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The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert.

In 1903, Epstein published his proof of meromorphic continuation and a functional equation for Dirichlet series associated with quadratic forms, now called Epstein zeta-functions. However, already in 1889 (or even earlier) Hurwitz was aware of these results as his mathematical diaries and some unpublished notes (in an almost final form) found in his estate at the ETH Zurich show. In this article we present and analyze Hurwitz's notes and compare his reasoning with Epstein's paper in detail.

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.

In Sarkozy (2001) the third author of this paper presented two conjectures on the additive decomposability of the sequence of ”smooth” (or ”friable”) numbers. Elsholtz and Harper (2015) proved (by using sieve methods) the second (less demanding) conjecture. The goal of this paper is to extend and sharpen their result in three directions by using a different approach (based on the theory of S -unit equations).

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ⁡ ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).

In this paper we treat realization results for operator-valued functions which are analytic in the complex sense or slice hyperholomorphic over the quaternions. In the complex setting, we prove a realization theorem for an operator-valued function analytic in a neighborhood of the origin with a coisometric state space operator thus generalizing an analogous result in the unitary case. A main difference with previous works is the use of reproducing kernel Krein spaces. We then prove the counterpart of this result in the quaternionic setting. The present work is the first paper which presents a realization theorem with a state space which is a quaternionic Krein space and may open new avenues of research in hypercomplex analysis.

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2 + 1 Levin–Wen model to obtain 1 + 1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

International audience; In this paper, we consider a semi-linear fourth order parabolic equation in a bounded smooth domain Ω with homogeneous Dirichlet and Neumann boundary conditions. The main result of this paper is the null controllability and the exact controllability to the trajectories at any time T > 0 for the associated control system with a control function acting at the interior.; Dans ce papier, on considère uneéquation parabolique semi-linéaire de quatrième ordre dans un domaine borné régulier Ω avec des conditions aux limites de type Dirichlet et Neumann homogènes. Le résultat principal de ce papier concerne la contrôlabilitéà zéro et la contrôlabilité exacte pour tout T > 0 du système de contrôle associé avec un contrôle agissantà l'interieur.

We investigate a steady planar flow of an ideal fluid in a bounded simply connected domain and focus on the vortex patch problem with prescribed vorticity strength. There are two methods to deal with the existence of solutions for this problem: the vorticity method and the stream function method. A long standing open problem is whether these two entirely different methods result in the same solution. In this paper, we will give a positive answer to this problem by studying the local uniqueness of the solutions. Another result obtained in this paper is that if the domain is convex, then the vortex patch problem has a unique solution.

The purpose of this paper is to show well-posedness results in L 2 -based Sobolev spaces for transmission, Dirichlet, Neumann, and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on a compact Riemannian manifold of dimension m ≥ 2 . The Dirichlet, transmission, and mixed problems for the nonlinear Darcy-Forchheimer-Brinkman system with L ∞ coefficients are also analyzed. First, we focus on the well-posedness of linear transmission, Dirichlet and mixed boundary value problems for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using a variational approach that reduces such a boundary value problem to a mixed variational formulation defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. Further, we show the equivalence between each boundary value problem for the Brinkman system with L ∞ coefficients and its mixed variational counterpart, and then the well posedness in L 2 -based Sobolev spaces by using the Necas-Babuska-Brezzi technique. The second goal of this paper is the construction of the Newtonian and layer potential operators for the Brinkman system with L ∞ coefficients in Lipschitz domains on compact Riemannian manifolds by using the well-posedness results for the analyzed linear transmission problems. Various mapping properties of these operators are also obtained and used to describe the weak solutions of the Poisson problems with Dirichlet and Neumann conditions for the nonsmooth Brinkman system in terms of such potentials. Finally, we combine the well-posedness results of the Poisson problems of Dirichlet, transmission, and mixed type for the nonsmooth Brinkman system with a fixed point theorem in order to show the existence of a weak solution of the Poisson problem of Dirichlet, transmission, or mixed type for the (nonlinear) Darcy-Forchheimer-Brinkman system with L ∞ coefficients in L 2 -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds of dimension m ∈ { 2 , 3 } .

