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The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for

We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008

The link between Vandiver’s conjecture and Gauss sums is well known since the papers of Iwasawa (Symposia Mathematica, vol 15, Academic Press, pp 447–459, 1975), Thaine (Mich Math J 42(2):311–344, 1995; Trans Am Math Soc 351(12):4769–4790, 1999) and Angles and Nuccio (Acta Arith 142(3):199–218, 2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver’s conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the p-th cyclotomic field (Sur la p-ramification abelienne (1984) vol. 20, pp. 1–26). Then we provide a specific use of Gauss sums of characters of order p of $${\mathbb {F}}_\ell ^\times $$ and prove new criteria for Vandiver’s conjecture to hold (Theorem 2 (a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in §5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as $$\ell $$ varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in the direction of the very probable truth of Vandiver’s conjecture and to inspire future research. The calculations with their PARI/GP programs are given in appendices.

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.

In 1874, Franz Mertens proved the so-called Mertens’ theorem, and in 1974, Kenneth S. Williams showed Mertens’ theorem associated with a character. In a previous paper, we presented a graph-theoretic analogue to Williams’ theorem. In this paper, we generalize our previous work in the sense that a character is extended to a representation. To our knowledge, a number-theoretic analogue to our result is not yet known. So, we expect that, by using our methods, it can be proven in the future.

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

In the paper under review, we analyze various types of degenerate abstract Volterra integrodifferential equations in sequentially complete locally convex spaces. From the theory of non-degenerate equations, it is well known that the class of (a,k)-regularized C-resolvent families provides an efficient tool for dealing with abstract Volterra integro-differential equations of scalar type. Following the approach of T.-J. Xiao and J. Liang [41]-[43], we introduce the class of degenerate exponentially equicontinuous (a,k)- regularized C-resolvent families and discuss its basic structural properties. In the final section of paper, we will look at generation of degenerate fractional resolvent operator families associated with abstract differential operators.

The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ ( 1 2 ) , and in 1925, S. Wigert obtained a generalization giving a formula for ζ ( 1 k ) for any even integer k ≥ 2 . We extend the work of Ramanujan and Wigert in this paper. The Voronoi summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σ s ( n ) . The modified Bessel functions K s ( x ) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σ s ( n ) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.

A quite general theorems on unconditional bases in separable Hilbert spaces, given in terms of values of entire operator valued vector-functions, were established in the papers [2, 3, 4]. In the present paper, we give a detailed analysis of the hypothesis of these theorems. We present examples of various classes of vector-functions that satisfy some of the hypothesis of the above theorems. On the other hand, we show that there exist natural classes of vector-functions that do not satisfy some of the hypothesis of Theorem A.

A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the $k$-SAT problem or the Ising model. Such models naturally induce a Gibbs measure on the set of assignments, which is characterised by its partition function. The present paper deals with the partition function of problems where the interactions between variables and constraints are induced by a sparse random (hyper)graph. According to physics predictions, a generic recipe called the "replica symmetric cavity method" yields the correct value of the partition function if the underlying model enjoys certain properties [Krzkala et al., PNAS 2007]. Guided by this conjecture, we prove general sufficient conditions for the success of the cavity method. The proofs are based on a "regularity lemma" for probability measures on sets of the form $\Omega^n$ for a finite $\Omega$ and a large $n$ that may be of independent interest., Comment: This version replaces version 1 and the RANDOM 2015 version of the paper, which contained critical errors affecting the main results

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).

In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.

In this paper, we introduce and study an explicit hybrid re- laxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solu- tion is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.

In a famous paper, Asai indicated how to develop a theory of Eisenstein series for arbitrary number fields, using hyperbolic 3-space to take care of the complex places. Unfortunately he limited himself to class number 1. The present paper gives a detailed exposition of the general case, to be used for many applications. First, it is shown that the Eisenstein series satisfy the authors’ definition of regularized products satisfying the generalized Lerch formula, and the basic axioms which allow the systematic development of the authors’ theory, including the Cramér theorem. It is indicated how previous results of Efrat and Zograf for the strict Hilbert modular case extend to arbitrary number fields, for instance a spectral decomposition of the heat kernel periodized with respect to SL2 of the integers of the number field. This gives rise to a theta inversion formula, to which the authors’ Gauss transform can be applied. In addition, the Eisenstein series can be twisted with the heat kernel, thus encoding an infinite amount of spectral information in one item coming from heat Eisenstein series. The main expected spectral formula is stated, but a complete exposition would require a substantial amount of space, and is currently under consideration.

