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It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coefficients of order of the Knudsen number ε >0. In the paper, we carry out the rigorous mathematical analysis of the compressible Navier-Stokes approximation for the Boltzmann equation regarding the initial-boundary value problems in general bounded domains. The main goal is to measure the uniform-in-time deviation of the Boltzmann solution with diffusive reflection boundary condition from a local Maxwellian with its fluid quantities given by the solutions to the corresponding compressible Navier-Stokes equations with consistent non-slip boundary conditions whenever ε > 0 is small enough. Specifically, it is shown that for well chosen initial data around constant equilibrium states, the deviation weighted by a velocity function is O(ε1/2) in L∞x,v and O(ε3/2) in L2x,v globally in time. The proof is based on the uniform estimates for the remainder in different functional spaces without any spatial regularity. One key step is to obtain the global-in-time existence as well as uniform-in-ε estimates for regular solutions to the full compressible Navier-Stokes equations in bounded domains when the parameter ε > 0 is involved in the analysis. [ABSTRACT FROM AUTHOR]

In the first author's Math. Res. Lett. paper (2014), the quasi-Frobenius-Lusztig kernel associated with 5/2 was constructed. In this paper we construct the quasi-Frobenius-Lusztig kernels associated with any simple Lie algebra g. [ABSTRACT FROM AUTHOR]

This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on C*-algebras satisfying a certain UHF-absorption condition. We show that given any finite group G and a separable, unital C*-algebra A that absorbs M|G|∞ tensorially, one can lift any group homomorphism G → Aut(A)/≈u to an honest Rokhlin action γ of G on A. Unitality may be dropped in favour of stable rank one or being stable. If A belongs to a certain class of C*-algebras that is classifiable by a suitable invariant (e.g. K-theory), then in fact every G-action on the invariant lifts to a Rokhlin action of G on A. For the crossed product C*-algebra A ⋊γ G of a Rokhlin action on a UHF-absorbing C*-algebra, an inductive limit decomposition is obtained in terms of A and γ. If G is assumed to be abelian, then the dual action ˆγ is locally representable in a very strong sense. We then show how some well-known constructions of finite group actions with certain predescribed properties can be recovered and extended by the main results of this paper when paired with known classification theorems. Among these is Blackadar's famous construction of symmetries on the CAR algebra whose fixed point algebras have non-trivial K1-groups. Lastly, we use the results of this paper to reduce the UCT problem for separable, nuclear C*-algebras to a question about certain finite group actions on O2. [ABSTRACT FROM AUTHOR]

In this paper we study various "abscissae" which one can associate to a given function f, or rather to the power moments of f. These are motivated by long-standing open problems in analytic number theory. We show how these abscissae connect to the distribution of values of f in a very elegant way using convex conjugates. This connection allows us to show which abscissae are realizable for both general and more specific arithmetical functions. Further it may give a new approach to, for example, Dirichlet's divisor problem. [ABSTRACT FROM AUTHOR]

In this paper we introduce new densities on the set of primes of a number field. If K/K0 is a Galois extension of number fields, we associate to any element x ∈ ... a density δK/K0,x on the primes of K. In particular, the density associated to x = 1 is the usual Dirichlet density on K. We also give two applications of these densities (for x ≠ 1): the first is a realization result à la the Grunwald-Wang theorem such that essentially, ramification is only allowed in a set of arbitrarily small (positive) Dirichlet density. The second concerns the so-called saturated sets of primes, introduced by Wingberg. [ABSTRACT FROM AUTHOR]

For a finite group G and an arbitrary commutative ring R, Broué has placed a Frobenius exact structure on the category of finitely generated RG-modules by taking the exact sequences to be those that split upon restriction to the trivial subgroup. The corresponding stable category is then tensor triangulated. In this paper we examine the case R = S/tn, where S is a discrete valuation ring having uniformising parameter t. We prove that the prime ideal spectrum (in the sense of Balmer) of this 'relative' version of the stable module category of RG is a disjoint union of n copies of that for kG, where k is the residue field of S. [ABSTRACT FROM AUTHOR]

