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This paper is concerned with transition fronts for reaction-diffusion equations of the Fisher-KPP type. Basic examples of transition fronts connecting the unstable steady state to the stable one are the standard traveling fronts, but the class of transition fronts is much larger and the dynamics of the solutions of such equations is very rich. In the paper, we describe the class of transition fronts and we study their qualitative dynamical properties. In particular, we characterize the set of their admissible asymptotic past and future speeds and their asymptotic profiles and we show that the transition fronts can only accelerate. We also classify the transition fronts in the class of measurable superpositions of standard traveling fronts. [ABSTRACT FROM AUTHOR]

In this paper, we give a new method to construct a compact symplectic manifold which does not satisfy the hard Lefschetz property. Using our method, we construct a simply connected compact Kähler manifold (M, ω, J) and a symplectic form s on M which does not satisfy the hard Lefschetz property, but is symplectically deformation equivalent to the Kαahler form M. As a consequence, we can give an answer to the question posed by Khesin and Mc- Duff as follows. According to symplectic Hodge theory, any symplectic form ω on a smooth manifold M defines symplectic harmonic forms on M. In a paper by D. Yan (1996), Khesin and McDuff posed a question whether there exists a path of symplectic forms {ωt} such that the dimension hkhr(M,ω) of the space of symplectic harmonic k-forms varies along t. By Yan and O. Mathieu, the hard Lefschetz property holds for (M,ω) if and only if hkhr(M, ω) is equal to the Betti number bk(M) for all k > 0. Thus our result gives an answer to the question. Also, our construction provides an example of a compact Kahler manifold whose Kahler cone is properly contained in the symplectic cone. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a notion of ampleness of a group action G on a right noetherian graded algebra A, and show that it is strongly related to the notion of AG to be a graded isolated singularity introduced by the second author of this paper. Moreover, if S is a noetherian AS-regular algebra and G is a finite ample group acting on S, then we will show that Db(tails SG) ≅= Db(mod∇S*G) where ∇S is the Beilinson algebra of S. We will also explicitly calculate a quiver QS,G such that Db(tails SG) ≅=Db(modkQS,G) when S is of dimension 2. [ABSTRACT FROM AUTHOR]

In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho. [ABSTRACT FROM AUTHOR]

In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A which depends only on the Grothendieck ring of H. When H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are. [ABSTRACT FROM AUTHOR]

This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, the "splitting" properties of affine supergroups are discussed, i.e., special kinds of factorizations they may admit -- either globally, or pointwise. Almost everything should be more or less known, but seems to be not as clear in the literature (to the author's knowledge) as it ought to. Second, a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs is presented (a method already introduced by Koszul, and later extended by other authors). Namely, a new functorial construction ψ is provided which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) -- thus setting a link with the first part of the paper. One knows that there exists a natural functor Φ from affine supergroups to super Harish-Chandra pairs. Then we show that the new functor ψ -- which goes the other way round -- is indeed a quasi-inverse to Φ, provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) Φ and ψ are equivalences between the categories of gs-split affine supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, via different approaches. Nevertheless, the novelty in the present paper lies in the construction of a different functor ψ and thus extends the result to a much larger setup, with a totally different, more geometrical method. In fact, this method (very concrete, indeed) is universal and characteristic-free and is presented here for the algebro-geometric setting, but actually it can be easily adapted to the frameworks of differential or complex analytic supergeometry. The case of linear supergroups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the notion of the Cohen-Macaulay cone of a Noetherian local domain R and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on R, the Grothendieck group G0(R) of finitely generated R-modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R is the cone in G0(R)ℝ spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers ∈i = 0,±1 (d/2 < i < d), we shall construct a d-dimensional Cohen-Macaulay local ring R (of characteristic p) and a maximal primary ideal I of R such that the function ℓR(R/...) is a polynomial in pn of degree d whose coefficient of (pn)i is the product of ∈i and a positive rational number for d/2 < i < d. The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist [ABSTRACT FROM AUTHOR]

