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A lattice of definability subspaces in the order of rational numbers is described. It is proved that this lattice consists of five subspaces defined in the paper that are generated by the following relations: “equality,” “less,” “between,” “cycle,” and “linkage.” For each of the subspaces, its width (the minimum number of arguments of a generating relation) is found and a convenient description of the automorphism group is given. Although the structure of this lattice was known previously, the proof in the paper is of syntactic nature and avoids the use of a group-theoretical method.

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.

Random sequences with random or stochastic indices controlled by a doubly stochastic Poisson process are considered in this paper. A Poisson stochastic index process (PSI-process) is a random process with the continuous time ψ(t) obtained by subordinating a sequence of random variables (ξj), j = 0, 1, …, by a doubly stochastic Poisson process Π1(tλ) via the substitution ψ(t) = $${{\xi }_{{{{\Pi }_{1}}(t\lambda )}}}$$ , t $$ \geqslant $$ 0, where the random intensity λ is assumed independent of the standard Poisson process Π1. In this paper, we restrict our consideration to the case of independent identically distributed random variables (ξj) with a finite variance. We find a representation of the fractional Ornstein–Uhlenbeck process with the Hurst exponent H ∈ (0, 1/2) introduced and investigated by R. Wolpert and M. Taqqu (2005) in the form of a limit of normalized sums of independent identically distributed PSI-processes with an explicitly given distribution of the random intensity λ. This fractional Ornstein–Uhlenbeck process provides a local, at t = 0, mean-square approximation of the fractional Brownian motion with the same Hurst exponent H ∈ (0, 1/2). We examine in detail two examples of PSI-processes with the random intensity λ generating the fractional Ornstein–Uhlenbeck process in the Wolpert and Taqqu sense. These are a telegraph process arising when ξ0 has a Rademacher distribution ±1 with the probability 1/2 and a PSI-process with the uniform distribution for ξ0. For these two examples, we calculate the exact and the asymptotic values of the local modulus of continuity for a single PSI-process over a small fixed time span.

A hypergraph $$H=(V,E)$$ has property $$B_k$$ if there exists a 2-coloring of the set $$V$$ such that each edge contains at least $$k$$ vertices of each color. We let $$m_{k,g}(n)$$ and $$m_{k,b}(n)$$ , respectively, denote the least number of edges of an $$n$$ -homogeneous hypergraph without property $$B_k$$ which contains either no cycles of length at least $$g$$ or no two edges intersecting in more than $$b$$ vertices. In the paper, upper bounds for these quantities are given. As a consequence, we obtain results for $$m^{*}_k(n)$$ , i.e., for the least number of edges of an $$n$$ -homogeneous simple hypergraph without property $$B_k$$ . Let $$\Delta(H)$$ be the maximal degree of vertices of a hypergraph $$H$$ . By $$\Delta_k(n,g)$$ we denote the minimal degree $$\Delta$$ such that there exists an $$n$$ -homogeneous hypergraph $$H$$ with maximal degree $$\Delta$$ and girth at least $$g$$ but without property $$B_k$$ . In the paper, an upper bound for $$\Delta_k(n,g)$$ is obtained.

In this paper, the concept of a fuzzy soft point is redefined, and the definition of a fuzzy soft net in a fuzzy soft topological space is given. On this basis, the convergence of a fuzzy soft net is defined by using the Q-neighborhood theory, and the continuity of fuzzy soft mappings is characterized by the net. The obtained results demonstrate that the concepts proposed in this paper are very suitable and will provide powerful research tools for further research in this field.