We show that general triple planes with genus and irregularity zero belong to at most 12 families, that we call surfaces of type I to XII, and we prove that the corresponding Tschirnhausen bundle is a direct sum of two line bundles in cases I, II, III, whereas it is a rank 2 Steiner bundle in the remaining cases. We also provide existence results and explicit descriptions for surfaces of type I to VII, recovering all classical examples and discovering several new ones. In particular, triple planes of type VII provide counterexamples to a wrong claim made in 1942 by Bronowski. Finally, in the last part of the paper we discuss some moduli problems related to our constructions. [ABSTRACT FROM AUTHOR]

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of R(ζ, q), where R(z, q) is the two-variable generating function of Dyson's rank function and ζ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon, and McIntosh, and motivated by Dyson's question, Bringmann, Ono, and Rhoades studied transformation properties of R(ζ, q). In this paper we strengthen and extend the results of Bringmann, Rhoades, and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo 5 from the Lost Notebook has an analogue for all primes greater than 3. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod 7. [ABSTRACT FROM AUTHOR]

In this paper, we derive a function field version of theWaldspurger formula for the central critical values of the Rankin-Selberg L-functions. This formula states that the central critical L-values in question can be expressed as the "ratio" of the global toric period integral to the product of the local toric period integrals. Consequently, this result provides a necessary and sufficient criterion for the non-vanishing of these central critical L-values, and supports the Gross-Prasad conjecture for SO(3) over function fields. [ABSTRACT FROM AUTHOR]

Let μ be a Borel measure on a compactum X. The main objects in this paper are s-ideals Ipdimq, J0pμq, Jf pμq of Borel sets in X that can be covered by countably many compacta which are finite-dimensional, or of μ-measure null, or of finite μ-measure, respectively. Answering a question of J. Zapletal, we shall show that for the Hilbert cube, the s-ideal Ipdimq is not homogeneous in a strong way. We shall also show that in some natural instances of measures μ with nonhomogeneous s-ideals J0pμq or Jf pμq, the completions of the quotient Boolean algebras BorelpXq{J0pμq or BorelpXq{Jf pμq may be homogeneous. We discuss the topic in a more general setting, involving calibrated s-ideals. [ABSTRACT FROM AUTHOR]

In the 1960s Maurice Auslander proved the following important result. Let R be the commutative polynomial ring C[x1, . . ., xn], and let G be a finite small subgroup of GLn(C) acting on R naturally. Let A be the fixed subring RG:= {a ∈ R|g(a) = a for all g ∈ G}. Then the endomorphism ring of the right A-module RA is naturally isomorphic to the skew group algebra R* G. In this paper, a version of the Auslander theorem is proven for the following classes of noncommutative algebras: (a) noetherian PI local (or connected graded) algebras of finite injective dimension, (b) universal enveloping algebras of finite-dimensional Lie algebras, and (c) noetherian graded down-up algebras. [ABSTRACT FROM AUTHOR]

A quasi-infinitely divisible distribution on R is a probability distribution whose characteristic function allows a L'evy-Khintchine type representation with a "signed L'evy measure", rather than a L'evy measure. Quasiinfinitely divisible distributions appear naturally in the factorization of infinitely divisible distributions. Namely, a distribution μ is quasi-infinitely divisible if and only if there are two infinitely divisible distributions μ1 and μ2 such that μ1 * μ = μ2. The present paper studies certain properties of quasiinfinitely divisible distributions in terms of their characteristic triplet, such as properties of supports, finiteness of moments, continuity properties, and weak convergence, with various examples constructed. In particular, it is shown that the set of quasi-infinitely divisible distributions is dense in the set of all probability distributions with respect to weak convergence. Further, it is proved that a distribution concentrated on the integers is quasi-infinitely divisible if and only if its characteristic function does not have zeroes, with the use of the Wiener-L'evy theorem on absolutely convergent Fourier series. A number of fine properties of such distributions are proved based on this fact. A similar characterization is not true for nonlattice probability distributions on the line. [ABSTRACT FROM AUTHOR]

