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We give the complete classification of Mukai pairs of dimension 4 and rank 2 with Picard number one, that is, pairs (X, E) where X is a Fano 4-fold with Picard number one, and E is an ample vector bundle of rank 2 on X with c1(X) = c1(E). Equivalently, the present paper completes the classification of ruled Fano 5-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007. [ABSTRACT FROM AUTHOR]

We give a characterization of the extremal sequences for the Bellman function of three variables of the dyadic maximal operator in relation to Kolmogorov's inequality. In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue. For this approach we use the one introduced in a paper by A. D. Melas and the author, where the respective Bellman function has been precisely evaluated. [ABSTRACT FROM AUTHOR]

We develop a (global) solvability theory for Euler type linear partial differential equations P(θ) on C∞(Ω), with Ω an open subset of Rd, i.e., for a special type of linear partial differential equation with polynomial coefficients. There is a natural closed upper bound C∞ I(P)(Ω) for the range of P(θ) on C∞(Ω). We characterize by P(θ)-convexity type conditions those Ω such that P(θ) is surjective on C∞ I(P)(Ω). We also clarify when all shifted operators P(c + θ) are surjective on C∞ I(P(c+ ·))(Ω). We classify in geometric terms those Ω with 0 ∈ Ω such that every Euler operator P(θ) is surjective on C∞ I(P)(Ω). Moreover, we determine the operators P(θ) which are surjective onto C∞ I(P)(Ω) for every open set Ω ⊆ Rd. Under some mild assumptions on Ω, we characterize those Euler operators which are invertible on C∞(Ω). Under the same assumptions we also calculate the spectrum of P(θ) on C∞(Ω). The results follow from the solvability theory for Hadamard type operators on the space of smooth functions and from a new general Mellin transform, both developed in this paper. [ABSTRACT FROM AUTHOR]

In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of mR-primary ideals in a Noetherian local ring R, generalizing the classical theory for mR-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of R is less than the dimension of R, which holds, for instance, if R is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of mR-primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and of Rees and Sharp. [ABSTRACT FROM AUTHOR]

In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales, each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi left continuous and admits the predictable representation property. Then we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature and has applications to stability results for backward stochastic differential equations with jumps, and to discretization schemes for stochastic systems. [ABSTRACT FROM AUTHOR]

In this paper we generalize a result of Galloway and Mendes in two different situations: in the first case for marginally outer trapped surfaces (MOTSs) of genus greater than 1 and, in the second case, for MOTSs of high dimension with negative σ-constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOTS which saturates a lower bound for the area (in dimension 2) or for the volume (in dimension ≥ 3). These results are extensions of theorems of Nunes and Moraru to general (non-time-symmetric) initial data sets. [ABSTRACT FROM AUTHOR]

In this paper we consider diffusions on the half line (0,∞) such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity which takes finite values at positive times. We study the behavior of hitting times of large barriers and, in a dual way, the behavior of the process starting at infinity for small time. In particular, we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctuations of the hitting times are asymptotically Gaussian. We also derive the tail of the distribution of the hitting time of the origin and a Yaglom limit for the diffusion starting from infinity. We finally prove that the distribution of this process killed at the origin is absolutely continuous with respect to the speed measure. The density is expressed in terms of the eigenvalues and eigenfunctions of the generator of the killed diffusion. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometry of the SYZ transform on a semiflat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and the extension of holomorphic vector bundles over the mirror elliptic curve, via the SYZ transform for immersed Lagrangian multisections defined by Arinkin and Joyce [Fukaya category and Fourier transform, AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 2001] and Leung, Yau, and Zaslow [Adv. Theor. Math. Phys. 4 (2000), no. 6, 1319-1341]. This study leads us to a new notion of equivalence between objects in the immersed Fukaya category of a general compact symplectic manifold (M, ω), under which the immersed Floer cohomology is invariant; in particular, this provides an answer to a question of Akaho and Joyce [J. Differential Geom. 86 (2010), no. 3, 831-500, Question 13.15]. Furthermore, if M admits a Lagrangian torus fibration over an integral affine manifold, we prove, under some additional assumptions, that this new equivalence is mirror to an isomorphism between holomorphic vector bundles over the dual torus fibration via the SYZ transform. [ABSTRACT FROM AUTHOR]

