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Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that $$P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}$$ , where for each j ∈ ℕ and x ∈ ℝ, $$S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}$$ . Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector $$(\frac{1}{2},\frac{1}{{{2^2}}},...)$$ , for all n ∈ ℕ, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In [BLu] we disproved this conjecture. In this paper we prove that for most groups $H$ the conjecture is actually true if we restrict to counting only non-uniform lattices., Comment: 23 pages, revised following referee's comments. Dedicated to Aner Shalev on his 60th birthday. This paper is related to our previous work arXiv:0905.1841 with which it shares some preliminaries

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications., Comment: 20 pp.23 Ref

Prime end boundaries $\partial_P\Omega$ of domains $\Omega$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincar\'e inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_P\Omega$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}\Omega$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_P\Omega\setminus\partial_{SP}\Omega$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples., Comment: 23 pages, 3 figures

In this paper, we prove a kernel formula of Bessel functions attached to irreducible smooth supercuspidal representations of p-adic $GL(n)$. We also show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell. As an application we give a proof of the local converse conjecture of Jacquet., Comment: This paper has been withdrawn by the author. By the author due to an error

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\Gamma$. We also investigate the situation where both $A,\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture., Comment: 14 pages

In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.

In this paper we introduce a natural model for the realization space of a polytope up to projective equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety determined by the slack ideal of the polytope. This is a saturated determinantal ideal that encodes the combinatorics of the polytope. We also derive a new model of the realization space of a polytope from the positive part of the variety of a related ideal. The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, non-prescribability of faces, and realizability of combinatorial polytopes., Comment: The original arXiv submission of this paper has now been split into two parts; this paper describing the slack realization space of a polytope and applications of the slack ideal, and a second paper focussing on toric slack ideals and projective uniqueness

For a Tychonoff space $X$, we denote by $(C(X), \tau_k, \tau_p)$ the bitopological space of all real-valued continuous functions on $X$ where $\tau_k$ is the compact-open topology and $\tau_p$ is the topology of pointwise convergence. In papers [5, 6, 13] variations of selective separability and tightness in $(C(X), \tau_k, \tau_p)$ were investigated. In this paper we continued to study the selective properties and the corresponding topological games in the space $(C(X), \tau_k, \tau_p)$., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1805.04363

In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $\nu = w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main result of the paper sentences that under different conditions on the weights we can obtain $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m}, \infty}(\nu v^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$ where $T$ is a multilinear Calder\'on-Zygmund operator. To obtain this result we first prove it for the $m$-fold product of the Hardy-Littlewood maximal operator $M$, and also for $\mathcal{M}(\vec{f})(x)$: the multi(sub)linear maximal function introduced in \cite{LOPTT}. As an application we also prove a vector-valued extension to the mixed weighted weak-type inequalities of multilinear Calder\'on-Zygmund operators., Comment: 10 pages

Two permutations of $[n]=\{1,2 \ldots n\}$ are \textit{$k$-neighbor separated} if there are two elements that are neighbors in one of the permutations and that are separated by exactly $k-2$ other elements in the other permutation. Let the maximal number of pairwise $k$-neighbor separated permutations of $[n]$ be denoted by $P(n,k)$. In a previous paper, the authors have determined $P(n,3)$ for every $n$, answering a question of K\"orner, Messuti and Simonyi affirmatively. In this paper we prove that for every fixed positive integer $\ell $, $$P(n,2^\ell+1) = 2^{n-o(n)}. $$ We conjecture that for every fixed even $k$, $P(n,k)=2^{n-o(n)}$. We also show that this conjecture is asymptotically true in the following sense $$\lim_{k \rightarrow \infty} \lim_{n \rightarrow \infty} \sqrt[n]{P(n,k)}=2.$$ Finally, we show that for even $n$, $P(n,n)= 3n/2$.

In this paper we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_{t}=\partial_{t}-\Delta+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case that $b$ belongs to the Lebesgue space $L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regularity of weak solutions to the parabolic equation $L_{t}u=0$ depends critically on the value of the parabolic exponent $\gamma=\frac{2}{l}+\frac{n}{q}$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $\gamma\leq1$, and the heat kernel estimate has been obtained as well, except for the case that $l=\infty,q=n$. The regularity of weak solutions was deemed not true for the critical case $L^{\infty}\left(0,T;L^{n}\left(\mathbb{R}^{n}\right)\right)$ for a general $b$, while it is true for the divergence-free case, and a written proof can be deduced from the results in [Semenov, 2006]. One of the results obtained in the present paper establishes the Aronson type estimate for critical and supercritical cases and for vector fields $b$ which are divergence-free. We will prove the best possible lower and upper bounds for the fundamental solution one can derive under the current approach. The significance of the divergence-free condition enters the study of parabolic equations rather recently, mainly due to the discovery of the compensated compactness. The interest for the study of such parabolic equations comes from its connections with Leray's weak solutions of the Navier-Stokes equations and the Taylor diffusion associated with a vector field where the heat operator $L_{t}$ appears naturally., Comment: 31 pages