In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in Rn: ... for any nonnegative functions ... and ... such that .... We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the ndimensional sphere Sn by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities. [ABSTRACT FROM AUTHOR]

In Theorem 3.1 of The rigidity theorems of self-shrinkers (2014), the author and Y. L. Xin proved a rigidity result for self-shrinkers under the integral condition on the norm of the second fundamental form. In this paper, we relax such a bound to any finite constant (see Theorem 4.4 for details). [ABSTRACT FROM AUTHOR]

In this paper we first construct a two-variable p-adic L-function for the standard representation associated with a Hida family of parallel weight genus g Siegel forms, using a method developed by Böcherer-Schmidt in one variable. When a form f has weight g + 1 a non-crystalline trivial zero could appear. In this case, using the two-variable p-adic L-function we have constructed, we can apply the method of Greenberg-Stevens to calculate the first derivative of the p-adic L-function for f and show that it has the form predicted by a conjecture of Greenberg on trivial zeros. [ABSTRACT FROM AUTHOR]

In [J. Inst. Math. Jussieu 14 (2015), 149-184] and [Int. Math. Res. Not. IMRN 7 (2017), 2014-2099] a family of Rankin-Selberg integrals was shown to represent the twisted standard L-function L(s, π, χ, st) of a cuspidal representation π of the exceptional group of type G2. These integral representations bind the analytic behavior of this L-function with that of a family of degenerate Eisenstein series for quasi-split forms of Spin8 associated to an induction from a character on the Heisenberg parabolic subgroup. This paper is divided into two parts. In Part 1 we study the poles of these degenerate Eisenstein series in the right half-plane Re(s) > 0. In Part 2 we use the results of Part 1 to prove the conjecture, made by J. Hundley and D. Ginzburg in [Israel J. Math. 207 (2015), 835-879], for stable poles and also to give a criterion for π to be a CAP representation with respect to the Borel subgroup of G2 in terms of the analytic behavior of L(s, π, χ, st) at s = 3/2. [ABSTRACT FROM AUTHOR]

Let G be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic p = 0. A closed subgroup H of G is called G-completely reducible (G-cr) if whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi subgroup of P. In this paper we determine the G-conjugacy classes of non-G-cr simple connected subgroups of G when p is good for G. For each such subgroup X, we determine the action of X on the adjoint module L(G) and the connected centraliser of X in G. As a consequence we classify all non-G-cr connected reductive subgroups of G, and determine their connected centralisers. We also classify the subgroups of G which are maximal among connected reductive subgroups, but not maximal among all connected subgroups. [ABSTRACT FROM AUTHOR]

In this paper we study some determinant inequalities and matrix inequalities which have a geometrical flavour. We first examine some inequalities which place work of Macbeath in a more general setting and also relate to recent work of Gressman. In particular, we establish optimisers for these determinant inequalities. We then use these inequalities to establish our Main Theorem, which gives a geometric inequality of matrix type which improves and extends some inequalities of Christ. [ABSTRACT FROM AUTHOR]

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al. and Shalev. The conjecture asserts that there is an absolute positive constant δ such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than δ. We show that with possibly finitely many exceptions, one can take δ = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements tends to 1 as n increases and we prove similar results for families of permutation representations. [ABSTRACT FROM AUTHOR]

This paper investigates lower bounds on the number of zeros and poles of a general Dirichlet series in a disk of radius r and gives, as a consequence, an affirmative answer to an open problem of Bombieri and Perelli on the bound. Applications will also be given to Picard type theorems, global estimates on the symmetric difference of zeros, and uniqueness problems for Dirichlet series. [ABSTRACT FROM AUTHOR]