In this paper, we study the notions of accretive operators, contraction mappings, non-expansive mappings using the idea of n (>1)-normed structures for some relevant results. Further, we define \(\left( \lambda ,q\right) \)-transform by utilizing the definition of the generating function of q-Genocchi polynomials with weight zero to construct interesting properties related to q-Genocchi polynomials with weight zero and Frobenius–Euler polynomials. Furthermore, we describe p-adic q-\( \lambda \)-transform of higher order and construct a link between q-Genocchi polynomials of higher order with weight zero and higher order Frobenius–Euler polynomials using this transform. Our applications in this paper provide a link between analytic numbers theory and n-normed spaces.

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.

In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$30$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .

In this paper, we study a uniqueness question of meromorphic functions of certain differential polynomials that share a nonzero finite value or have the same fixed points with the same of L-functions. The results in this paper extend the corresponding results from Steuding [12, p. 152], Li [7], Fang [1] and Yang-Hua [14].

In a recent paper, the first two authors proved that the generating series of the Poincare polynomials of the quasihomogeneous Hilbert schemes of points in the plane has a simple decomposition in an infinite product. In this paper, we give a very short geometrical proof of that formula.

Approximate random $k$-colouring of a graph $G$ is a well studied problem in computer science and statistical physics. It amounts to constructing a $k$-colouring of $G$ which is distributed close to {\em Gibbs distribution} in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erd\H{o}s-R\'enyi random graph $G(n,d/n)$, where $d$ is a sufficiently large constant. We propose a novel efficient algorithm for approximate random $k$-colouring $G(n,d/n)$ for any $k\geq (1+\epsilon)d$. To be more specific, with probability at least $1-n^{-\Omega(1)}$ over the input instances $G(n,d/n)$ and for $k\geq (1+\epsilon)d$, the algorithm returns a $k$-colouring which is distributed within total variation distance $n^{-\Omega(1)}$ from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a "simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution., Comment: This paper is accepted for publication in SIAM Journal on Computing. This is the journal version of two papers of the author in SODA'12 and ESA'14

The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations., Comment: 33 pages, 10 figures. This is the second paper of the series "Hook formulas for skew shapes". Most of Sections 8 and 9 in this paper used to be part of arxiv:1512.08348 (v1,v2); v2 fixed several typos; v3 made precision in definition of flagged tableaux in Section 3.2 and fixed typo in Example 3.3; v4 fixed small typos in proof of Corollary 7.6

For a flnite group G let i(G) denote the graph deflned on the non- identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph i(G) contains a Hamiltonian cycle for many flnite groups G. In the literature many deep results about flnite simple groups G can equivalently be stated as theorems about i(G). Three examples are given. Guralnick and Shalev (10) showed that for su-ciently large G the graph i(G) has diameter at most 2. Guralnick and Kantor (9) showed that there is no isolated vertex in i(G). Finally, Breuer, Guralnick, Kantor (4) showed that the diameter of i(G) is at most 2 for all G. In this paper those flnite groups G are considered for which i(G) contains a Hamiltonian cycle. The following proposition reduces the investigations to those non-solvable groups G for which G=N is cyclic for any non-trivial normal subgroup N of G. Proposition 1.1. Let G be a flnite solvable group that has at least 4 elements. Then the graph i(G) contains a Hamiltonian cycle if and only if G=N is cyclic for all non-trivial normal subgroups N of G. The three main results of this paper are Theorems 1.2, 1.3, and 1.4.

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.

This paper is concerned with the performance of Orthogonal Matching Pursuit (OMP) algorithms applied to a dictionary $\mathcal{D}$ in a Hilbert space $\mathcal{H}$. Given an element $f\in \mathcal{H}$, OMP generates a sequence of approximations $f_n$, $n=1,2,\dots$, each of which is a linear combination of $n$ dictionary elements chosen by a greedy criterion. It is studied whether the approximations $f_n$ are in some sense comparable to {\em best $n$ term approximation} from the dictionary. One important result related to this question is a theorem of Zhang \cite{TZ} in the context of sparse recovery of finite dimensional signals. This theorem shows that OMP exactly recovers $n$-sparse signal, whenever the dictionary $\mathcal{D}$ satisfies a Restricted Isometry Property (RIP) of order $An$ for some constant $A$, and that the procedure is also stable in $\ell^2$ under measurement noise. The main contribution of the present paper is to give a structurally simpler proof of Zhang's theorem, formulated in the general context of $n$ term approximation from a dictionary in arbitrary Hilbert spaces $\mathcal{H}$. Namely, it is shown that OMP generates near best $n$ term approximations under a similar RIP condition., 12 pages