In this paper we engage in a general study of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten-Reshetikhin-Turaev topological quantum field theory via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions on the mapping class group elements that allow us to provide a full asymptotic expansion. In particular, we show that our results apply to all pseudo-Anosov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hitherto successfully considered assumptions in this context. The proof of our main theorem relies on our new results regarding asymptotic expansions of oscillatory integrals, which allows us to go significantly beyond the standard transversely cut out assumption on the fixed point set. This makes use of the Picard-Lefschetz theory for Laplace integrals. [ABSTRACT FROM AUTHOR]

In this paper we extend a recent breakthrough of Chen and Cheng on the existence of a constant scalar Kahler metric on a compact Kahler manifold to Calabi's extremal metric. There are no new a priori estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector V if and only if the modified Mabuchi energy is proper, modulo the action of the subgroup in the identity component of the automorphism group which commutes with the flow of V . We introduce two essentially equivalent notions, called reductive properness and reduced properness. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy. [ABSTRACT FROM AUTHOR]

There exists a holomorphic quadratic differential defined on any H-surface immersed in the homogeneous space E(κ t) given by U. Abresch and H. Rosenberg, called the Abresch-Rosenberg differential. However, there was no Codazzi pair on such an H-surface associated with the Abresch-Rosenberg differential when t ≠ 0. The goal of this paper is to find a geometric Codazzi pair defined on any H-surface in E(κ, τ), when t ≠ 0, whose (2, 0)-part is the Abresch-Rosenberg differential. We denote such a pair as (I, IIAR), were I is the usual first fundamental form of the surface and IIAR is the Abresch-Rosenberg second fundamental form. In particular, this allows us to compute a Simons' type equation for H-surfaces in E(κ, τ). We apply such Simons' type equation, first, to study the behavior of complete H-surfaces S of finite Abresch-Rosenberg total curvature immersed in E(κ, τ). Second, we estimate the first eigenvalue of any Schrodinger operator L = Δ+V, V continuous, defined on such surfaces. Finally, together with the Omori-Yau maximum principle, we classify complete H-surfaces in E(κ, τ), τ ≠ 0, satisfying a lower bound on H depending on κ, τ, and an upper bound on the norm of the traceless IIAR, a gap theorem. [ABSTRACT FROM AUTHOR]

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research toward showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. More than 25 years ago, Erdõs, Faudree, and Sós conjectured that in any C-Ramsey graph there are Ω(n5/2) induced subgraphs, no pair of which have the same numbers of vertices and edges. Improving on earlier results of Alon, Balogh, Kostochka, and Samotij, in this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

In this paper we tackle a question raised by N. Templier and A. Saha concerning the size of Whittaker new vectors appearing in infinitedimensional representations of GL2 over nonarchimedean fields. We derive precise bounds for such functions in all possible situations. Our main tool is the p-adic method of stationary phase. [ABSTRACT FROM AUTHOR]

This paper is dedicated to providing new tools and methods for studying the trend to equilibrium of gradient flows in metric spaces (M, d) in the entropy and metric sense, to establish decay rates and finite time of extinction, and to characterize Lyapunov stable equilibrium points. More precisely, our main results are as follows. [ABSTRACT FROM AUTHOR]

It is known that a holomorphic motion (an analytic version of an isotopy) of a set X in the complex plane C always extends to a holomorphic motion of the entire plane. In the topological category, it was recently shown that an isotopy h : X×[0, 1] → C, starting at the identity, of a plane continuum X also always extends to an isotopy of the entire plane. Easy examples show that this result does not generalize to all plane compacta. In this paper we will provide a characterization of isotopies of uniformly perfect plane compacta X which extend to an isotopy of the entire plane. Using this characterization, we prove that such an extension is always possible provided the diameters of all components of X are uniformly bounded away from zero. [ABSTRACT FROM AUTHOR]

In this paper we describe compactified universal Jacobians, i.e., compactifications of the moduli space of line bundles on smooth curves obtained as moduli spaces of rank 1 torsion-free sheaves on stable curves, using an approach due to Oda-Seshadri. We focus on the combinatorics of the stability conditions used to define compactified universal Jacobians. We explicitly describe an affine space, the stability space, with a decomposition into polytopes such that each polytope corresponds to a proper Deligne-Mumford stack that compactifies the moduli space of line bundles. We apply this description to describe the set of isomorphism classes of compactified universal Jacobians (answering a question of Melo) and to resolve the indeterminacy of the Abel- Jacobi sections (addressing a problem raised by Grushevsky-Zakharov). [ABSTRACT FROM AUTHOR]