Let X be a finite set such that |X| = n and let i ≤ j ≤ n. A group G ≤ Sn is said to be (i, j)-homogeneous if for every I, J ⊆ X, such that |I| = i and |J| = j, there exists g ∈ G such that Ig ⊆ J. (Clearly (i, i)-homogeneity is i-homogeneity in the usual sense.) A group G ≤ Sn is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P of X into k blocks, there exists g ∈ G such that Ig is a section for P. (That is, the orbit of each k-subset of X contains a section for each k-partition of X.) In this paper we classify the groups with the k-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k-1, k)-homogeneous groups (for 2 < k ≤ [n+1/2]). As a corollary of the classification we prove that a (k - 1, k)-homogeneous group is also (k - 2, k - 1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the k-universal transversal property have the (k - 1)-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank k transformation on X generate a regular semigroup (for 1 ≤ k ≤ [n+1/2]) The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory. [ABSTRACT FROM AUTHOR]

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly. Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable. [ABSTRACT FROM AUTHOR]

This is the first of two papers devoted to the study of the properties of the blow-up surface for the N dimensional semilinear wave equation with subconformal power nonlinearity. In a series of papers, we have clarified the situation in one space dimension. Our goal here is to extend some of the properties to higher dimension. In dimension one, an essential tool was to study the dynamics of the solution in similarity variables, near the set of non-zero equilibria, which are obtained by a Lorentz transform of the spaceindependent solution. As a matter of fact, the main part of this paper is to study similar objects in higher dimensions. More precisely, near that set of equilibria, we show that solutions are either non-global or go to zero or converge to some explicit equilibrium. We also show that the first case cannot occur in the characteristic case and that only the third possibility occurs in the non-characteristic case, thanks to the non-degeneracy of the blow-up limit, another new result in our paper. As a by-product of our techniques, we obtain the stability of the zero solution. [ABSTRACT FROM AUTHOR]

This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(א0,D) of א0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters and thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by non-low theories. Assuming κ > א0 is measurable, we construct a regular ultrafilter on λ ≥ 2κ which is flexible but not good, and which moreover has large lcf(N0) but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving non-low does not imply maximal. Since flexible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, OK need not imply good, addressing a problem from Dow (1985). Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for D regular on κ and M a model of an unstable theory, Mκ/D is not (2κ)+- saturated; and for M a model of a non-simple theory and λ = λ<λ, Mλ/D is not λ++-saturated. In the third part of the paper, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on SOP2. We prove that if D is a κ-complete ultrafilter on κ, any ultrapower of a sufficiently saturated model of linear order will have no (κ, κ)-cuts, and that if D is also normal, it will have a (κ+, κ+)-cut. We apply this to prove that for any n < ω, assuming the existence of n measurable cardinals below λ, there is a regular ultrafilter D on λ such that any D-ultrapower of a model of linear order will have n alternations of cuts, as defined below. Moreover, D will λ+-saturate all stable theories but will not (2κ)+-saturate any unstable theory, where κ is the smallest measurable cardinal used in the construction. [ABSTRACT FROM AUTHOR]

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification. [ABSTRACT FROM AUTHOR]

Let G be a compact connected Lie group. The question of when a weighted Fourier algebra on G is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on G with the order of growth strictly bigger than half of the dimension of the group. The case of SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of the Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the quasihyperbolic metrics are quasiinvariant under a quasisymmetric mapping between two suitable metric spaces. Meanwhile, we also show that quasi-invariance of the quasihyperbolic metrics implies that the corresponding map is quasiconformal. At the end of this paper, as an application of these theorems, we prove that the composition of two quasisymmetric mappings in metric spaces is a quasiconformal mapping. [ABSTRACT FROM AUTHOR]

In a series of papers by the authors we introduced full quivers and pseudo-quivers of representations of algebras, and used them as tools in describing PI-varieties of algebras. In this paper we apply them to obtain a complete proof of Belov's solution of Specht's problem for affine algebras over an arbitrary Noetherian ring. The inductive step relies on a theorem that enables one to find a "q-characteristic coefficient-absorbing polynomial in each T-ideal I", i.e., a nonidentity of the representable algebra A arising from Γ, whose ideal of evaluations in A is closed under multiplication by q-powers of the characteristic coefficients of matrices corresponding to the generators of A, where q is a suitably large power of the order of the base field. The passage to an arbitrary Noetherian base ring C involves localizing at finitely many elements a kind of C, and reducing to the field case by a local-global principle. [ABSTRACT FROM AUTHOR]