We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors., Comment: 65 pages, 4 figures; a number of typos fixed; the exposition has been significantly revised, fixing a technical error in the non-compact case in the process; this paper is now restricted to the simple normal crossings case; the arbitrary normal crossings case will be detailed in a followup paper

From spline theory it is well-known that univariate cubic spline interpolation, if carried out in its natural Hilbert space W 2 2 [ a , b ] and on point sets with fill distance h , converges only like O ( h 2 ) in L 2 [ a , b ] if no additional assumptions are made. But superconvergence up to order h 4 occurs if more smoothness is assumed and if certain additional boundary conditions are satisfied. This phenomenon was generalized in 1999 to multivariate interpolation in Reproducing Kernel Hilbert Spaces on domains Ω ⊂ R d for continuous positive definite Fourier-transformable shift-invariant kernels on R d . But the sufficient condition for superconvergence given in 1999 still needs further analysis, because the interplay between smoothness and boundary conditions is not clear at all. Furthermore, if only additional smoothness is assumed, superconvergence is numerically observed in the interior of the domain, but a theoretical foundation still is a challenging open problem. This paper first generalizes the “improved error bounds” of 1999 by an abstract theory that includes the Aubin–Nitsche trick and the known superconvergence results for univariate polynomial splines. Then the paper analyzes what is behind the sufficient conditions for superconvergence. They split into conditions on smoothness and localization, and these are investigated independently. If sufficient smoothness is present, but no additional localization conditions are assumed, it is numerically observed that superconvergence always occurs in the interior of the domain, and some supporting arguments are provided. If smoothness and localization interact in the kernel-based case on R d , weak and strong boundary conditions in terms of pseudodifferential operators occur. A special section on Mercer expansions is added, because Mercer eigenfunctions always satisfy the sufficient conditions for superconvergence. Numerical examples illustrate the theoretical findings.

For any symplectic manifold $${(M,\omega )}$$ , the set of Hamiltonian diffeomorphisms $${{\text {Ham}}^c(M,\omega )}$$ forms a group and $${{\text {Ham}}^c(M,\omega )}$$ contains an important subset $${{\text {Aut}}(M,\omega )}$$ which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that $${{\text {Aut}}(M,\omega )}$$ is a very small subset of $${{\text {Ham}}^c(M,\omega )}$$ . In this paper, we estimate the size of the subset $${{\text {Aut}}(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $${H^\infty}$$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where $${e_{\ell}, {\ell} = 1, 2, 3}$$ are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables $${x = (x_{1}, x_{2}, x_{3})}$$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version $${T^{\alpha}, {\rm for} \alpha \in (0, 1)}$$ , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although prior work has established convergence results for algorithms that integrate both descent and negative curvature steps, there has not yet been extensive numerical evidence showing that such methods offer consistent performance improvements. In this paper, we present new frameworks for combining descent and negative curvature directions: alternating two-step approaches and dynamic step approaches. The aspect that distinguishes our approaches from ones previously proposed is that they make algorithmic decisions based on (estimated) upper-bounding models of the objective function. A consequence of this aspect is that our frameworks can, in theory, employ fixed stepsizes, which makes the methods readily translatable from deterministic to stochastic settings. For deterministic problems, we show that instances of our dynamic framework yield gains in performance compared to related methods that only follow descent steps. We also show that gains can be made in a stochastic setting in cases when a standard stochastic-gradient-type method might make slow progress.