In this paper, we study punctured spheres in two dimensional ball quotient compactifications (X,D). For example, we show that smooth toroidal compactifications of ball quotients cannot contain properly holomorphically embedded 3-punctured spheres. We also use totally geodesic punctured spheres to prove ampleness of KX + αD for α ∈ (1/4, 1), giving a sharp version of a theorem of the first author with G. Di Cerbo. Finally, we produce the first examples of bielliptic ball quotient compactifications modeled on the Gaussian integers. [ABSTRACT FROM AUTHOR]

In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions such as being totally real or p-adic, improving on earlier work of Bombieri and Zannier in the totally p-adic case. These bounds applied as the degree of the algebraic number over the rationals tended towards infinity. In this paper, we use discrete energy approximation techniques on the Berkovich projective line to make the dependence on the degree in these bounds explicit, and we establish lower bounds for algebraic numbers which depend only on local properties of the numbers. [ABSTRACT FROM AUTHOR]

The main result of this paper is the following: if F is any field and R any F-subalgebra of the algebra Mn(F) of n × n matrices over F with Lie nilpotence index m, then. dimFR ≼ M(m + 1, n), where M(m + 1, n) is the maximum of 1/2(n² -Σ m+1 i=1 k² i) + 1 subject to the constraint Σ m+1 i=1 ki = n and k1, k2, . . . , km+1 nonnegative integers. This answers in the affirmative a conjecture by the first and third authors. The case m = 1 reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if F is an algebraically closed field of characteristic zero and R is any commutative F-subalgebra of Mn(F), then dimFR ≼ n²/4+ 1. Examples constructed from block upper triangular matrices show that the upper bound of M(m+1, n) cannot be lowered for any choice of m and n. An explicit formula for M(m + 1, n) is also derived. [ABSTRACT FROM AUTHOR]

In the present paper, we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators L = -(∇-ia)TA(∇-ia)+V. The latter class includes, in particular, the magnetic Schrödinger operator -(∇-ia)² + V and the generalized electric Schrödinger operator -divA∇ + V. Our exponential decay bounds rest on a generalization of the Fefferman-Phong uncertainty principle to the present context and are governed by the Agmon distance associated with the corresponding maximal function. In the presence of a scale-invariant Harnack inequality--for instance, for the generalized electric Schrödinger operator with real coefficients--we establish both lower and upper estimates for fundamental solutions, thus demonstrating the sharpness of our results. The only previously known estimates of this type pertain to the classical Schrödinger operator -∆+V. [ABSTRACT FROM AUTHOR]

This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary, cornered asymptotically hyperbolic manifolds, and proves a theorem of Cartan-Hadamard-type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asymptotically hyperbolic normal form near the infinite boundary. [ABSTRACT FROM AUTHOR]

The dynamical stability of solitary lattice waves in non-integrable FPUT chains is a long-standing open problem and has been solved so far only in a certain asymptotic regime, namely by Friesecke and Pego for the KdV limit, in which the waves propagate with near sonic speed, have large wave length, and carry low energy. In this paper we derive a similar result in a complementary asymptotic regime related to fast and strongly localized waves with high energy. In particular, we show that the spectrum of the linearized FPUT operator contains asymptotically no unstable eigenvalues except for the neutral ones that stem from the shift symmetry and the spatial discreteness. This ensures that high-energy waves are linearly stable in some orbital sense, and the corresponding nonlinear stability is granted by the general, non-asymptotic part of the seminal Friesecke-Pego result and the extension by Mizumachi. Our analytical work splits into two principal parts. First we refine two-scale techniques that relate high-energy waves to a nonlinear asymptotic shape ODE and provide accurate approximation formulas. In this way we establish the existence, local uniqueness, smooth parameter dependence, and exponential localization of fast lattice waves for a wide class of interaction potentials with algebraic singularity. Afterwards we study the crucial eigenvalue problem in exponentially weighted spaces, so that there is no unstable essential spectrum. Our key argument is that all proper eigenfunctions can asymptotically be linked to the unique bounded and normalized solution of the linearized shape ODE, and this finally enables us to disprove the existence of unstable eigenfunctions in the symplectic complement of the neutral ones. [ABSTRACT FROM AUTHOR]