In this paper we introduce Hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions. These topologies are strong duals of Frechet-Schwartz space topologies and even strong duals of nuclear Fréchet space topologies. In particular, any bounded set is relatively compact and one benefits from all deep properties of nuclearity. These algebras of generalized functions contain most of the classical irregular functions and distributions. They are obtained by replacing the mathematical tool of C∞ functions in the original version of nonlinear generalized functions by the far more evolved tool of holomorphic functions. This paper continues the nonlinear theory of generalized functions in which such locally convex topological properties were strongly lacking up to now. [ABSTRACT FROM AUTHOR]

We correct mistakes in a 2004 paper by the second author and report on recent new developments which settle cases left open in that paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation for certain random environments which are strongly mixing. [ABSTRACT FROM AUTHOR]

In a previous paper, we applied Alexandrov geometry methods to study infinite semiplanar graphs with nonnegative combinatorial curvature. We proved the weak relative volume comparison and the Poincare inequality on these graphs to obtain a dimension estimate for polynomial growth harmonic functions which is asymptotically quadratic in the growth rate. In the present paper, instead of using volume comparison on the graph, we translate the problem to a polygonal surface by filling polygons into the graph with edge lengths 1. This polygonal surface then is an Alexandrov space of nonnegative curvature. From a harmonic function on the graph, we construct a function on the polygonal surface that is not necessarily harmonic, but satisfies crucial estimates. Using the arguments on the polygonal surface, we obtain the optimal dimension estimate for polynomial growth harmonic functions on the graph which is linear in the growth rate. [ABSTRACT FROM AUTHOR]

The purpose of this paper and its prequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties. While the focus of the prequel is on how to work with toric stacks, the focus of this paper is how to show a stack is toric. For toric varieties, a classical result says that a finite type scheme with an action of a dense open torus arises from a fan if and only if it is normal and separated. In the same spirit, the main result of this paper is that any Artin stack with an action of a dense open torus arises from a stacky fan under reasonable hypotheses. [ABSTRACT FROM AUTHOR]

The purpose of this paper and its sequel is to introduce and develop a theory of toric stacks which encompasses and extends several notions of toric stacks defined in the literature, as well as classical toric varieties. In this paper, we define a toric stack as the stack quotient of a toric variety by a subgroup of its torus (we also define a generically stacky version). Any toric stack arises from a combinatorial gadget called a stacky fan. We develop a dictionary between the combinatorics of stacky fans and the geometry of toric stacks, stressing stacky phenomena such as canonical stacks and good moduli space morphisms. We also show that smooth toric stacks carry a moduli interpretation extending the usual moduli interpretations of Pn and [A¹/Gm]. Indeed, smooth toric stacks precisely solve moduli problems specified by (generalized) effective Cartier divisors with given linear relations and given intersection relations. Smooth toric stacks therefore form a natural closure to the class of moduli problems introduced for smooth toric varieties and smooth toric DM stacks in papers by Cox and Perroni, respectively. We include a plethora of examples to illustrate the general theory. We hope that this theory of toric stacks can serve as a companion to an introduction to stacks, in much the same way that toric varieties can serve as a companion to an introduction to schemes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation ..., where ( -- ?)s is the fractional Laplace operator, s ∈ (0, 1), O is an open bounded set of Rn, n > 2s, with Lipschitz boundary, ?> 0 is a real parameter and 2* = 2n/(n -- 2s) is a fractional critical Sobolev exponent. In this paper we first study the problem in a general framework; indeed we consider the equation ..., where LK is a general non-local integrodifferential operator of order s and f is a lower order perturbation of the critical power |u|2* -2u. In this setting we prove an existence result through variational techniques. Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if ?1,s is the first eigenvalue of the non-local operator (--?)s with homogeneous Dirichlet boundary datum, then for any X ∈ (0,?1,s) there exists a non-trivial solution of the above model equation, provided n = 4s. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators. [ABSTRACT FROM AUTHOR]