This paper is concerned with the asymptotic behavior of bounded solutions of the Cauchy problem { u t = u x x + f ( t , u ) , x ∈ R , t > 0 , u ( x , 0 ) = u 0 , x ∈ R , where u 0 is a nonnegative bounded function with compact support and f is a rather general nonlinearity that is periodic in t and satisfies f ( ⋅ , 0 ) = 0 . In the autonomous case where f = f ( u ) , the convergence of every bounded solution to an equilibrium has been established by Du and Matano (2010). However, the presence of periodic forcing makes the problem significantly more difficult, partly because the structure of time periodic solutions is much less understood than that of steady states. In this paper, we first prove that any ω-limit solution is either spatially constant or symmetrically decreasing. Furthermore, we show that the set of ω-limit solutions either consists of a single time-periodic solution or it consists of multiple time-periodic solutions and heteroclinic connections among them. Next, under a mild non-degenerate assumption on the corresponding ODE, we prove that the ω-limit set is a singleton, which implies the solution converges to a time-periodic solution. Lastly, we apply these results to equations with bistable nonlinearity and combustion nonlinearity, and specify more precisely which time-periodic solutions can possibly be selected as the limit.

The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.

Though the boundary layer formation in the chemotactic process has been observed in experiment (cf. [63] ), the mathematical study on the boundary layer solutions of chemotaxis models is just in its infant stage. Apart from the sophisticated theoretical tools involved in the analysis, how to impose/derive physical boundary conditions is a state-of-the-art in studying the boundary layer problem of chemotaxis models. This paper will proceed with a previous work [24] in one dimension to establish the convergence of boundary layer solutions of the Keller–Segel model with singular sensitivity in a two-dimensional space (half-plane) with respect to the chemical diffusion rate denoted by e ≥ 0 . Compared to the one-dimensional boundary layer problem, there are many new issues arising from multi-dimensions such as possible Prandtl type degeneracy, curl-free preservation and well-posedness of large-data solutions. In this paper, we shall derive appropriate physical boundary conditions and gradually overcome these barriers and hence establish the convergence of boundary layer solutions of the singular Keller–Segel system in the half-plane as the chemical diffusion rate vanishes. Specially speaking, we justify that the boundary layer converges to the outer layer (solution with e = 0 ) plus the inner layer as e → 0 , where both outer and inner layer profiles are precisely derived and well understood. By doing this, the structure of boundary layer solutions is clearly characterized. We hope that our results and methods can shed lights on the understanding of underlying mechanisms of the boundary layer patterns observed in the experiment for chemotaxis such as the work by Tuval et al. [63] , and open a new window in the future theoretical study of chemotaxis models.

Our main purpose in this paper is to establish the existence and nonexistence of extremal functions (also known as maximizers) and symmetry of extremals for several Trudinger-Moser type inequalities on the entire space R n , including both the critical and subcritical Trudinger-Moser inequalities (see Theorems 1.1, 1.2, 1.3, 1.4, 1.5). Most of earlier works on existence of maximizers in the literature rely on the complicated blow-up analysis of PDEs for the associated Euler-Lagrange equations of the corresponding Moser functionals. The new approaches developed in this paper are using the identities and relationship between the supremums of the subcritical Trudinger-Moser inequalities and the critical ones established by the same authors in [25] , combining with the continuity of the supremum function that is observed for the first time in the literature. These allow us to establish the existence and nonexistence of the maximizers for the Trudinger-Moser inequalities in different ranges of the parameters (including those inequalities with the exact growth). This method is considerably simpler and also allows us to study the symmetry problem of the extremal functions and prove that the extremal functions for the subcritical singular Truddinger-Moser inequalities are symmetric. Moreover, we will be able to calculate the exact values of the supremums of the Trudinger-Moser type in certain cases. These appear to be the first results in this direction.