We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. Next we study in arbitrary rings those elements whose powers are regular, and provide a method for constructing inner inverses which satisfy many additional strong relations. As a corollary we show that if each of the powers a, a², ..., an is a regular element in some ring R (for some n ≥ 1), then there exists w ∈ R such that akwkak = ak and wkakwk = wk for 1 ≤ k ≤ n. Similar statements are also obtained for unit-regular elements. The paper ends with a large number of examples elucidating further connections (and disconnections) between cleanness, regularity, and unit-regularity. [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]

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]

This paper gives two results on the simple modules for the Brauer algebra Bn(d) over the complex field. First we describe the module structure of the restriction of all simple Bn(d)-modules to Bn-1(d). Second we give a new geometrical interpretation of Ram and Wenzl's construction of bases for "d-permissible" simple modules. [ABSTRACT FROM AUTHOR]

This paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that nongeometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces. [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]

Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness at finite and infinite minimal Martin boundary points for a large class of purely discontinuous symmetric Lévy processes. [ABSTRACT FROM AUTHOR]

This paper begins the study of relations between Riemannian geometry and contact topology on (2n + 1)-manifolds and continues this study on 3-manifolds. Specifically we provide a lower bound for the radius of a geodesic ball in a contact (2n+ 1)-manifold (M, ξ) that can be embedded in the standard contact structure on R2n+1, that is, on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form a for ξ. In dimension 3, this further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curve techniques to provide a lower bound for the radius of a PS-tight ball. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear Schrödinger equation iut + Δu = λ|u| 4/N-4u in RN, N = 5, with λ ∈ C. We prove local well-posedness (local existence, unconditional uniqueness, continuous dependence) in the critical space H²(RN). [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three- dimensional generic shrinking Ricci soliton is given by quotients of either S³, RS² or R³ under some very weak conditions on the vector field X generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the X-Laplacian on every generic Ricci soliton without any assumption on X. [ABSTRACT FROM AUTHOR]

This paper contains the construction, examples and properties of a trace and a trace pairing for certain morphisms in a monoidal category with switching isomorphisms. Our construction of the categorical trace is a common generalization of the trace for endomorphisms of dualizable objects in a balanced monoidal category and the trace of nuclear operators on a topological vector space with the approximation property. In a forthcoming paper, applications to the partition function of super-symmetric field theories will be given. [ABSTRACT FROM AUTHOR]

In this paper we study entire solutions of the Allen-Cahn equation Δu - F'(u) = 0, where F is an even, bistable function. We are particularly interested in the description of the moduli space of solutions which have some special structure at infinity. The solutions we are interested in have their zero set asymptotic to 2k, k&#8805 2 oriented affine half-lines at infinity and, along each of these affine half-lines, the solutions are asymptotic to the one-dimensional heteroclinic solution: such solutions are called multiple-end solutions, and their set is denoted by MM2k. The main result of our paper states that if u ∈ M2k is nondegenerate, then locally near u the set of solutions is a smooth manifold of dimension 2k. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen-Cahn equation in dimension 2, for k ≥ 2. [ABSTRACT FROM AUTHOR]

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schützenberger) could be used to define a new basis for the ring of quasisymmetric functions we call ``Quasisymmetric Schur functions'' (QS functions for short). In this paper we develop the combinatorics of these polynomials further, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coefficients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis. [ABSTRACT FROM AUTHOR]

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group $ W$ $ \pi_\downarrow^c$ to the unique maximal $ c$ $ \pi_\downarrow^c$-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of $ \pi_\downarrow^c$ [ABSTRACT FROM AUTHOR]

We correct two errors in the statement and proof of a theorem in an earlier paper (2007), and at the same time extend that result to a more general theorem about extensions of $p$-local finite groups. Other special cases of this theorem have already been shown in two later papers, so we feel it will be useful to have this more general result in the literature. [ABSTRACT FROM AUTHOR]