A universal geometric cluster algebra over an exchange matrix B is a universal object in the category of geometric cluster algebras over B related by coefficient specializations. (Following an earlier paper on universal geometric cluster algebras, we broaden the definition of geometric cluster algebras relative to the definition originally given by Fomin and Zelevinsky.) The universal objects are closely related to a fan FB called the mutation fan for B. In this paper, we consider universal geometric cluster algebras and mutation fans for cluster algebras arising from marked surfaces. We identify two crucial properties of marked surfaces: The Curve Separation Property and the Null Tangle Property. The latter property implies the former. We prove the Curve Separation Property for all marked surfaces except once-punctured surfaces without boundary components, and as a result we obtain a construction of the rational part of FB for these surfaces. We prove the Null Tangle Property for a smaller family of surfaces and use it to construct universal geometric coefficients for these surfaces. [ABSTRACT FROM AUTHOR]

For 2g - 2 + n > 0, the Teichmüller modular group Γg,n of a compact Riemann surface of genus g with n points removed Sg,n is the group of homotopy classes of diffeomorphisms of Sg,n which preserve the orientation of Sg,n and a given order of its punctures. Let Γg,n be the fundamental group of Sg,n, with a given base point, and ... its profinite completion. There is then a natural faithful representation Γg,n ↪ Out(...). The procongruence completion ... of the Teichmüller group is defined to be the closure of the Teichmüller group Γg,n inside the profinite group Out(...). In this paper, we begin a systematic study of the procongruence completion .... The set of profinite Dehn twists of ... is the closure, inside this group, of the set of Dehn twists of Γg,n. The main technical result of the paper is a parametrization of the set of profinite Dehn twists of ... and the subsequent description of their centralizers. This is the basis for the Grothendieck-Teichmüller Lego with procongruence Teichmüller groups as building blocks. As an application, in Section 7, we prove that some Galois representations associated to hyperbolic curves over number fields and their moduli spaces are faithful. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra - the Lie algebra VectTN of vector fields on an N-dimensional torus for N > 1. The case N = 1 gives a famous Virasoro algebra (or its centerless version - the Witt algebra). The algebra VectTN has a natural class of tensor modules parametrized by finitedimensional modules of glN. Tensor modules can be used in turn to construct bounded irreducible modules for VectTN+1 (induced from VectTN), which are the focus of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible Ω1 (TN+1) / dΩ0 (TN+1) ⋊ VectTN+1-modules constructed in a paper by the first author. These modules remain irreducible when restricted to the subalgebra VectTN+1, unless they belong to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob (1999). [ABSTRACT FROM AUTHOR]

Let A be a commutative Banach algebra with a BAI (=bounded approximate identity). We equip A** with the (first) Arens multiplication. To each idempotent element u of A** we associate the closed ideal Iu = {a ∊ A : au = 0} in A. In this paper we present a characterization of the closed ideals of A with BAI's in terms of idempotent elements of A**. The main results are: a) A closed ideal I of A has a BAI iff there is an idempotent u ∊ A** such that I = Iu and the subalgebra Au is norm closed in A**. b) For any closed ideal I of A with a BAI, the quotient algebra A/I is isomorphic to a subalgebra of A**. We also show that a weak* closed invariant subspace X of the group von Neumann algebra VN(G) of an amenable group G is naturally complemented in VN(G) iff the spectrum of X belongs to the closed coset ring Rc(Gd) of Gd, the discrete version of G. This paper contains several applications of these results. [ABSTRACT FROM AUTHOR]

In this paper we present a new family of non-compact properly embedded, self-shrinking, asymptotically conical, positive mean curvature ends Σn ⊆ Rn+1 that are hypersurfaces of revolution with circular boundaries. These hypersurface families interpolate between the plane and half-cylinder in Rn+1, and any rotationally symmetric self-shrinking non-compact end belongs to our family. The proofs involve the global analysis of a cubic-derivative quasi-linear ODE. We also prove the following classification result: a given complete, embedded, self-shrinking hypersurface of revolution Σn is either a hyperplane Rn, the round cylinder R × Sn-1 of radius √2(n - 1), the round sphere Sn v of radius √2n, or is diffeomorphic to an S1 × Sn-1 (i.e. a "doughnut" as in the paper by Sigurd B. Angenent, 1992, which when n = 2 is a torus). In particular, for self-shrinkers there is no direct analogue of the Delaunay unduloid family. The proof of the classification uses translation and rotation of pieces, replacing the method of moving planes in the absence of isometries. [ABSTRACT FROM AUTHOR]