We determine the Gross–Hopkins duals of certain higher real K– theory spectra. More specifically, let p be an odd prime, and consider the Morava E–theory spectrum of height n = p−1. It is known, in expert circles, that for certain finite subgroups G of the Morava stabilizer group, the homotopy fixed point spectra EhGn are Gross–Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups G which contain p–torsion. This generalizes previous results for n = 2 and p = 3. [ABSTRACT FROM AUTHOR]

In 1994, Becker conjectured that if F(z) is a k-regular power series, then there exists a k-regular rational function R(z) such that F(z)/R(z) satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies a0(z) = 1. In this paper, we prove Becker’s conjecture in the best-possible form; we show that the rational function R(z) can be taken to be a polynomial zγQ(z) for some explicit nonnegative integer γ and such that 1/Q(z) is k-regular. [ABSTRACT FROM AUTHOR]

Cochran, Orr, and Teichner developed a filtration of the knot concordance group indexed by half integers called the solvable filtration. Its terms are denoted by Fn. It has been shown that Fn/Fn.5 is a very large group for n ≥ 0. For a generalization to the setting of links the third author showed that Fn.5/Fn+1 is non-trivial. In this paper we provide evidence for knots F0.5 = F1. In particular we prove that every genus 1 algebraically slice knot is 1-solvable. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of commutative algebra and combinatorics. It covers all previous major results in this topic and has several interesting consequences. [ABSTRACT FROM AUTHOR]

The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two S-resolvent operators and the S-spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the S-spectrum of a quaternionic linear operator T is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S-spectrum and of slice hyperholomorphicity of the S-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator. [ABSTRACT FROM AUTHOR]

Given a measure ν on a regular planar domain D, the Gaussian multiplicative chaos measure of ν studied in this paper is the random measure ṽ obtained as the limit of the exponential of the γ-parameter circle averages of the Gaussian free field on D weighted by ν. We investigate the dimensional and geometric properties of these random measures. We first show that if ν is a finite Borel measure on D with exact dimension α > 0, then the associated GMC measure ṽ is nondegenerate and is almost surely exact dimensional with dimension α − γ²/2, provided γ²/2 < α. We then show that if νt is a Hölder-continuously parameterized family of measures, then the total mass of ṽt varies Hölder-continuously with t, provided that γ is sufficiently small. As an application we show that if γ < 0.28, then, almost surely, the orthogonal projections of the γ-Liouville quantum gravity measure ũ on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with H¨older continuous densities. Furthermore, ũ has positive Fourier dimension almost surely. [ABSTRACT FROM AUTHOR]

In this paper, we prove an exponential integral formula for the Fourier transform of Bessel functions over complex numbers, along with a radial exponential integral formula. The former will enable us to develop the complex spectral theory of the relative trace formula for the Shimura–Waldspurger correspondence and extend the Waldspurger formula from totally real fields to arbitrary number fields. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive group, and let X be a smooth affine spherical G-variety, both defined over the complex numbers. A well-known theorem of I. Losev’s says that X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to prove an orbit theorem and to apply it to obtain a result that can be regarded as a significant step towards the solution of Gluck’s conjecture on large character degrees of finite solvable groups. [ABSTRACT FROM AUTHOR]

Let R = C[x1, x2, . . ., xn]/(f1, . . ., fm) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there are no negative weight derivations when R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R when the degrees of f1, . . .,fm are bounded below by a constant C depending only on the weights of x1, . . ., xn. Moreover this bound C is improved in several special cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation γΔ²u − Δu + αu = |u|2σu, u ∈ H²(RN), under the constraint ∫RN |u|² dx = c > 0. We assume that γ > 0,N ≥ 1, 4 ≤ σN < 4N/(N−4)+, whereas the parameter α ∈ R will appear as a Lagrange multiplier. Given c ∈ R+, we consider several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation. [ABSTRACT FROM AUTHOR]

In this paper we characterize the zero sets of functions from ℓAp (the analytic functions on the open unit disk D whose Taylor coefficients form an ℓp sequence) by developing a concept of an “inner function” modeled by Beurling’s discussion of the Hilbert space ℓA², the classical Hardy space. The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, we give an alternative proof of a result of Vinogradov [Dokl. Akad. Nauk SSSR 160 (1965), pp. 263–266] which says that when p > 2, there are zero sets for ℓAp which are not Blaschke sequences. [ABSTRACT FROM AUTHOR]