Quasi-elliptic cohomology is a variant of elliptic cohomology theories. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. Thus, the constructions on it can be made in a neat way. This theory reflects the geometric nature of the Tate curve. In this paper we provide a systematic introduction of its construction and definition., Comment: Final Version. 26 pages. To appear in Advances in Mathematics. In this paper we generalize the construction in arXiv:1612.00930. The subtle point of this generalization is explained in Section 2

In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives with these kernels possess the integrable singularities at the point zero, the kernels of the general fractional integrals can be - depending on their order - both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved., 15 pages

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary., 23 pages

International audience; Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g\ge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Hölder property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera., Corrects the formulation of Proposition 5.4 in the published version (reversed order of quantifiers); the results and proofs are unchanged, with a minor rewording

We develop a theory of $p$-adic automorphic forms on unitary groups that allows $p$-adic interpolation in families and holds for all primes $p$ that do not ramify in the reflex field $E$ of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if $p$ splits completely in $E$), we recover Hida's theory of $p$-adic automorphic forms, which is defined over the ordinary locus. More generally, we work over the $\mu$-ordinary locus, which is open and dense. By eliminating the splitting condition on $p$, our framework should allow many results employing Hida's theory to extend to infinitely many more primes. We also provide a construction of $p$-adic families of automorphic forms that uses differential operators constructed in the paper. Our approach is to adapt the methods of Hida and Katz to the more general $\mu$-ordinary setting, while also building on papers of each author. Along the way, we encounter some unexpected challenges and subtleties that do not arise in the ordinary setting., Comment: 44 pages. Accepted for publication in the American Journal of Mathematics

Complex linear differential equations with entire coefficients are studied in the situation where one of the coefficients is an exponential polynomial and dominates the growth of all the other coefficients. If such an equation has an exponential polynomial solution $f$, then the order of $f$ and of the dominant coefficient are equal, and the two functions possess a certain duality property. The results presented in this paper improve earlier results by some of the present authors, and the paper adjoins with two open problems., 17 pages

This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies. This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope (FBE), which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka-\L ojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.

In this paper, we study the algebraic relations satisfied by the solutions of q-difference equations and their transforms with respect to an auxiliary operator. Our main tools are the parametrized Galois theories developed in Hardouin and Singer (Math Ann 342(2):333–377, 2008) and Ovchinnikov and Wibmer (Int Math Res Not 12:3962–4018, 2015). The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals with a $$\mathbf {q}$$ -difference operator. In both cases, we give criteria to guarantee the algebraic independence of a series, solution of a q-difference equation, with either its successive derivatives or its $$\mathbf {q}$$ -transforms. We apply our results to q-hypergeometric series.

In this paper we use the blow-up surgery introduced in [G] to produce new higher dimensional partially hyperbolic flows. The main contribution of the paper is the slow-down construction which accompanies the blow-up construction. This new ingredient allows to dispose of a rather strong domination assumption which was crucial for results in [G]. Consequently we gain more flexibility which allows to construct new volume-preserving partially hyperbolic flows as well as new examples which are not fiberwise Anosov. The latter are produced by starting with the geodesic flow on complex hyperbolic manifold which admits a totally geodesic complex curve. Then by performing the slow-down first and the blow-up second we obtain a new (volume-preserving) partially hyperbolic flows., Comment: 16 pages. This is part II. Part I being arXiv:1609.05925

In 1981, Foias, Guillop\'e and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier-Stokes equation. Such bounds are instructive in the numerical investigation of intermittency often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier-Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. Construction relies on space analyticity of the solutions at almost all times. An auxiliary problem is introduced, and a Sobolev norm of its solutions bounds from below the size in $C^3$ of the region of space analyticity of the solutions to the original problem. We recover the exponents obtained earlier for the hydrodynamic problem. The same approach is also followed here to derive and prove similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions. This paper is dedicated to Professor Uriel Frisch on the occasion of his 80th anniversary as a sign of appreciation of the Scientist and the Teacher., Comment: 29 pages, 1 figure. The definitive version of the paper (formatting faults and some misprints in the published version corrected); more misprints corrected in version 2

The paper is concerned with the problem of identifying the norm attaining operators in the von Neumann algebra generated by two orthogonal projections on a Hilbert space. This algebra contains every skew projection on that Hilbert space and hence the results of the paper also describe functions of skew projections and their adjoints that attain the norm., Comment: 6 pages

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional surface of constant curvature: plane, sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in $\mathbb R^3$ equipped with appropriate quadratic form. Namely, we show that a billiard is polynomially integrable, if and only if its boundary is a union of confocal conical arcs and appropriate geodesic segments. We also present a complexification of these results. These are joint results of Mikhail Bialy, Andrey Mironov and the author. The proof is split into two parts. The first part is given by Bialy and Mironov in their two joint papers. They considered the tautological projection of the boundary to $\mathbb{RP}^2$ and studied its orthogonal-polar dual curve, which is piecewise algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's theorem, it suffices to show that each non-linear complex irreducible component of the dual curve is a conic. They have proved that all its singularities and inflection points (if any) lie in the projectivized zero locus of the corresponding quadratic form on $\mathbb C^3$. The present paper provides the second part of the proof: we show that each above irreducible component is a conic and finish the solution of the Algebraic Birkhoff Conjecture in constant curvature., To appear in the Journal of the European Mathematical Society (JEMS), 69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and misprint corrections

Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N < \infty$. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity. We establish this result by showing that $W_2$-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolation. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain: - equiboundedness of the densities along the entropic interpolations, - local equi-Lipschitz continuity of the Schr��dinger potentials, - a uniform weighted $L^2$ control of the Hessian of such potentials. Finally, the techniques adopted in this paper can be used to show that in the $RCD$ setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case. With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality., arXiv admin note: substantial text overlap with arXiv:1701.03932

A nonlinear Markov chain is a discrete time stochastic process whose transitions depend on both the current state and the current distribution of the process. The nonlinear Markov chain over an infinite state space can be identified by a continuous mapping (the so-called nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex). In the present paper, we consider a continuous analogue of the mentioned mapping acting on L 1 -spaces. Main aim of the current paper is to investigate projective surjectivity of quadratic stochastic operators (QSO) acting on the set of all probability measures. To prove the main result, we study the surjectivity of infinite dimensional nonlinear Markov operators and apply them to the projective surjectivity of the considered QSO. Furthermore, the obtained results are applied to the existence of the positive solution of some Hammerstein integral equations.

The so called Landis conjecture states that if a solution of the equation $$\Delta u+V(x)u=0$$ in an exterior domain decays faster than $e^{-\kappa|x|}$, for some $\kappa>\sqrt{\sup |V|}$, then it must be identically equal to $0$. This property can be viewed as a unique continuation at infinity (UCI) for solutions satisfying a suitable exponential decay. The Landis conjecture was disproved by Meshkov in the case of complex-valued functions, but it remained open in the real case. In the 2000s, several papers have addressed the issue of the UCI for linear elliptic operators with real coefficients. The results that have been obtained require some kind of sign condition, either on the solution or on the zero order coefficient of the equation. The Landis conjecture is still open nowadays in its general form. In the present paper, we start with considering a general (real) elliptic operator in dimension $1$. We derive the UCI property with a rate of decay $\kappa$ which is sharp when the coefficients of the operator are constant. In particular, we prove the Landis conjecture in dimension $1$, and we can actually reach the threshold value $\kappa=\sqrt{\sup |V|}$. Next, we derive the UCI property -- and then the Landis conjecture -- for radial operators in arbitrary dimension. Finally, with a different approach, we prove the same result for positive supersolutions of general elliptic equations.

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman's "potential theoretic" method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to slightly generalise Lutz and Stull's theorems: the versions in this paper apply to orthogonal projections to $m$-planes in $\mathbb{R}^{n}$, for all $0 < m < n$., 11 pages. v2: Incorporated referee suggestions

In this paper we study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that H\"ormander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper., Comment: 23 pages

In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper., Comment: 36 pages, 16 figures. Portions of this work previously appeared as an appendix to arXiv:1603.03822, but has evolved into its own work and has been accepted for publication separately. Final version to appear in Acta Mathematica

In 1970, Chou showed there are $|\mathbb{N}^*| = 2^{2^\mathbb{N}}$ topologically invariant means on $L_\infty(G)$ for any noncompact, $\sigma$-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on $L_\infty(G)$ and $VN(G)$ were determined for any locally compact group. Each paper on a new case reached the same conclusion -- "the cardinality is as large as possible" -- but a unified proof never emerged. In this paper, I show $L_1(G)$ and $A(G)$ always contain orthogonal nets converging to invariance. An orthogonal net indexed by $\Gamma$ has $|\Gamma^*|$ accumulation points, where $|\Gamma^*|$ is determined by ultrafilter theory. Among a smattering of other results, I prove Paterson's conjecture that left and right topologically invariant means on $L_\infty(G)$ coincide iff $G$ has precompact conjugacy classes., Comment: 10 pages, completely rewritten from v2

In part I of this paper we studied additive decomposability of the set F y of the y -smooth numbers and the multiplicative decomposability of the shifted set G y = F y + { 1 } . In this paper, focusing on the case of ’large’ functions y , we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of k -term sumsets, for arbitrary k .