Fix $m\geq 0$, and let $A=\left( A_{ij}\left( x\right) \right) _{1\leq i\leq N,1\leq j\leq M}$ be a matrix of semialgebraic functions on $\mathbb{R}^{n}$ or on a compact subset $E \subset \mathbb{R}^n$. Given $f=\left( f_{1},\cdots ,f_{N}\right) \in C^{\infty }\left( \mathbb{R}^{n},\mathbb{R}^{N}\right) $, we consider the following system of equations \begin{equation} \sum_{j=1}^{M}A_{ij}\left( x\right) F_{j}\left( x\right) =f_{i}\left( x\right) \text{ }\left( i=1,\cdots ,N\right) \text{.} \end{equation} In this paper, we give algorithms for computing a finite list of linear partial differential operators such that $AF= f$ admits a $C^m(\mathbb{R}^n, \mathbb{R}^M)$ solution $F=(F_1,\cdots, F_M)$ if and only if $f=(f_1,\cdots, f_N)$ is annihilated by the linear partial differential operators., 48 pages, see also the related paper "Generators for the $C^m$-closures of Ideals"

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [{\it Invent. Math.} {\bf 138}(3) (1999), 451--494] resolving Sprind\v zuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. {\it Selecta Math.} {\bf 10} (2004), 479--523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson--Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In the first of this series of papers [{\it Selecta Math.} {\bf 24}(3) (2018), 2165--2206], we introduce and develop a systematic account of two classes of measures, which we call {\it quasi-decaying} and {\it weakly quasi-decaying}. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ``inherited exponent of irrationality'' version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson--Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying., Comment: Link to Part I: arXiv:1504.04778

A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.

The aim of this (mostly expository) article is to show a connection between the finiteness conditions arising in twisted K-theory. There are two different conditions arising naturally in two main approaches to the problem of computing the index of the appropriate family of elliptic operators (the approach of Nistor and Troitsky and the approach of Mathai, Melrose, and Singer). These conditions are formulated absolutely differently, but in some sense they should be close to each other. In this paper, we find this connection and prove the corresponding formal statement. Thereby it is shown that these conditions map to each other. This opens a possibility to synthesize these approaches. It is also shown that the finiteness condition arising in the paper of Nistor and Troitsky is a special case of the finiteness condition that appears in the paper of Emerson and Meyer, where the theorem of Nistor and Troitsky is proved not only for the case of a bundle of Lie groups, but also for the case of a general groupoid.

Recently, a unified integral operator has been introduced by Farid, 2020, which produces several kinds of known fractional and conformable integral operators defined in recent decades (Kwun, 2019, Remarks 6 and 7). The aim of this paper is to establish bounds of this unified integral operator by means of α,m-convex functions. The resulting inequalities provide the bounds of all associated fractional and conformable integral operators in a compact form. Also, the results of this paper hold for different kinds of convex functions connected with α,m-convex functions.

We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.

The paper concerns with properties of approximate Benson and Henig proper efficient solutions of vector equilibrium problems. Relationships between both kinds of solutions and the approximate weak efficient solutions of the problem are stated. Roughly speaking, it is proved that they coincide provided that the ordering cone of the problem can be separated from another closed cone by an approximating sequence of dilating cones. As a result, the limit behavior of approximate Benson and Henig proper efficient solutions when the error tends to zero is studied. It is shown that efficient and proper efficient solutions of the problem can be obtained as the limit of approximate Benson and Henig proper efficient solutions whenever a suitable domination set is considered. Finally, optimality conditions for approximate Benson and Henig proper efficient solutions are derived by approximate solutions of scalar equilibrium problems. They are formulated via linear scalarization and the necessary conditions hold true in problems satisfying a kind of nearly subconvexlikeness condition. The main results of the paper generalize some recent ones because they involve weaker assumptions.

In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbed nonhomogeneous equations in Orlicz–Sobolev spaces and to nonlocal problems in fractional Sobolev spaces.

In this paper we consider one dimensional mean-field backward stochastic differential equations (BSDEs) under weak assumptions on the coefficient. Unlike [3], the generator of our mean-field BSDEs depends not only on the solution (Y, Z) but also on the law PY of Y. The first part of the paper is devoted to the existence and uniqueness of solutions in Lp, 1 < p ≤ 2, where the monotonicity conditions are satisfied. Next, we show that if the generator f is uniformly continuous in (μ, y, z), uniformly with respect to (t, ω), and if the terminal value ξ belongs to Lp(Ω, F, P) with 1 < p ≤ 2, the mean-field BSDE has a unique Lp solution.