In this paper, we consider a time periodic non-monotone and non-local delayed reaction-diffusion population model with stage structure. We first prove the existence of the asymptotic speed c∗ of spread by virtue of two auxiliary equations and comparison arguments. By the method of super- and sub-solutions and the fixed point theorem, as applied to the truncated problem on a finite interval, and the limiting arguments, we then establish the existence of time periodic traveling wave solutions of the model system with wave speed c > c∗. We further use the results of the asymptotic speed of spread to obtain the non-existence of traveling wave solutions for wave speed c < c∗. Finally, we prove the existence of the critical periodic traveling wave with wave speed c = c∗. It turns out that the asymptotic speed of spread coincides with the minimal wave speed for positive periodic traveling waves. These results are also applied to the model system with two prototypical birth functions. [ABSTRACT FROM AUTHOR]

Frank Calegari was kind enough to point out a phenomenon overlooked in the paramodular conjecture in our paper. We propose a modification and prove that the phenomenon occurs infrequently. [ABSTRACT FROM AUTHOR]

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a byproduct, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on multiple hyperplanes and prove the existence of such bundles that do not split if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most 3/2 on the double plane and describe the family of isomorphism classes of them. [ABSTRACT FROM AUTHOR]

In this paper we prove that, under mild conditions, an equicharacteristic integrally closed domain which is a finite abelian extension of a regular domain has the uniform symbolic topology property. [ABSTRACT FROM AUTHOR]

We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. One main aim of this paper is to show that in many contexts, transverse surgery is a more natural tool than surgery on Legendrian knots. We reinterpret contact (±1)-surgery on Legendrian knots as transverse surgery on transverse push-offs, allowing us to give simpler proofs of known results. We give the first result on the tightness of inadmissible transverse surgery (cf. contact (+1)-surgery) for contact manifolds with vanishing Heegaard Floer contact invariant. In particular, inadmissible transverse r-surgery on the connected binding of a genus g open book that supports a tight contact structure preserves tightness if r > 2g - 1. [ABSTRACT FROM AUTHOR]

Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √ nZn and Gn in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance, and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq²/n goes to zero, and not so if (p, q) sits on the curve pq² = σn. A previous work by Jiang (2006) shows that the total variation distance goes to zero if both p/ √ n and q/ √ n go to zero, and it is not true provided p = c √ n and q = d √ n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ. [ABSTRACT FROM AUTHOR]

The Turán inequalities and the higher order Turán inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre–Pólya class. A sequence {an}n≥0 of real numbers is said to satisfy the Turán inequalities or to be log-concave if for n ≥ 1, an² − an−1an+1 ≥ 0. It is said to satisfy the higher order Turán inequalities if for n ≥ 1, 4(an² − an−1an+1)(an+1² − anan+2) − (anan+1 − an−1an+2)² ≥ 0. For the partition function p(n), DeSalvo and Pak showed that for n > 25, the sequence {p(n)}n>25 is log-concave, that is, p(n)² − p(n − 1)p(n + 1) > 0 for n > 25. It was conjectured by the first author that p(n) satisfies the higher order Turán inequalities for n ≥ 95. In this paper, we prove this conjecture by using the Hardy–Ramanujan–Rademacher formula to derive an upper bound and a lower bound for p(n + 1)p(n − 1)/p(n)². Consequently, for n ≥ 95, the Jensen polynomials p(n − 1) + 3p(n)x + 3p(n + 1)x² + p(n + 2)x³ have only distinct real zeros. We conjecture that for any positive integer m ≥ 4 there exists an integer N(m) such that for n ≥ N(m), the Jensen polynomial associated with the sequence (p(n), p(n+1), . . ., p(n+m)) has only real zeros. This conjecture was posed independently by Ono. [ABSTRACT FROM AUTHOR]

The classical Hartman-Grobman Theorem states that a smooth diffeomorphism F(x) near its hyperbolic fixed point x is topological conjugate to its linear part DF(x) by a local homeomorphism Φ(x). In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth F(x) is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a C∞ diffeomorphism F(x), the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a C¹ diffeomorphism F(x) with DF(x) being α-Hölder continuous at the fixed point that the local homeomorphism Φ(x) is differentiable at the fixed point. Here, α > 0 depends on the bands of the spectrum of F¹(x) for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on F(x) cannot be lowered to C¹. [ABSTRACT FROM AUTHOR]