In an upcoming paper by Lee and Wang, we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak formulation of the mean curvature flow. Moreover, there is no mass loss along the Brakke flow. In this paper, we generalize these results to higher dimensions. We construct new higher-dimensional Hamiltonian stationary cones of different topology as generalizations of the Schoen-Wolfson cones. Hamiltonian stationary self-shrinkers and self-expanders that are asymptotic to these Hamiltonian stationary cones are constructed as well. They can also be glued together to produce eternal solutions of the Brakke flow without mass loss. Finally, we show that the same conclusion holds for those Lagrangian self-similar examples recently found by Joyce, Tsui and the first author. [ABSTRACT FROM AUTHOR]

Let $Omega subset mathbb {R}^N$ be a compact imbedded Riemannian manifold of dimension~$dge 1$ and define the $(d+1)$-dimensional Riemannian manifold $mathcal {M}:={(x,r(x)omega ) : xin mathbb {R}, omega in Omega }$ with $r>0$ and smooth, and the natural metric $ds^2=(1+r'(x)^2)dx^2+r^2(x)ds_Omega ^2$. We require that $mathcal {M}$ has conical ends: $r(x)=|x| + O(x^{-1})$ as $xto pm infty $. The Hamiltonian flow on such manifolds always exhibits trapping. Dispersive estimates for the Schrödinger evolution $e^{itDelta _mathcal {M}}$ and the wave evolution $e^{itsqrt {-Delta _mathcal {M}}}$ are obtained for data of the form $f(x,omega )=Y_n(omega ) u(x)$, where $Y_n$ are eigenfunctions of~$Delta _Omega $. This paper treats the case $d=1$, $Y_0=1$. In Part II of this paper we provide details for all cases $d+n>1$. Our method combines two main ingredients: par (A) A detailed scattering analysis of Schrödinger operators of the form $-partial _xi ^2 + V(xi )$ on the line where $V(xi )$ has inverse square behavior at infinity. par (B) Estimation of oscillatory integrals by (non)stationary phase. [ABSTRACT FROM AUTHOR]

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation $mathcal {F}$ on a closed manifold $M$. The essential transverse category $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is introduced in this paper, and we prove that $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category $operatorname {cat}_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is finite, and thus $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F}) = operatorname {cat}_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ holds. par A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ into a standard problem about $mathbf O(q)$-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated $mathbf O(q)$-manifold $widehat W$, we have that $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F}) = operatorname {cat}_{mathbf O(q)}(widehat W)$. Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. par A generalization of the Lusternik-Schnirelmann theorem is derived: given a $C^1$-function $f colon M to mathbb {R}$ which is constant along the leaves of a Riemannian foliation $mathcal {F}$, the essential transverse category $operatorname {cat}^e_{mathbin {cap {mkern -9mu}mid } }(M,mathcal {F})$ is a lower bound for the number of critical leaf closures of $f$. [ABSTRACT FROM AUTHOR]

A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen (2000, 2007) that a connection exists between discrete Painlevé equations and (finite order) Nevanlinna theory is an estimate of the integrated average of $log ^+|f(z+1)/f(z)|$ on $|z|=r$. We obtained essentially the same estimate in our previous paper (2008) independent of Halburd et al. (2006). We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order less than and greater than one. We have established a ``difference'' analogue of the classical Wiman-Valiron type estimates for meromorphic functions with order less than one, which allow us to prove that all entire solutions of linear difference equations (with polynomial coefficients) of order less than one must have positive rational order of growth. We have also established that any entire solution to a first order algebraic difference equation (with polynomial coefficients) must have a positive order of growth, which is a ``difference'' analogue of a classical result of Pólya. [ABSTRACT FROM AUTHOR]

By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in $mathbb {R}^d$. In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear, this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like {bf 0}-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on {bf 0}-symmetric convex surfaces in $mathbb {R}^d$ can be approximated by sums of 2 homogeneous polynomials. This conjecture has not yet been resolved, but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) $d=2$; 2) convex surfaces in $mathbb {R}^d$ with $C^{1窛⊖ }$ boundary. [ABSTRACT FROM AUTHOR]