This article gives a class of Nullstellensätze for noncommutative polynomials. The singularity set of a noncommutative polynomial $f=f(x_1,\dots ,x_g)$ is $\mathscr{Z}(\,f)=(\mathscr{Z}_n(\,f))_n$, where $\mathscr{Z}_n(\,f)=\{X \in{\operatorname{M}}_{n}({\mathbb{C}})^g \colon \det f(X) = 0\}.$ The 1st main theorem of this article shows that the irreducible factors of $f$ are in a natural bijective correspondence with irreducible components of $\mathscr{Z}_n(\,f)$ for every sufficiently large $n$. With each polynomial $h$ in $x$ and $x^*$ one also associates its real singularity set $\mathscr{Z}^{{\operatorname{re}}}(h)=\{X\colon \det h(X,X^*)=0\}$. A polynomial $f$ that depends on $x$ alone (no $x^*$ variables) will be called analytic. The main Nullstellensatz proved here is as follows: for analytic $f$ but for $h$ dependent on possibly both $x$ and $x^*$, the containment $\mathscr{Z}(\,f) \subseteq \mathscr{Z}^{{\operatorname{re}}} (h)$ is equivalent to each factor of $f$ being “stably associated” to a factor of $h$ or of $h^*$. For perspective, classical Hilbert-type Nullstellensätze typically apply only to analytic polynomials $f,h $, while real Nullstellensätze typically require adjusting the functions by sums of squares of polynomials (sos). Since the above “algebraic certificate” does not involve a sos, it seems justified to think of this as the natural determinantal Hilbert Nullstellensatz. An earlier paper of the authors (Adv. Math. 331 (2018): 589–626) obtained such a theorem for special classes of analytic polynomials $f$ and $h$. This paper requires few hypotheses and hopefully brings this type of Nullstellensatz to near final form. Finally, the paper gives a Nullstellensatz for zeros ${\mathcal{V}}(\,f)=\{X\colon f(X,X^*)=0\}$ of a hermitian polynomial $f$, leading to a strong Positivstellensatz for quadratic free semialgebraic sets by the use of a slack variable.

In this paper, we study the regularity of the free boundaries of the parabolic double obstacle problem for the heat operator and fully nonlinear operator. The result in this paper are generalizations of the theory for the elliptic problem in Lee et al. (Calc Var Partial Differ Equ 58(3):104, 2019) and Lee and Park (The regularity theory for the double obstacle problem for fully nonlinear operator, , 2018) to parabolic case and also the theory for the parabolic single obstacle problem in Caffarelli et al. (J Am Math Soc 17(4):827–869, 2004) to double obstacle case. New difficulties in the theory which are generated by the characteristic of parabolic PDEs and the existence of the upper obstacle are discussed in detail. Furthermore, the thickness assumptions to have the regularity of the free boundary are carefully considered.

In the classical theory of solid mechanics, the behavior of solids is described by partial differential equations (PDE) through Newton’s second law of motion. However, when spontaneous cracks and fractures exist, such PDE models are inadequate to characterize the discontinuities of physical quantities such as the displacement field. Recently, a peridynamic continuum model was proposed which only involves the integration over the differences of the displacement field. A linearized peridynamic model can be described by the integro-differential equation with initial values. In this paper, we study the well-posedness and regularity of a linearized peridynamic model with singular kernel. The novelty of the paper is that the singular kernel is represented as the Laplacian of a regular function. This let to convert the model to an operator valued Volterra integral equation. Then the existence and regularity of the solution of the peridynamics problem are established through the study of the Volterra integral equation.

The paper deals with the abelian cancellative semigroups and the reduced semigroup C*-algebras. It is supposed that there exist epimorphisms from the semigroups onto the group of integers modulo n. For these semigroups we study the structure of the reduced semigroup C*-algebras which are also called the Toeplitz algebras. Such a C*-algebra can be defined for any non-abelian left cancellative semigroup. It is a very natural object in the category of C*-algebras because this algebra is generated by the left regular representation of a semigroup. In the paper, by a given epimorphism σ we construct the grading of a semigroup C*-algebra. To this aim the notion of the σ-index of a monomial is introduced. This notion is the main tool in the construction of the grading. We make use of the σ-index to define the linear independent closed subspaces in the semigroup C*-algebra. These subspaces constitute the C*-algebraic bundle, or the Fell bundle, over the group of integers modulo n. Moreover, it is shown that this grading of the reduced semigroup C*-algebra is topological. As a corollary, we obtain the existence of the contractive linear operators that are non-commutative analogs of the Fourier coefficients. Using these operators, we prove the result on the geometry of the underlying Banach space of the semigroup C*-algebra

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.