We study the convexity of the area functional for the graphs of maps with respect to the singular values of their differentials. Suppose that f is a solution to the Dirichlet problem for the minimal surface system and the area functional is convex at f. Then the graph of f is stable. New criteria for the stability of minimal graphs in any co-dimension are derived in the paper by this method. Our results in particular generalize the co-dimension one case, and improve the condition in the 2003 paper of the first author and M.-T. Wang from ∣Λ² df∣ ≤ 1/p-1 to ∣ Λ² df∣ ≤ 1/√ p-1 , where p is an upper bound of the rank of df , and the condition in the 2008 paper of the first author and M.-T. Wang from √det(I + (df )T df ) ≤ 43/40 to √det(I + (df )T df ) ≤ 2. [ABSTRACT FROM AUTHOR]

We extend to the case of moving solitons, the result on asymptotic stability of ground states of the NLS obtained by the author in an earlier paper. For technical reasons we consider only smooth solutions. The proof is similar to the earlier paper. However now the flows required for the Darboux Theorem and the Birkhoff normal forms, instead of falling within the framework of standard theory of ODE's, are related to quasilinear hyperbolic symmetric systems. It is also not obvious that the Darboux Theorem can be applied, since we need to compare two symplectic forms in a neighborhood of the ground states not in H¹(R¹), but rather in the space S where also the variance is bounded. But the NLS does not preserve small neighborhoods of the ground states in S. [ABSTRACT FROM AUTHOR]

In his pioneering work, H. Garland constructed Eisenstein series on affine Kac-Moody groups over the field of real numbers. He established the almost everywhere convergence of these series, obtained a formula for their constant terms, and proved a functional equation for the constant terms. In his subsequent paper, the convergence of the Eisenstein series was obtained. In this paper, we define Eisenstein series on affine Kac-Moody groups over global function fields using an adelic approach. In the course of proving the convergence of these Eisenstein series, we also calculate a formula for the constant terms and prove their convergence and functional equations. [ABSTRACT FROM AUTHOR]

Finitely presented algebraic systems, such as groups and semigroups, are of foundational interest in algebra and computation. Finitely presented algebraic systems necessarily have a computably enumerable (c.e. for short) word equality problem and these systems are finitely generated. Call finitely generated algebraic systems with a c.e. word equality problem computably enumerable. Computably enumerable finitely generated algebraic systems are not necessarily finitely presented. This paper is concerned with finding finitely presented expansions of finitely generated c.e. algebraic systems. The method of expansions of algebraic systems, such as turning groups into rings or distinguishing elements in the underlying algebraic systems, is an important method used in algebra, model theory, and in various areas of theoretical computer science. Bergstra and Tucker proved that all c.e. algebraic systems with decidable word problem possess finitely presented expansions. Then they, and, independently, Goncharov asked if every finitely generated c.e. algebraic system has a finitely presented expansion. In this paper we build examples of finitely generated c.e. semigroups, groups, and algebras that fail to possess finitely presented expansions, thus answering the question of Bergstra- Tucker and Goncharov for the classes of semigroups, groups and algebras. We also construct an example of a residually finite, infinite, and algorithmically finite group, thus answering the question of Miasnikov and Osin. Our constructions are based on the interplay between key concepts and known results from computability theory (such as simple and immune sets) and algebra (such as residual finiteness and the theorem of Golod-Shafaverevich). [ABSTRACT FROM AUTHOR]