In this paper, as a main result, we derive a Chung-Fuchs type condition for the recurrence of Feller processes associated with pseudo-differential operators. In the Lévy process case, this condition reduces to the classical and well-known Chung-Fuchs condition. Further, we also discuss the recurrence and transience of Feller processes with respect to the dimension of the state space and Pruitt indices and the recurrence and transience of Feller-Dynkin diffusions and stable-like processes. Finally, in the one-dimensional symmetric case, we study perturbations of Feller processes which do not affect their recurrence and transience properties, and we derive sufficient conditions for their recurrence and transience in terms of the corresponding Lévy measure. In addition, some comparison conditions for recurrence and transience also in terms of the Lévy measures are obtained. [ABSTRACT FROM AUTHOR]

A geometric categorification is given for arbitrary-large-finitedimensional quotients of quantum osp(1/2) and tensor products of its simple modules. The modified quantum osp(1/2) of Clark-Wang, a new version in this paper and the modified quantum sl(2) are shown to be isomorphic to each other over a field containing Q(v) and √-1. [ABSTRACT FROM AUTHOR]

Let $G$ be a reductive $ \mathfrak{p}$-adic group. In his paper, ``The Plancherel Formula for Reductive $\mathfrak{p}$-adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for $G$ and sketched a proof of the Plancherel theorem for $G$. One step in the proof, stated as Theorem 11 in Harish-Chandra's paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem. [ABSTRACT FROM AUTHOR]

In this paper, by combining modularity of the Witten genus and the modular forms constructed by Liu and Wang, we establish mod 3 congruence properties of certain twisted signatures of 24 dimensional string manifolds. [ABSTRACT FROM AUTHOR]

This paper concerns the first passage times to a point a > 0, denoted by σa, of Bessel processes. We are interested in the case when the process starts at x > a and we compute the densities of the distributions of aa to obtain the exact asymptotic forms of them as t → < ∞ that are valid uniformly in x > a for every order of the Bessel process. [ABSTRACT FROM AUTHOR]

Let F be a quadratic form in N ≥ 2 variables defined on a vector space V ⊆ KN over a global field K, and Ƶ ⊆ KN be a finite union of varieties defined by families of homogeneous polynomials over K. We show that if V \ Ƶ contains a nontrivial zero of F, then there exists a linearly independent collection of small-height zeros of F in V \ Ƶ, where the height bound does not depend on the height of Ƶ, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace W of the quadratic space (V, F) such that W is not contained in Ƶ. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit. [ABSTRACT FROM AUTHOR]

In 2007 Phillips and Weaver showed that, assuming the Continuum Hypothesis, there exists an outer automorphism of the Calkin algebra. (The Calkin algebra is the algebra of bounded operators on a separable complex Hilbert space, modulo the compact operators.) In this paper we establish that the analogous conclusion holds for a broad family of quotient algebras. Specifically, we will show that assuming the Continuum Hypothesis, if A is a separable algebra which is either simple or stable, then the corona of A has nontrivial automorphisms. We also discuss a connection with cohomology theory, namely, that our proof can be viewed as a computation of the cardinality of a particular derived inverse limit. [ABSTRACT FROM AUTHOR]

The object of study in this paper is the on-shell scattering matrix S(E) of the Schrödinger operator with the potential satisfying assumptions typical in the theory of shape resonances. We study the spectrum of S(E) in the semiclassical limit when the energy parameter E varies from Eres - e to Eres + e, where Eres is a real part of a resonance and e is sufficiently small. The main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle. This result is closely related to the Breit-Wigner effect. [ABSTRACT FROM AUTHOR]

In this paper, we show that the chordal Loewner differential equation with C° driving function generates a C°1/2 slit for 1/2 < ° ≼ 2, except when ° = 3/2 the slit is only proved to be weakly C1,1. [ABSTRACT FROM AUTHOR]

Let A be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that A0 is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described. [ABSTRACT FROM AUTHOR]