An open problem proposed by Rhoades is the following. Is there a contractive condition which guarantees the existence of a fixed point, but does not require the mapping to be continuous at the point? In this paper, we generalize a celebrated result of Eshaghi et al., [On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18 (2017), 569–578], which allows us to find a new solution to this open problem. Furthermore we show that a claim of the aforementioned paper, that Banach’s fixed point theorem cannot be applied in their application, is incorrect. Finally, as an application, we prove that a multivalued function satisfying a general linear functional inclusion admits a unique selection fulfilling the corresponding functional equation.

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $\psi$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $\gamma$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $\gamma$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3\pi g}}\cdot\frac{1}{4^g}$ times less frequent., Comment: The current paper (as well as the companion paper arXiv:2007.04740) has grown from arxiv:1908.08611. The conjectures stated in arXiv:1908.08611 are proved by A. Aggarwal in arXiv:2004.05042

In this paper, we suggest a new type of tasks for control theory for stochastic dynamical systems — programmed control with Probability 1 (PCP1). PCP1 is an application of an invariant theory. We use the PCP1 concept for dynamical processes described by a system of Ito differential equations with jump-diffusion (GSDES). The considered equations include the drift, the diffusion, and the jumps, together or not. Features of our approach are both a wide set of dynamical systems and investigation of such systems for their unique trajectories. Our method is based on the concept of a stochastic first integral (SFI) for GSDES and its equations which author studied before. The purpose of the present paper is to construct a differential equation system (both stochastic and deterministic) using a known set of FIs for the investigating process. Several examples are given.

Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.

In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod-Gaussian under an appropriate renormalisation. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod-Gaussian moduli space., Comment: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures

This paper is a follow up of [6]. We investigate the boundedness of the bilinear fractional integral operator introduced by Grafakos in [3]. When the local integrability index $s$ falls 1 with weights and $t$ exceeds 1, He and Yan obtained some results on this operator was worked on Morrey spaces earlier in [7]. Later in the paper [6], we considered the case $t=1$. This paper handles the remaining case $0, Comment: 9 PAGES

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrodinger quantization of an infinite-dimensional Hamiltonian system is often defined using a sigma-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized Lebesgue measure, which is translation-invariant. In implicit form, such a measure was used in the first paper published by Feynman (1948). In this situation, pseudodifferential operators whose symbols are classical Hamiltonian functions are formally defined as in the finite-dimensional case. In particular, they use unitary Fourier transforms which map functions (on a finite-dimensional space) into functions. Such a definition of the infinite-dimensional unitary Fourier transforms has not been used in the literature.

In the present paper, the notion of (additive) local multiplier on the ring of matrices over an arbitrary unital associative ring is introduced and investigated. It is proved that every local left multiplier on the matrix ring over a division ring is a left multiplier. Also, in the present paper, the notion of (additive) local Jordan multiplier on a Jordan ring of symmetric matrices is introduced and investigated. It is proved that every local Jordan multiplier on the Jordan ring of symmetric matrices over the field of rational numbers is a Jordan multiplier. Also corollaries of these results are proved.

All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a−1bma = bn〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G(pr, −pr), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q-finite for no prime q. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p-finite for some prime p, which was announced in the same paper of McCarron, is also given.

In this paper, the fixed-point theorem for monotone contraction mappings in the setting of a uniformly convex smooth Banach space is studied. This paper provides a version of the Banach fixed-point theorem in a complete metric space.