In the first part of this article, we complete the program announced in the preliminary note by the author and Malchiodi by proving a conjecture presented in a previous paper that states the equivalence of contractibility and p1-stability for generalized spaces of formal barycenters σρ,ɑ and hence we get purely algebraic conditions for the solvability of the singular Liouville equation on Riemann surfaces. This relies on a structure decomposition theorem for σρ,ɑ in terms of maximal strata, and on elementary combinatorial arguments based on the selection rules that define such spaces. Moreover, we also show that these solvability conditions on the parameters are not only sufficient, but also necessary, at least when for some i ε {1, . . .,m} the value ɑi approaches -1. This disproves a conjecture made in Section 3 of a paper by Tarantello and gives the first non-existence result for this class of PDEs without any genus restriction. The argument we present is based on a combined use of the maximum/comparison principle and of a Pohozaev type identity and applies for an arbitrary choice both of the underlying metric g of σ and of the datum h. [ABSTRACT FROM AUTHOR]

In this paper the stochastic version of the Evans-Aronsson problem is studied. Both for the mechanical case and two dimensional problems we prove the existence of smooth solutions. We establish that the corresponding effective Lagrangian and Hamiltonian are smooth. We study the limiting behavior and the convergence of the effective Lagrangian and Hamiltonian, Mather measures and minimizers. We end the paper with applications to stationary mean-field games. [ABSTRACT FROM AUTHOR]

The primary theorem of this paper concerns the Poincaré (Hilbert) series for the cohomology ring of a finite group G with coefficients in a prime field of characteristic p. This theorem is proved using the ideas of equivariant cohomology whereby one considers more generally the cohomology ring of the Borel construction H*(EG ×G X), where X is a manifold on which G acts. This work results in a formula that computes the "degree" of the Poincaré series in terms of corresponding degrees of certain subgroups of the group G. In this paper, we discuss the theorem and the method of proof. [ABSTRACT FROM AUTHOR]

In this paper, we extend the method in a recent paper of Tian and Zhu to study the energy level L(·) of Perelman's entropy ?(·) for the Kähler-Ricci flow on a Fano manifold M. We prove that L(·) is independent of the initial metric of the Kähler-Ricci flow under an assumption that the modified Mabuchi's K-energy is bounded from below on M. As an application of the above result, we give an alternative proof to the main theorem about the convergence of Kähler-Ricci flow found in a 2007 paper by Tian and Zhu. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a unitary invariant G : [B(H)n]1-? N8 × N8 × N8, N8:= N ? {8}, defined in terms of the characteristic function TT, the noncommutative Poisson kernel KT, and the defect operator ?T associated with T ∈ [B(H)n] - 1 . We show that the map G detects the pure row isometries in the closed unit ball of B(H)n and completely classify them up to a unitary equivalence. We also show that G detects the pure row contractions with polynomial characteristic functions and completely noncoisometric row contractions, while the pair (G,TT ) is a complete unitary invariant for these classes of row contractions. The unitary invariant G is extracted from the theory of characteristic functions and noncommutative Poisson transforms, and from the geometric structure of row contractions with polynomial characteristic functions which are studied in this paper. As an application, we characterize the row contractions with constant characteristic function. In particular, we show that any completely noncoisometric row contraction T with constant characteristic function is homogeneous, i.e., T is unitarily equivalent to ?(T) for any free holomorphic automorphism ? of the unit ball of B(H)n. Under a natural topology, we prove that the free holomorphic automorphism group Aut(B(H)n1 ) is a metrizable, s-compact, locally compact group, and provide a concrete unitary projective representation of it in terms of noncommutative Poisson kernels. [ABSTRACT FROM AUTHOR]

We continue our previous study of improved Hardy, Rellich and uncertainty principle inequalities on a Riemannian manifold M, started in our earlier paper from 2009. In the present paper we prove new weighted Hardy-Poincar'e, Rellich type inequalities as well as an improved version of our uncertainty principle inequalities on a Riemannian manifold M. In particular, we obtain sharp constants for these inequalities on the hyperbolic space Hn. [ABSTRACT FROM AUTHOR]

This paper concerns homotopically stratified spaces. These were defined by Frank Quinn in his paper Homotopically Stratified Sets. His definition of stratified space is very general and relates strata by "homotopy rather than geometric conditions". This makes homotopically stratified spaces the ideal class of stratified spaces on which to define and study stratified homotopy theory. We will define stratified analogues of the usual definitions of maps, homotopies and homotopy equivalences. Then we will provide an elementary criterion for deciding when a strongly stratified map is a stratified homotopy equivalence. This criterion states that a strongly stratified map is a stratified homotopy equivalence if and only if the induced maps on strata and holink spaces are homotopy equivalences. Using this criterion we will prove that any homotopically stratified space is stratified homotopy equivalent to a homotopically stratified space where neighborhoods of strata are mapping cylinders. Finally, we will develop categorical descriptions of the class of homotopically stratified spaces up to stratified homotopy. [ABSTRACT FROM AUTHOR]