In this paper, we study the impact of geodesic vector fields (vector fields whose trajectories are geodesics) on the geometry of a Riemannian manifold. Since, Killing vector fields of constant lengths on a Riemannian manifold are geodesic vector fields, leads to the question of finding sufficient conditions for a geodesic vector field to be Killing. In this paper, we show that a lower bound on the Ricci curvature of the Riemannian manifold in the direction of geodesic vector field gives a sufficient condition for the geodesic vector field to be Killing. Also, we use a geodesic vector field on a 3-dimensional complete simply connected Riemannian manifold to find sufficient conditions to be isometric to a 3-sphere. We find a characterization of an Einstein manifold using a Killing vector field. Finally, it has been observed that a major source of geodesic vector fields is provided by solutions of Eikonal equations on a Riemannian manifold and we obtain a characterization of the Euclidean space using an Eikonal equation.

Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .

In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .

This is the first in a series of three papers where we study the integral manifolds of the charged three-body problem. The integral manifolds are the fibers of the map of integrals. Their topological type may change at critical values of the map of integrals. Due to the non-compactness of the integral manifolds one has to take into account besides ‘ordinary’ critical points also critical points at infinity. In the present paper we concentrate on ‘ordinary’ critical points and in particular elucidate their connection to central configurations. In a second paper we will study critical points at infinity. The implications for the Hill regions, i.e. the projections of the integral manifolds to configuration space, are the subject of a third paper.

The paper concerns operators in ideal topological spaces. Some characterizations of Hayashi–Samuel spaces, Ψ - C sets and semi-open sets of ∗ -topology are investigated. Continuity and decomposition are also part of this paper.

Jleli and Samet (2018) introduced a new metric space and named it as F -space. In this paper we consider the notion of α - ψ -contraction in the setting of F -metric spaces. We present some fixed point and coupled fixed point results in the generalized setting. Moreover, our purpose in this paper is to concerned with the solution of nonlinear neutral differential equation x ′ ( t ) = − a ( t ) x ( t ) + b ( t ) g ( x ( t − r ( t ) ) ) + c ( t ) x ′ ( t − r ( t ) ) with unbounded delay using fixed point theory in F -metric space.

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

This paper is the second part of our comprehensive study on the structure of the solutions for the following Chern–Simons–Higgs equation: (0.1) { Δ u + 1 e 2 e u ( 1 − e u ) = 4 π ∑ j = 1 N δ p j , in Ω , u is doubly periodic on ∂ Ω , where Ω is a parallelogram in R 2 and e > 0 is a small parameter. In part 1 [29] , we proved the non-coexistence of different bubbles in the bubbling solutions and obtained an existence result for the Chern–Simons type bubbling solutions under some nearly necessary conditions. Mean field type bubbling solutions for (0.1) have been constructed in [27] . In this paper, we shall study two other important issues for the mean field type bubbling solutions: the necessary conditions for the existence and the local uniqueness. The results in this paper lay the foundation to find the exact number of solutions for (0.1) .

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.

In this paper we extend the idea of weighted density of Balcerzak et al. (2015) by using a modulus function and introduce the idea f -density of weight g of subsets of ω ≔ { 0 , 1 , … } (at the same time extending the notion of f -density (Aizpuru et al., 2014)), which we name d g f where g : ω → [ 0 , ∞ ) satisfies g ( n ) → ∞ and n ∕ g ( n ) ↛ 0 and f is a modulus function. The aim of this paper is to show that we can get new ideals Z g ( f ) consisting of sets A ⊂ ω for which d g f ( A ) = 0 different from all the previously constructed ideals Z g of Balcerzak et al. (2015) and moreover they retain all the nice properties of the ideals Z g .

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators { A ( t ) : t ∈ [ 0 , b ] } generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C 0 − semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.

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

This paper deals with the existence of multiple solutions for the quasilinear equation − div A ( x , ∇ u ) + V ( x ) | u | α ( x ) − 2 u = f ( x , u ) in R N , which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like | ξ | q ( x ) − 2 ξ for small | ξ | and like | ξ | p ( x ) − 2 ξ for large | ξ | , where 1 α ( ⋅ ) ≤ p ( ⋅ ) q ( ⋅ ) N . Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works A. Azzollini et al. (2014) [4] and N. Chorfi and V. Radulescu (2016) [11] from cases where the exponents p and q are constant, to the case where p ( ⋅ ) and q ( ⋅ ) are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the Cerami compactness condition.