This paper is devoted to the Lp (p > 1) solutions of one-dimensional backward stochastic differential equations (BSDEs for short) with general time intervals and generators satisfying some non-uniform conditions in t and ω. An existence and uniqueness result, a comparison theorem and an existence result for the minimal solutions are respectively obtained, which considerably improve some known works. Some classical techniques used to deal with the existence and uniqueness of Lp (p > 1) solutions of BSDEs with Lipschitz or linear-growth generators are also developed in this paper.

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

In this paper, the structure of relative disposition of 3-vertex cutsets in a 3-connected graph is studied. All such cutsets are divided into structural units – complexes of flowers, of cuts, of single cutsets, and trivial complexes. The decomposition of the graph by a complex of each type is described in detail. It is proved that for any two complexes C1 and C2 of a 3-connected graph G there is a unique part of the decomposition of G by C1 that contains C2. The relative disposition of complexes is described with the help of a hypertree T (G) – a hypergraph any cycle of which is a subset of a certain hyperedge. It is also proved that each nonempty part of the decomposition of G by the set of all of its 3-vertex cutsets is either a part of the decomposition of G by one of the complexes or corresponds to a hyperedge of T (G). This paper can be considered as a continuation of studies begun in the joint paper by D. V. Karpov and A. V. Pastor “On the structure of a 3-connected graph,” published in 2011. Bibliography: 10 titles.

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation., Comment: 23 pages. Section 5.4 corrected

In the study of the constant in Ahlfors' second fundamental theorem involving a set E_{q} of q points, branch values of covering surfaces outside E_{q} bring a lot of troubles. To avoid this situation, for a given surface S, it is useful to construct a new surface So such that L(So) =H(S), and all branch values of So are contained in E_{q}. The goal of this paper is to prove the existence of such So, which generalizes Lemma 9.1 and Theorem 10.1 in Zhang G.Y.: The precise bound for the area-length ratio in Ahifors' theory of covering surfaces. Invent math 191:197-253(2013), This paper is a preparation for the further research on Ahlfors' theorey of covering surfaces

Our paper begins with a revision of the spectral theory for commutative Banach algebras, which enables us to prove the $$L^p_{\omega }$$ -conjecture for locally compact abelian groups. We follow an alternative approach to the one known in the literature. In particular, we do not resort to any structural theorems for locally compact groups at this stage. Subsequently, we discuss nilpotent, locally compact groups. The climax of the paper is the proof of the $$L^p_{\omega }$$ -conjecture for these groups.

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.

This paper presents a brief survey for the theory of stability of weakly hyperbolic invariant sets. It has been proved in several papers that I published along with Pliss and Sell that a weakly hyperbolic invariant set is stable even if the Lipschitz condition fails to hold. However, the uniqueness of leaves of a weakly hyperbolic invariant set of a perturbed system remains an open question. We show that this problem is connected to the so-called plaque expansivity conjecture in the theory of dynamical systems.

This paper deals with properties of the ultraproducts for various structures. We introduce and study the concept of the ergodic action of a group with respect to a normal state on an abelian von Neumann algebra. In particular, we provide an example showing that the ultraproduct of ergodic states, generally speaking, is not ergodic. The ultraproduct of the Radon measures on a compact convex subset of a locally convex space is also investigated in the paper. As is well-known, the study of the extreme points in the state set for a $$C^{*}-$$ algebra is a very interesting problem in itself. Considering the ultraproducts of $$C^{*}$$ -algebras and the states on these algebras, we get quite nontrivial results.

In the paper, the elementary net closure problem is considered. An elementary net (net without a diagonal) σ = (σij)i ≠ j of additive subgroups σij of field k is called “closed” if elementary net group E(σ) does not contain new elementary transvections. Elementary net σ = (σij) is called “supplemented” if table (with a diagonal) σ = (σij), 1 ≤ i, j ≤ n, is a (full) net for some additive subgroups σii of field k. The supplemented elementary nets are closed. The necessary and sufficient condition for the supplementarity of elementary net σ = (σij) is the implementation of inclusions σijσjiσij ⊆ σij (for any i ≠ j). The question (Kourovka Notebook, Problem 19.63) is investigated of whether it true that, for closure of elementary net σ = (σij) it suffices to implement inclusions $$\sigma _{{ij}}^{2}{{\sigma }_{{ji}}}$$ ⊆ σji for any i ≠ j (here, ($$\sigma _{{ij}}^{2}$$ denotes the additive subgroup of field k generated by the squares from σij). The elementary nets for which the latter inclusions are satisfied are called “weakly supplemented elementary nets.” The concepts of supplemented and weakly supplemented elementary nets coincide for fields of odd characteristic. Thus, the aforementioned question of the sufficiency of weak supplementarity for the closure of an elementary net is relevant for the fields of characteristics 0 and 2. In this paper, examples of weakly supplemented but not supplemented elementary nets are constructed for the fields of characteristics 0 and 2. An example of a closed elementary net that is not weakly supplemented is constructed.