In this paper we obtain first a gap theorem for a class of conformally compact Einstein manifolds with a renormalized volume that is close to its maximum value. We also use a blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The major part of this paper is the study of how a property of the conformal infinity influences the geometry of the interior of a conformally compact Einstein manifold. Specifically we are interested in conformally compact Einstein manifolds with conformal infinity whose Yamabe invariant is close to that of the round sphere. Based on the approach initiated by Dutta and Javaheri we present a complete proof of the relative volume inequality ... for conformally compact Einstein manifolds. This leads not only to the complete proof of the rigidity theorem for conformally compact Einstein manifolds in arbitrary dimension without spin assumption but also a new curvature pinching estimate for conformally compact Einstein manifolds with conformal infinities having large Yamabe invariant. We also derive curvature estimates for such manifolds. [ABSTRACT FROM AUTHOR]

In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self-evident. The first is a genuine, 5-line proof, for the isoperimetric inequality for k-cycles in Rn, with a constant differing from the optimal constant by a factor of only √k, as opposed to a factor of kk produced by all of the other soft methods. The second is a 3-line proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of Giga and Yama-uchi (1993). We then turn to use the above-mentioned relation to prove a bound on the parabolic Hausdorff measure of the space-time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon isoperimetric inequality. To prove it, we are led to study the geometric measure theory of Euclidean rectifiable sets in parabolic space and prove a co-area formula in that setting. This formula, the proof of which occupies most of this paper, may be of independent interest. [ABSTRACT FROM AUTHOR]

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasiordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin's axiom for σ-centered posets. In his 1973 paper he showed under this assumption that both ω1 and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin's axiom for σ-centered posets implies that the Boolean algebra P(ω)/FIN equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility. [ABSTRACT FROM AUTHOR]

This paper is motivated by the results and questions of Jason P. Bell and Kevin G. Hare in 2009. Let O be a finitely generated Z-algebra that is an integrally closed domain of characteristic zero. We investigate the following two problems: (A) Fix q and r that are integral over O and describe all pairs (m, n) ∊ N2 such that O[qm] = O[rn]. (B) Fix r that is integral over O and describe all q such that O[q] = O[r]. In this paper, we solve Problem (A), present a solution of Problem (B) by Evertse and Gy"ory, and explain their relation to the paper of Bell and Hare. In the following, c1 and C2 are effectively computable constants with a very mild dependence on O, q, and r. For (B), Evertse and Gy"ory show that there are N ⩽ C2 elements s1,..., sN such that O[si] = O[r] for every i, and for every q such that O[q] = O[r], we have q-usi ∊ O for some 1 ⩽ i ⩽ N and u ∊ O*. This immediately answers two questions about Pisot numbers by Bell and Hare. For (A), we show that except for some "degenerate" cases that can be explicitly described, there are at most c1 such pairs (m, n). This significantly strengthens some results of Bell and Hare. We also make some remarks on effectiveness and discuss further questions at the end of the paper. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of stability, uniqueness and recurrence of generalized traveling waves of reaction-diffusion equations in time heterogeneous media of ignition type, whose existence has been proven by the authors of the present paper in a previous work. It is first shown that generalized traveling waves exponentially attract wave-like initial data. Next, properties of generalized traveling waves, such as space monotonicity and exponential decay ahead of interface, are obtained. Uniqueness up to space translations of generalized traveling waves is then proven. Finally, it is shown that the wave profile and the front propagation velocity of the unique generalized traveling wave are of the same recurrence as the media. In particular, if the media is time almost periodic, then so are the wave profile and the front propagation velocity of the unique generalized traveling wave. [ABSTRACT FROM AUTHOR]

This is the third in a series of four papers in which we prove a conjecture made by Boston et al. and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields. [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]

Whether, in the presence of a boundary, solutions of the Navier-Stokes equations converge to a solution to the Euler equations in the vanishing viscosity limit is unknown. In a seminal 1983 paper, Tosio Kato showed that the vanishing viscosity limit is equivalent to having sufficient control of the gradient of the Navier-Stokes velocity in a boundary layer of width proportional to the viscosity. In a 2008 paper, the present author showed that the vanishing viscosity limit is equivalent to the formation of a vortex sheet on the boundary. We present here several observations that follow from these two papers. [ABSTRACT FROM AUTHOR]