Andrei Andreevich Markov Jr. (born September 22, 1903, in St. Petersburg; died Oc tober 11, 1979, in Moscow) was the late and only child of the great Russian mathe matician Andrei Andreevich Markov Sr. (born June 14, 1856, in Ryazan7; died July 20, 1922, in Petrograd), universally recognized, in particular, for his contributions to the theory of probability (e.g., Markov chains and Markov processes). At his father's suggestion, young Andrei entered the chemistry section of the School of Physics and Mathematics at the University of Petrograd (formerly St. Petersburg, then Leningrad, and now again St. Petersburg). The young man was fascinated by chemistry and al ready by 1920 had taken part in chemical research. The results of his and his coau thor's work were published in 1924. Thus Markov's first paper dealt with chemistry. In his sophomore year he became interested in theoretical physics, and he graduated in 1924 with a physics degree. Markov's publications in chemistry were followed by a series of papers on the three body problem and dynamical systems (1926-1937), and a paper on Schr?dinger's quantum mechanics. The latter was one of the first papers on quantum mechanics published in the U.S.S.R., appearing less than a year after Schr?dinger's own ground breaking series of publications. In this connection it should be noted that it was Markov who, in 1931, introduced the concept of an abstract (topological) dynamical system. In 1932 Markov published an intriguing paper (in German) on relativity titled "Deriving a World Metric from the Relation 'Earlier Than.' " His interest in abstract mathematics is represented by series of papers on topology, algebra, analysis, and geometry. After World War II Markov's interests turned to axiomatic set theory, mathematical logic, and the foundations of mathematics. He founded the Russian school of con structive mathematics in the late 1940s and early 1950s. But in private conversations Markov often said that he had nurtured constructivist convictions for a very long time, in fact, long before the war. The Moscow mathematical school had been interested in constructivism, especially intuitionism, since its inception in the 1920s. It is enough to mention the 1925 work of Kolmogorov on intuitionistic logic [10]. It may be that this interest was due, at

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following: A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(log⁡n)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(log⁡n)1−o(1) such that not all graphs in the family have O(n) edges. Trevisan and Arora-Barak-Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges. Sudakov and the second author have recently shown that every graph with average degree d contains an n-vertex subgraph with average degree at least (1−o(1))d and vertex expansion 1/(log⁡n)1+o(1). We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1). The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super-linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.

In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogs of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts., Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive linear recurrence relations, distribution of gaps, longest gap, Markov processes

We prove that in the Wasserstein space built over R the subset of measures that does not charge the non-differentiability set of convex functions is not displacement convex. This completes the study of Gigli on the geometric structure of measures meeting the sharp hypothesis of the refined version of Brenier’s Theorem. Optimal transport is nowadays a central tool in many fields of analysis, differential geometry and probability theory (see e.g. the books by Villani [11, 12]). It takes its origin in the problem of Monge, asking for the shortest way to displace an amount of soil from one place of the Euclidean space to a heap of soil at another place. This problem induces a very natural distance between probability measures – the Wasserstein distance – where the measures represent the heaps of soil. Wasserstein distance is somewhat complementary with the Lebesgue L norms because it corresponds to an “horizontal” displacement: in first approximation, for two localized probability measures on R, the Wasserstein distance is the distance between their barycenters. Brenier’s Theorem [4] on monotone rearrangement of maps of R has become the very core of the theory of optimal transport. It gives a representation of the optimal transport map in term of gradient of convex functions. A very enlightening heuristic on (P2(R ),W2) is proposed in [7] where it appears with an infinite differential structure and the Wasserstein distance is seen as a Riemannian-like distance. This point of view has raised a lot of developments, among which the gradient flow theory presented in [2]. In [6], Gigli explores the Riemannian-like structure and proposes to think of the measures meeting the hypothesis of a refined version of Brenier’s Theorem as the regular points of the Wasserstein space. In this paper we show that the set of those transport-regular measures is not geodesically convex (Theorems 2.2 and 2.5). It is quite surprising because it is well-known that the subset made of absolutely continuous measures (the most notorious transport-regular measures) is geodesically convex. It answers a question suggested by Gigli [6, Remark 2.12]. The paper is built as follows: we first recall some main results on the quadratic Monge(-Kantorovich) problem. The key idea of our result can be found in Lemma 1.6 while Proposition 1.8 gives a way, together with Proposition 1.7, to characterize transport-regular measures. In the second (and last) part, we prove the main theorem (Theorem 2.2) and its generalization (Theorem 2.5) thanks to explicit constructions. 2010 Mathematics Subject Classification. Primary 28A75.

