler characteristic and finiteness obstruction determine the universal functorial additive invariant for finite, respectively finitely dominated, G-CW-complexes. The fifth section contains some algebraic computations of Wa in terms of reduced projective class groups of certain integral group rings. In the nonequivariant case Wall's algebraic approach and our geometric one agree. Finally, in the sixth section, the results of the second and fourth sections are used to state an abstract product formula, a restriction formula, and a diagonal product formula. We make some remarks about the simple-homotopy approach to the finiteness obstruction due to Ferry. The treatment by Ferry in [7] is extended by Kwasik in [14] to the equivariant case. In § 6 we construct geometrically an injection I(Y): W(Y)^>Wh(YxS) into the equivariant Whitehead group of Y x S sending our geometric finiteness obstruction to that of Kwasik. A compact Lie group is denoted by G.