Decomposition spaces are simplicial $\infty$-groupoids subject to a certain exactness condition, needed to induce a coalgebra structure on the space of arrows. Conservative ULF functors (CULF) between decomposition spaces induce coalgebra homomorphisms. Suitable added finiteness conditions define the notion of M\"obius decomposition space, a far-reaching generalisation of the notion of M\"obius category of Leroux. In this paper, we show that the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but is not induced by a M\"obius category), can be realised as the homotopy cardinality of a M\"obius decomposition space $U$ of all M\"obius intervals, and that in a certain sense $U$ is universal for M\"obius decomposition spaces and CULF functors., Comment: 35 pages. This paper is one of six papers that formerly constituted the long manuscript arXiv:1404.3202. v3: minor expository improvements. Final version to appear in Adv. Math

This is a chiefly expository paper on the subject of the title which, in our view, has not received a detailed treatment in the literature which is commensurate with its importance. We expect the results presented here to be useful in a number of contexts. For example, several of them will be applied in a forthcoming paper by the authors., Comment: 25 pages

This paper establishes a version of Nevanlinna theory based on Askey–Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane C . A second main theorem that we have derived allows us to define an Askey–Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole-scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey–Wilson type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey–Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey–Wilson. This paper concludes with an application to difference equations generalising the Askey–Wilson second-order divided difference equation.

Let M m be a closed smooth manifold equipped with a smooth involution having fixed point set of the form F n ∪ F 3 , where F n and F 3 are submanifolds with dimensions n and 3, respectively, where 3 n m and with the normal bundles over F n and F 3 being nonbounding. The authors of this paper, together with Patricia E. Desideri, previously showed that, when n is even, then m ≤ n + 4 , which we call a small codimension phenomenon. Further, they showed that this small bound is best possible. In this paper we study this problem for n odd, which is much more complicated, requiring more sophisticated techniques involving characteristic numbers. We show in this case that m ≤ M ( n − 3 ) + 6 , where M ( n ) is the Stong–Pergher number (see the definition of M ( n ) in Section 1). Further, we show that this bound is almost best possible, in the sense that there exists an example with m = M ( n − 3 ) + 5 , which means that for n odd the small codimension phenomenon does not occur and the bound in question is meaningful. The existence of these bounds is guaranteed by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5 2 n .

In this paper, we describe a surprising link between the theory of the Goldman–Turaev Lie bialgebra on surfaces of genus zero and the Kashiwara–Vergne (KV) problem in Lie theory. Let Σ be an oriented 2-dimensional manifold with non-empty boundary and K a field of characteristic zero. The Goldman–Turaev Lie bialgebra is defined by the Goldman bracket { − , − } and Turaev cobracket δ on the K -span of homotopy classes of free loops on Σ. Applying an expansion θ : K π → K 〈 x 1 , … , x n 〉 yields an algebraic description of the operations { − , − } and δ in terms of non-commutative variables x 1 , … , x n . If Σ is a surface of genus g = 0 the lowest degree parts { − , − } − 1 and δ − 1 are canonically defined (and independent of θ). They define a Lie bialgebra structure on the space of cyclic words which was introduced and studied by Schedler [31] . It was conjectured by the second and the third authors that one can define an expansion θ such that { − , − } = { − , − } − 1 and δ = δ − 1 . The main result of this paper states that for surfaces of genus zero constructing such an expansion is essentially equivalent to the KV problem. In [24] , Massuyeau constructed such expansions using the Kontsevich integral. In order to prove this result, we show that the Turaev cobracket δ can be constructed in terms of the double bracket (upgrading the Goldman bracket) and the non-commutative divergence cocycle which plays the central role in the KV theory. Among other things, this observation gives a new topological interpretation of the KV problem and allows to extend it to surfaces with arbitrary number of boundary components (and of arbitrary genus, see [2] ).

Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. In the case if Dolgachev surfaces were of type $(2,3)$, we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exist a nontrivial phantom category in the derived category., Comment: 35 pages; 3 figures; exposition improved; Adv. Math. (to appear)

This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12] . We show that the isotropic discrepancy of a lattice point set is at most d 2 2 ( d + 1 ) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $\epsilon$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms., Comment: v3: Final version; to appear in Advances in Mathematics

Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, therefore $f$ is well defined by the $f(p)$, $p$ prime. Assuming that $f$ is $CM$, give conditions on the $f(p)$, either necessary or sufficient, both is possible, for $f$ being $CMO$ : that is the general purpose of the authors. The $CMO$ character of $f$ is invariant under slight modifications of the sequence $(f(p))$ (theorem~3). The same idea applies also in a more general context (theorem~4). After general statements of that sort, including examples of $CMO$ (theorem~5), the paper is devoted to ``small'' functions, that is, functions of the form $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem~8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ and $f(2)\not=-2$ (theorem~7). A third result gives only a sufficient condition (theorem~9). The three results apply to the particular case $f(p)=-1$, the historical example of Euler. Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Hal\'asz, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Ska\l ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the multiplicative convolution (theorems~10~and~11).

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert $A$-modules over locally $C^{*}$-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon--Nikodym theorem for this type of completely positive $n\times n$ matrices., 17 pages; The paper was revised and some material is added to it. To appear in Indag. Math

We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages

The year 2017 marks the 80th anniversary of Witt’s famous paper containing key results, including the Witt cancellation theorem, which form the foundation for the algebraic theory of quadratic forms. We pay homage to this paper by presenting a transparent and algebraic proof of the Witt cancellation theorem, which itself is based on a cancellation. We also present an overview of some recent spectacular work which is still building on Witt’s original creation of the algebraic theory of quadratic forms.

Motivated by the paper Gerard-Varet and Dormy (2010) [6] [JAMS, 2010] about the linear ill-posedness for the Prandtl equations around a shear flow with exponential decay in normal variable, and the recent study of well-posedness on the Prandtl equations in Sobolev spaces, this paper aims to extend the result in [6] to the case when the shear flow has general decay. The key observation is to construct an approximate solution that captures the initial layer to the linearized problem motivated by the precise formulation of solutions to the inviscid Prandtl equations.

It is well-known that dyadic martingale transforms are a good model for Calderon–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A 2 matrix weight, then the weighted L 2 -norm of a Calderon–Zygmund operator with cancellation has the same dependence on the A 2 characteristic of W as the weighted L 2 -norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderon–Zygmund operators on the A 2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderon–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytonen to extend the result to general Calderon–Zygmund operators.

In this paper, the Taylor sequence space t p θ of non-absolute type is introduced which consists of all sequences whose Taylor transforms of order θ , ( 0 θ 1 ) , are in the space l p . It is shown that the space t p θ is a normed space which includes the space l p where 1 ≤ p ≤ ∞ . Moreover, in this paper a general upper estimate is obtained for the norm of Hausdorff matrices and Norlund matrices as operators from l p into the Taylor sequence space t p θ . Finally, the results are extended to the matrix domain of an arbitrary invertible matrix E in the sequence space l p .

In a recent paper [5] S. Solecki proved a finite self-dual Ramsey theorem that extends simultaneously the classical finite Ramsey theorem [4] and the Graham–Rothschild theorem [2] . In this paper we prove the corresponding infinite dimensional version of the self-dual theorem. As a consequence, we extend the classical infinite Ramsey theorem [4] and the Carlson–Simpson theorem [1] .

In this expository paper, we give a complete proof of van den Essen’s theorem that the de Rham cohomology spaces of a holonomic D -module are finite-dimensional in the case of a formal power series ring over a field of characteristic zero. This proof requires results from at least five of van den Essen’s papers as well as his unpublished thesis, and until now has not been available in a self-contained document.

In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.