Let X={X(t),t∈ℝ N } be a Gaussian random field with values in ℝ d defined by $$X(t) = (X_1(t), \ldots, X_d(t)),\quad t \in {\mathbb{R}}^N,$$ where X 1,…,X d are independent copies of a real-valued, centered, anisotropic Gaussian random field X 0 which has stationary increments and the property of strong local nondeterminism. In this paper we determine the exact Hausdorff measure function for the range X([0,1] N ). We also provide a sufficient condition for a Gaussian random field with stationary increments to be strongly locally nondeterministic. This condition is given in terms of the spectral measures of the Gaussian random fields which may contain either an absolutely continuous or discrete part. This result strengthens and extends significantly the related theorems of Berman (Indiana Univ. Math. J. 23:69–94, 1973, Stochast. Process. Appl. 27:73–84, 1988), Pitt (Indiana Univ. Math. J. 27:309–330, 1978) and Xiao (Asymptotic Theory in Probability and Statistics with Applications, pp. 136–176, 2007, A Minicourse on Stochastic Partial Differential Equations, Lecture Notes in Math, vol. 1962, pp. 145–212, 2009), and will have wider applicability beyond the scope of the present paper.

We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo­ geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.

The Pascal triangle, with its associated properties, binomial coefficients, and Fibonacci numbers, is surely among the most familiar mathematical objects. Its "extended" versions, which arise in a fairly natural way, have entries and properties that are the natural generalizations of the original, and although not commonly known, are in many ways equally usefuil and interesting. In what follows we introduce these extended triangles and discuss a few of their properties and applications. The extended Pascal triangle, Tl,,, is the (left-justified) array of coefficients in the expansion of (1 + x + x2 + xx"'-), for m n > O. These arrays may have been first explicitly discussed by J. E. Freund in a 1956 paper [1], where they arise in the solution of a restricted occupancy problem. There are no references to any previous similar development in that paper, and no connection with a generating function. N. Ya. Vilenkin's 1971 book on combinatorics [2, Chap. 5] discusses these generalizations (there called m-arithmetical triangles), in this case arriving at them through chessboard problems; again, they'are not connected with a generating function, and there are no references to previous work. S. J. Turner introduced these triangles (there called Pascal-T triangles, a name now also frequently used) in a 1979 paper [3] on a probability problem. In 1984 [4] and 1986 [5] the author proved a number of theorems on the counting properties of these arrays and gave examples of their use in combinatorics and reliability theory. Some problems related to the kinds of results discussed in this paper recently appear (although without an explicit connection to these triangles) in a collection [6] of problems on combinatorics by I. Tomescu (see, e.g., such problems as 1.9, 1.10, 1.11, 1.19).

We characterize the polynomial decay of orbits of Hilbert space $C_0$-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in the paper by C.J.K.Batty and T.Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces (J. Evol. Eq. 2008), are sharp. This settles a conjecture posed in the paper by C.J.K.Batty and T.Duyckaerts., 21 pages, to appear in Matematische Annalen

In the present paper, we introduce a new concept of positive p-majorizing operators as a dual notion of positive p-summing operators and generalize the concept of majorizing operators introduced by Schaefer (Isr J Math 13:400–415, 1972). We introduce the concept of positive (p, q)-dominated operators and prove a positive version of the famous Kwapien’s factorization theorem for (p, q)-dominated operators via positive p-majorizing operators. We also introduce the notion of disjoint p-summing operators which is a new larger class of operators than positive p-summing operators and use it to characterize the Radon–Nikodým property. Finally, we investigate the maximal properties of these four classes of operators and prove that they are maximal in corresponding sense.