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.

This paper deals with normal submanifolds immersed in a Riemannian manifold $${\overline{M}}$$ . We generalized some recent results of surfaces in space forms obtained by Hernandez-Lamoneda and Ruiz-Hernandez (Bull Braz Marh Soc (NS) 49:447–462, 2018) to arbitrary submanifolds. More precisely, given a submanifold M in $${\overline{M}}$$ , we study the submanifolds formed by orthogonal geodesics to M, and call it a ruled normal submanifold to M. In the first part of this paper, we analyze these submanifolds and establish some geometric properties of them. Furthermore, we extend some properties about the lines of curvature and using the ideas of [3] also give an extension of the classical Theorem of Bonnet to hypersurfaces of $${\overline{M}}$$ .

In this paper we consider a link $$B_{n,\rho }$$Bn,ρ between Baskakov type operators $$B_{n,\infty }$$Bn,∞ and genuine Baskakov–Durrmeyer type operators $$ B_{n,1}$$Bn,1 depending on a positive real parameter $$\rho $$ρ. The topic of the present paper is the pointwise limit relation $$\left( B_{n,\rho }f\right) \left( x\right) \rightarrow \left( B_{n,\infty }f\right) \left( x\right) $$Bn,ρfx→Bn,∞fx as $$\rho \rightarrow \infty $$ρ→∞ for $$x\ge 0.$$x≥0. As a main result we derive uniform convergence on each compact subinterval of the positive real axis for all continuous functions f of polynomial growth.

The paper describes the results of a functional-geometric study of the necessary and sufficient conditions for the existence of a differential realization in the terms of the tensor product of real Hilbert spaces. There are considered continuous infinite-dimensional dynamical system in the class of controlled bilinear nonstationary ordinary differential equations of the second order (including quasi-linear hyperbolic models) in a separable Hilbert space. Therefore the topological and metric conditions for the continuity of the RayleighRitz operator with the calculation of the fundamental group of its image are analytically substantiated. The results of paper give incentives for generalizations in the qualitative theory of nonlinear structural identification of higher order multi-linear differential models.

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere ℂ ⁢ ℙ 1 ∖ { 0 , 1 , ∞ } {\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of ℂ ⁢ ℙ 1 ∖ { a 1 , … , a n } {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}} , n ≥ 3 {n\geq 3} , as well, and the metric on ℂ ⁢ ℙ 1 ∖ { 0 , 1 3 , - 1 6 ± 3 6 ⁢ i } {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.

This paper is devoted to some counting functions of level one and level three in the case of quotient space generated by some strictly hyperbolic Fuchsian group and the upper half-plane. Each of the functions is represented as a sum of some explicit part plus the error term. The explicit part is indexed over singularities of the corresponding Selberg zeta function. In particular, the obtained error term is not larger than O ( x 3/4) . The method applied in this paper follows traditional approach for achieving the error terms in the case of locally symmetric spaces of real rank one. In order to establish an analogy with the classical case, we consider the counting functions divided by x and x3, respectively.

In this paper we generalize the result (obtained by the author earlier in the paper titled “Products and coproducts in the category $${{\mathcal {S}}}(B)$$ of Segal topological algebras”) about the existence of the coproduct of objects of the category $${\mathcal S}(B)$$ from finite collection of objects of $${{\mathcal {S}}}(B)$$ to any collection of objects of $${{\mathcal {S}}}(B)$$ .