In this paper we deal with an invariant ergodic hyperbolic measure μ for a diffeomorphism f, assuming that f is either C1+α or C1 and the Oseledec splitting of μ is dominated. We show that this system (f,μ) satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers. Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincaré recurrence, removing the assumption of the non-uniform specification property in the main theorem of "Recurrence and Lyapunov exponents" by Saussol, Troubetzkoy and Vaienti that establishes an inequality between Lyapunov exponents and local recurrence properties. Another consequence is the fact that any such measure is the weak limit of averages of Dirac measures at periodic points, as in a paper by Sigmund. One can show that the topological pressure can be calculated by considering the convenient weighted sums on periodic points whenever the dynamic is positive expansive and every measure with pressure close to the topological pressure is hyperbolic. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive algebraic group over an algebraically closed field k. In a recent paper, Bate, Martin, Röhrle and Tange show that every (smooth) subgroup of G is separable provided that the characteristic of k is very good for G. Here separability of a subgroup means that its scheme-theoretic centralizer in G is smooth. Serre suggested extending this result to arbitrary, possibly non-smooth, subgroup schemes of G. The aim of this paper is to prove this more general result. Moreover, we provide a condition on the characteristic of k that is necessary and sufficient for the smoothness of all centralizers in G. We finally relate this condition to other standard hypotheses on connected reductive groups. [ABSTRACT FROM AUTHOR]

By a classical heuristics, lacunary function systems exhibit many asymptotic properties which are typical for systems of independent random variables. For example, for a large class of functions f the system (f(nkx))k=1, where (nk)k=1 is a lacunary sequence of integers, satisfies a law of the iterated logarithm (LIL) of the form ... where c1, c2 are appropriate positive constants. In a previous paper we gave a criterion, formulated in terms of the number of solutions of certain linear Diophantine equations, which guarantees that the value of the lim sup in (1) equals the L²2-norm of f for a.e. x, which is exactly what one would also expect in the case of i.i.d. random variables. This result can be used to prove a precise LIL for the discrepancy of (nkx)k=1, which corresponds to the Chung-Smirnov LIL for the Kolmogorov-Smirnov-statistic of i.i.d. random variables. In the present paper we give a full solution of the problem in the case of "stationary" Diophantine behavior, by this means providing a unifying explanation of the aforementioned "regular" LIL behavior and the "irregular" LIL behavior which has been observed by Kac, Erdős, Fortet and others. [ABSTRACT FROM AUTHOR]

In this paper we prove (integral) homological stability for the sequences of spaces C+n (M,X). These are the spaces of configurations of n points in a connected manifold of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an orientation -- an ordering of the points up to even permutations. They are double covers of the unordered configuration spaces Cn(M,X), and indeed to prove our result we adapt methods from a paper of Randal- Williams, which proves homological stability in the unordered case. Interestingly the oriented configuration spaces stabilise more slowly than the unordered ones: the stability slope we obtain is ⅓, compared to ½ in the unordered case (and these are the best possible slopes in their respective cases). This result can also be interpreted as homological stability for the unordered configuration spaces with certain twisted Z ⊕ Z-coefficients. [ABSTRACT FROM AUTHOR]

In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed Fibonacci conjecture by M. Niebrzydowski and J. H. Przytycki from their 2009 paper. Further, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Finally, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle. [ABSTRACT FROM AUTHOR]