International audience; In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian. In the study of Lagrangian systems, John Mather introduced several invariant sets composed of globally minimizing extremals. He developed methods to construct several orbits undergoing interesting behaviors in phase space under some assumptions on these invariant sets, see [14]. In order to pursue this theory and to apply it on examples, it is necessary to have tools to describe precisely the invariant sets. At least two points of view can be adopted. One can study the invariant set from a purely topological point of view in the style of Conley as compact metric spaces with flows, and study their transitive components. One can also study these set from the point of view of action minimization, and decompose them in invariant subsets that have been called static classes. These points of view are very closely related, but each of them has specific features. For example, understanding the decomposition in static classes is necessary for the variational construction of interesting orbits, while the topological decomposition behaves well under perturbations. Our goal in the present paper is to explicit the links between these two decompositions. We explain that the topological decomposition is finer than the variational one, and that they coincide for most (but not all) systems. As an application, we prove a result of semi-continuity of the so-called Aubry set as a function of the Lagrangian, under certain non-degeneracy hypotheses. The semi-continuity of the Aubry set is a subtle problem, which has remained open for several years, until John Mather gave a counter example, see §18 in [16]. In the same paper, he also states without proof that semi-continuity holds under appropriate hypotheses. Our result extends the one of Mather. The methods we use are inspired from the recent work of Fathi, Figalli and Rifford, [9].

In this paper, we mainly investigate two versions of the Bohr theorem for slice regular functions over the largest alternative division algebras of octonions $\mathbb {O}$. To this end, we establish the coefficient estimates for self-maps of the unit ball of $\mathbb {O}$ and the Carathéodory class in this setting. As a further application of the coefficient estimate, the 1/2-covering theorem is also proven for slice regular functions with convex image.

Let C be a nonempty closed convex subset of a real Hilbert space H and $$T: C\rightarrow CB(H)$$ T : C → C B ( H ) be a multi-valued Lipschitz pseudocontractive nonself mapping. A Halpern–Ishikawa type iterative scheme is constructed and a strong convergence result of this scheme to a fixed point of T is proved under appropriate conditions. Moreover, an iterative method for approximating a fixed point of a k-strictly pseudocontractive mapping $$T: C\rightarrow Prox(H)$$ T : C → P r o x ( H ) is constructed and a strong convergence of the method is obtained without end point condition. The results obtained in this paper improve and extend known results in the literature.

Fomin and Kirillov initiated a line of research into the realization of the cohomology and K-theory of generalized flag varieties G/B as commutative subalgebras of certain noncommutative algebras. This approach has several advantages, which we discuss. This paper contains the most comprehensive result in a series of papers related to the mentioned line of research. More precisely, we give a model for the T -equivariant K-theory of a generalized flag variety KT (G/B) in terms of a certain braided Hopf algebra called the Nichols-Woronowicz algebra. Our model is based on the Chevalley-type multiplication formula for KT (G/B) due to the first author and Postnikov; this formula is stated using certain operators defined in terms of so-called alcove paths (and the corresponding affine Weyl group). Our model is derived using a type-independent and concise approach. Dedicated to Professor Kenji Ueno on the occasion of his sixtieth birthday

In this paper, we mainly discuss the constructions of some new K-g-frames which differ from the existing methods. Meanwhile, we use the relation between a positive operator and the frame operator of a K-g-frame to yield a new K-g-frame. We also obtain a necessary and sufficient condition to generate a new K-g-frame. In addition, we correct some recent results which were obtained by Huang and Leng. In the end, we give an equivalent characterization to construct some new tight K-g-frames by two given g-Bessel sequences. Our results generalize and improve some remarkable results.

It is known that under the Dirichlet product, the set of arithmetic functions in several variables becomes a unique factorization domain. A. Zaharescu and M. Zaki proved an analog of the ABC conjecture in this ring and showed that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$). It is also known that under the Cauchy product, the set of arithmetic functions becomes a unique factorization domain. In this paper, we consider the ring of arithmetic functions in several variables under the Cauchy product and prove an analog of the ABC conjecture in this ring to show that there exists a non-trivial solution to the Fermat equation $$z^n=x^n+y^n$$ ($$n\ge 3$$).