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable., Final version

In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic relativistic Calogero-Moser type. We focused on the first steps of the scheme in Part I, and on the cases $N=2$ and $N=3$ in Part II. In this paper, we determine the dominant asymptotics of a similarity transformed function $\rE_N(b;x,y)$ for $y_j-y_{j+1}\to\infty$, $j=1,\ldots, N-1$, and thereby confirm the long standing conjecture that the particles in the hyperbolic relativistic Calogero-Moser system exhibit soliton scattering. This result generalizes a main result in Part II to all particle numbers $N>3$., 21 pages

In this paper, we discuss the generalization of finitary $2$-representation theory of finitary $2$-categories to finitary birepresentation theory of finitary bicategories. In previous papers on the subject, the classification of simple transitive $2$-representations of a given $2$-category was reduced to that for certain subquotients. These reduction results were all formulated as bijections between equivalence classes of $2$-representations. In this paper, we generalize them to biequivalences between certain $2$-categories of birepresentations. Furthermore, we prove an analog of the double centralizer theorem in finitary birepresentation theory., Significant revision of the original version

We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen., V.3, the paper has been splitted into two papers, this is the first part on amalgamated free products

In this paper we will show how to constructed an Skew-Field with trace-preserving endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of endomorphisms algebra and trace-preserving endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving endomorphism, with the help of which we can construct the inverse trace-preserving endomorphisms of every trace-preserving endomorphism. At the end of this paper we have proven that the set of trace-preserving endomorphisms together with the actions of 'addition' and 'composition' (which is in the role of 'multiplication') forms a skew-field., 11 pages. arXiv admin note: substantial text overlap with arXiv:2003.09528

We show in this paper a Lipschitz stability result for a crack inverse problem in half space. The direct problem is a Laplace equation with zero Neumann condition on the top boundary. The forcing term is a discontinuity across the crack. This formulation can be related to geological faults in elastic media or to irrotational incompressible flows in a half space minus an inner wall. The direct problem is well posed in an appropriate functional space. We study the related inverse problem where the jump across the crack is unknown, and more importantly, the geometry and the location of the crack are unknown. The data for the inverse problem is of Dirichlet type over a portion of the top boundary. We prove that this inverse problem is uniquely solvable under some assumptions on the geometry of the crack. The highlight of this paper is showing a stability result for this inverse problem. Assuming that the crack is planar, we show that reconstructing the plane containing the crack is Lipschitz stable despite the fact that the forcing term for the underlying PDE is unknown. This uniform stability result holds under the assumption that the forcing term is bounded above and the Dirichlet data is bounded below away from zero in appropriate norms., Comment: 1 figure

The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the category $\mathbf{A}$ at level $-N-e-1$. To prove this, we use the following fact: a category with an action of $\widetilde{\mathfrak sl}_{e+1}$ contains a (canonically defined) subcategory with an action of $\widetilde{\mathfrak sl}_{e}$. We also prove a general statement that says that in some general situation a functor that satisfies a list of axioms is automatically Koszul dual to some sort of Zuckerman functor., 71 pages. This represents a portion of arXiv:1512.04878 which was split into two parts, this is the second part. This paper is rewritten (compared to arXiv:1512.04878) in a way that we never use KLR algebras explicitly. This makes the paper more independent from the first part

In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $$E(\Lambda ,\varphi ) = \{e^{2\pi i \lambda \cdot \varphi (x)}: \lambda \in \Lambda \}$$ where the phase function $$\varphi $$ is a Borel measurable which is not necessarily linear. A complete characterization of pairs $$(\Lambda ,\varphi )$$ for which $$E(\Lambda ,\varphi )$$ is an orthogonal basis or a frame for $$L^{2}(\mu )$$ is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when $$\mu $$ is the Lebesgue measure on [0, 1] and $$\Lambda = {{\mathbb {Z}}},$$ we show that only the standard phase functions $$\varphi (x) = \pm x$$ are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions $$\varphi $$ defined on $${{\mathbb {R}}}^{d}$$ such that the system $$E(\Lambda ,\varphi )$$ is an orthonormal basis for $$L^{2}[0,1]^{d}$$ when $$d\ge 2.$$ Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.

In this paper, we study large $m$ asymptotics of the $l^1$ minimal $m$-partition problem for Dirichlet eigenvalue. For any smooth domain $\Omega\in \mathbb{R}^n$ such that $|\Omega|=1$, we prove that the limit $\lim\limits_{m\rightarrow\infty}l_m^1(\Omega)=c_0$ exists, and the constant $c_0$ is independent of the shape of $\Omega$. Here $l_m^1(\Omega)$ denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any $m$-partition of $\Omega$., Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematics

In this paper we study the problem of stopping a Brownian bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $$\sup_{0\le \tau\le 1}\mathsf{E}[\mathrm{e}^{X_\tau}]$$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 2015, pp. 419--423] and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we develop techniques that use pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory in order to find the optimal stopping rule and to show regularity of the value function., Comment: 22 pages, 6 figures