Unimodular fans are central to toric algebraic geometry, where they correspond to non-singular toric varieties. The Schauder bases mentioned in the title may be described as the standard bases of the free Z-module of support functions (=invariant Cartier divisors) of a unimodular (a.k.a. regular) fan. An abstract, purely algebraic version of Schauder bases was investigated in the first part of the present paper, with motivations coming from the theory of lattice-ordered Abelian groups. The main result obtained there is that such abstract Schauder bases can be characterised in terms of the maximal spectral space of lattice-ordered Abelian groups, with no reference to polyhedral geometry. The results in the present paper will show that abstract Schauder bases can in fact be characterised in the language of lattice-ordered groups by means of an elementary algebraic notion that we call regularity, with no reference to either maximal spectral spaces or to polyhedral geometry. We prove that finitely generated projective lattice-ordered Abelian groups are precisely the lattice-ordered Abelian groups that have a finite, regular set of positive generators. This theorem complements Beynon's well-known 1977 result that the finitely generated projective lattice-ordered Abelian groups are precisely the finitely presented ones; and the core of the proof consists in showing that finite, regular sets of positive generators are the same thing as abstract Schauder bases. We give three applications of the main result. First, we establish a necessary and sufficient criterion for the lattice-group isomorphism of two lattice-ordered Abelian groups with finite, regular sets of positive generators. Next, we classify in elementary terms (i.e. without reference to spectral spaces) all finitely generated projective lattice-ordered Abelian groups whose maximal spectrum is a closed topological surface. Finally, we show how to explicitly construct Z-module bases of any finitely generated projective lattice-ordered Abelian group. [ABSTRACT FROM AUTHOR]

Strongly positive discrete series represent a particularly important class of irreducible square-integrable representations of p-adic groups. Indeed, these representations are used as basic building blocks in known constructions of general discrete series. In this paper, we explicitly describe Jacquet modules of strongly positive discrete series. The obtained description of Jacquet modules, which relies on the classification of strongly positive discrete series given in our earlier paper on metaplectic groups, is valid in both the classical and the metaplectic cases. We expect that our results, besides being interesting by themselves, should be relevant to some potential applications in the theory of automorphic forms, where both representations of metaplectic groups and the structure of Jacquet modules play an important part. [ABSTRACT FROM AUTHOR]

We formulate the measure analogue of generically stable types in first order theories with NIP (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and p-adic fields. [ABSTRACT FROM AUTHOR]

Regular polygonal complexes in euclidean 3-space E³ are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The present paper and its predecessor describe a complete classification of regular polygonal complexes in E³. In Part I we established basic structural results for the symmetry groups, discussed operations on their generators, characterized the complexes with face mirrors as the 2-skeletons of the regular 4-apeirotopes in E³, and fully enumerated the simply flag-transitive complexes with mirror vector (1, 2). In this paper, we complete the enumeration of all regular polygonal complexes in E³ and in particular describe the simply flagtransitive complexes for the remaining mirror vectors. It is found that, up to similarity, there are precisely 25 regular polygonal complexes which are not regular polyhedra, namely 21 simply flag-transitive complexes and 4 complexes which are 2-skeletons of regular 4-apeirotopes in E³. [ABSTRACT FROM AUTHOR]

Extending the methods developed in the author's recent paper and using some techniques from a paper by Sogge and Stein in conjunction with various facts about adapted coordinate systems in two variables, an Lp boundedness theorem is proven for maximal operators over hypersurfaces in R³ when p > 2. When the best possible p is greater than 2, the theorem typically provides sharp estimates. This gives another approach to the subject of recent work of Ikromov, Kempe, and Müller (2010). [ABSTRACT FROM AUTHOR]

In this paper a family of linear coagulation models is solved. These models arise in the analysis of the asymptotic behaviour of coagulation equations yielding gelation for large particles. The tools and techniques that are developed in this paper are based on the definition of a class of weighted Sobolev spaces that take into account the characteristic time scales associated to the coagulation equation for large particles, as well as in the continuation argument introduced by Schauder to prove well posedness of general classes of elliptic and parabolic equations. [ABSTRACT FROM AUTHOR]

Iterated function systems have been most extensively studied when the functions are affine transformations of Euclidean space and, more recently, projective transformations on real projective space. This paper investigates iterated function systems consisting of Möbius transformations on the extended complex plane or, equivalently, on the Riemann sphere. The main result is a characterization, in terms of topological, geometric, and dynamical properties, of Möbius iterated function systems that possess an attractor. The paper also includes results on the duality between the attractor and repeller of a Möbius iterated function system. [ABSTRACT FROM AUTHOR]