Let V be a finite dimensional vector space over a local field. Let us say that a complex function on V is elementary if it is a product of the additive character of a rational function Q on V and multiplicative characters of polynomials on V. In this paper we study when the Fourier transform of an elementary function is elementary. If Q has a nonzero Hessian, a necessary condition for this is that the Legendre transform Q_* of Q is rational. The basic example is a nondegenerate quadratic form. We study such functions Q, give examples, and find all of them such that both Q and Q_* are of the form f(x)/t, where f is a cubic form in many variables (the simplest case after quadratic forms). It turns out that this classification is closely related to Zak's classification of Severi varieties. The second half of the paper is devoted to finding and classifying elementary functions with elementary Fourier transforms when Q is a fixed function with rational Q_*. We consider the simplest case when Q is a monomial, and classify combinations of multiplicative characters that can arise. The answer (for real and complex fields) is given in terms of exact covering systems. We also describe examples related to prehomogeneous vector spaces. Finally, we consider examples over p-adic fields, and in particular give a local proof of an integral formula of D.K. that could previously be proved only by a global method., Comment: 31 pages, latex; the proof of Theorem 3.10 given in the previous version was found to be incomplete. We have added a remark about this and a reference to a paper by Chaput and Sabatino that gives a complete proof

The main goal of the present paper is two-fold. First we extend the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to the class of toroidal varieties with stratifications (which is the main body of the paper). Second we give a proof of the following weak factorization theorem as an application and illustration of the theory: A birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow ups and blow downs with smooth centers. Another proof of the weak factorization theorem appeared in a joint paper with Abramovich, Karu and Matsuki (math.AG/9904135) In that paper the theorem is stated and proven in general for proper algebraic and analytic spaces., Comment: 69 pages, a revised version with conceptual simplifications. The present version is self-contained and it does not rely on weak factorization theorem for toric varieties

In the present paper, we face the problem of classifying classes of orientable PL 5-manifolds M5 with h ≥ 1 boundary components, by making use of a combinatorial invariant called regular genusG(M5). In particular, a complete classification up to regular genus five is obtained: where denotes the regular genus of the boundary ∂M5 and denotes the connected sumof h ≥ 1 orientable 5-dimensional handlebodies 𝕐αi of genus αi ≥ 0 (i = 1, . . . ,h), so that .Moreover, we give the following characterizations of orientable PL 5-manifolds M5 with boundary satisfying particular conditions related to the “gap” between G(M5) and either G(∂M5) or the rank of their fundamental group rk(π1(M5)): Further, the paper explains how the above results (together with other known properties of regular genus of PL manifolds) may lead to a combinatorial approach to 3-dimensional Poincaré Conjecture.

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which underlies the R-matrix actions arising from the Levi subalgebras of U_h(g) and the quantum Weyl group action of the generalised braid group B_g can be transferred to integrable, category O representations of g. We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, R--matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when g is semisimple, and is used in arXiv:1512.03041 to describe the monodromy of the rational Casimir connection of g in terms of the quantum Weyl group operators of U_h(g). Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the PROP describing a Lie bialgebra graded by the non-negative roots of g., Expanded Introduction and Sec. 5 to discuss convolution product (5.11), cosimplicial structure on basis elements (5.13) and module structure on coinvariants (5.15). Minor revisions in Sec. 7.1 (gradings), 7.4 (deformation DY modules), 9.7 (exposition), 15.7 (Drinfeld double) and 15.15 (rigidity for diagrammatic KM algebras). Final version, to appear in Adv. Math. 81 pages

This paper presents a method for deriving optimal lower and upper bounds for probabilities and conditional probabilities (given a σ-field) for various combinations of events. The optimality is understood as the possibility that inequalities become equalities for some sets of events. New generalizations of the Jordan formula and the Bonferroni inequalities are obtained. The corresponding conditional versions of these results are also considered.

Given a fixed sigma-finite measure space $$\left( X,\mathscr {B},\nu \right) $$ , we shall study an associated family of positive definite kernels K. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $$\nu $$ is infinite, but sigma-finite. We introduce such positive definite kernels $$K\left( \cdot ,\cdot \right) $$ with the two variables from the subclass of the sigma-algebra $$\mathscr {B}$$ whose elements are sets with finite $$\nu $$ measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for K, and we give necessary and sufficient conditions for K to have realizations and factorizations in $$L^{2}\left( \nu \right) $$ . Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.