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The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.

In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.

In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space L loc 1 . The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms $$L_q$$ , $$1\le q

In this paper, we consider boundary-value problems for a new class of hybrid systems of differential equations whose coefficients contain the Dirac delta-function. Hybrid systems are systems that contain both ordinary and partial differential equations; such systems appear, for example, when equations of motion of mechanical systems of rigid bodies attached to a beam by elastic bonds are derived from the Hamilton–Ostrogradsky variational principle. We present examples that lead to such systems and introduce the notions of generalized solutions and eigenvalues of a boundary-value problem. We also compare results of numerical simulations based on methods proposed in this paper with results obtained by previously known methods and show that our approach is reliable and universal.

Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(y)considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.

The first aim of this paper is to investigate the growth of the entire function defined by the Laplace-Stieltjes transform converges on the whole complex plane. By introducing the concept of generalized order, we obtain two equivalence theorems of Laplace-Stieltjes transforms related to the generalized order, A * and λn. The second purpose of this paper is to study the problem on the approximation of this Laplace-Stieltjes transform. We also obtain some theorems about the generalized order, the error, and the coefficients of Laplace-Stieltjes transforms, which are generalization and improvement of the previous results.

In this paper we study multi-dimensional mean-field backward doubly stochastic differential equations (BDSDEs), that is, BDSDEs whose coefficients depend not only on the solution processes but also on their law. The first part of the paper is devoted to the comparison theorem for multi-dimensional mean-field BDSDEs with Lipschitz conditions. With the help of the comparison result for the Lipschitz case we prove the existence of a solution for multi-dimensional mean-field BDSDEs with an only continuous drift coefficient of linear growth, and we also extend the comparison theorem to such BDSDEs with a continuous coefficient.

A sequence $$(v_1,\ldots ,v_k)$$ of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex $$v_i$$ in the sequence totally dominates at least one vertex that was not totally dominated by $$\{v_1,\ldots , v_{i-1}\}$$ and $$\{v_1,\ldots ,v_k\}$$ is a total dominating set of G. The length of a shortest such sequence is the total domination number of G ( $$\gamma _{t}(G)$$ ), while the length of a longest such sequence is the Grundy total domination number of G ( $$\gamma _{gr}^t(G)$$ ). In this paper we study graphs with equal total and Grundy total domination numbers. We characterize bipartite graphs with both total and Grundy total dominations number equal to 4, and show that there is no connected chordal graph G with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=4$$ . The main result of the paper is a characterization of bipartite graphs with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=6$$ proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.

A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.

In order to construct global solutions to two-dimensional (2D for short) Riemann problems for nonlinear hyperbolic systems of conservation laws, it is important to study various types of wave interactions. This paper deals with two types of wave interactions for a 2D nonlinear wave system with a nonconvex equation of state: Rarefaction wave interaction and shock-rarefaction composite wave interaction. In order to construct solutions to these wave interactions, the authors consider two types of Goursat problems, including standard Goursat problem and discontinuous Goursat problem, for a 2D self-similar nonlinear wave system. Global classical solutions to these Goursat problems are obtained by the method of characteristics. The solutions constructed in the paper may be used as building blocks of solutions of 2D Riemann problems.

In this paper, we consider the operator sheaf $$ -\Delta +V+\varepsilon {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right)+{\lambda}^2 $$ in the space L2(ℝ2), where the real-valued potential V depends only on the first variable x1, e is a small positive parameter, λ is the spectral parameter, $$ {\mathrm{\mathcal{L}}}_{\varepsilon}\left(\lambda \right) $$ is a localized operator bounded with respect to the Laplacian −Δ, and the essential spectrum of this operator is independent of e and contains certain critical points defined as isolated eigenvalues of the operator $$ -\frac{d^2}{dx_1^2}+V\left({x}_1\right) $$ in L2(ℝ). The basic result obtained in this paper states that for small values of e, in neighborhoods of critical points mentioned, isolated eigenvalues of the sheaf considered arise. Sufficient conditions for the existence or absence of such eigenvalues are obtained. The number of arising eigenvalues is determined, and in the case where they exist, the first terms of their asymptotic expansions are found.

In this paper, we investigate a mixed fractional integral boundary value problem with p(t)-Laplacian operator. Firstly, we derive the Green function through the direct computation and obtain the properties of Green function. For $$p(t)\ne $$ constant, under the appropriate conditions of the nonlinear term, we establish the existence result of at least one positive solution of the above problem by means of the Leray–Schauder fixed point theorem. Meanwhile, we also obtain the positive extremal solutions and iterative schemes in view of applying a monotone iterative method. For $$p(t)=$$ constant, by using Guo–Krasnoselskii fixed point theorem, we study the existence of positive solutions of the above problem. These results enrich the ones in the existing literatures. Finally, some examples are included to demonstrate our main results in this paper and we give out an open problem.

A goal of the paper is to prove new inequalities connecting some functionals of probability distribution functions. These inequalities are based on the strict convexity of functions used in the definition of the functionals. The starting point is the paper “Cramer–von Mises distance: probabilistic interpretation, confidence intervals and neighborhood of model validation” by Ludwig Baringhaus and Norbert Henze. The present paper provides a generalization of inequality obtained in probabilistic interpretation of the Cramer–von Mises distance. If the equality holds there, then a chance to give characterization of some probability distribution functions appears. Considering this fact and a special character of the functional, it is possible to create a class of free-of-distribution two sample tests.

In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.

International audience; Sampled-data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled-data control systems and certain generalizations. In this paper we derive a Pontryagin maximum principle for general nonlinear finite-dimensional optimal sampled-data control problems with running inequality state constraints. In particular, we obtain a nonpositive averaged Hamiltonian gradient condition with the adjoint vector being a function of bounded variations. Our proof is based on the Ekeland variational principle. In general, optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities (the jumps and the singular part) of the adjoint vector. However in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times. We call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper. Finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions.

In this paper, we prove the weak convergence of a modified extragradient algorithm for solving a variational inequality problem involving a pseudomonotone operator in an infinite dimensional Hilbert space. Moreover, we establish further the R-linear rate of the convergence of the proposed algorithm with the assumption of error bound. Several numerical experiments are performed to illustrate the convergence behaviour of the new algorithm in comparisons with others. The results obtained in the paper have extended some recent results in the literature.

This paper aims to discuss the mathematical details in Lewis’ model by considering the analyticity and integrability conditions of characteristic functions and payoff functions of contingent claims. In his seminal paper, Lewis shows that it is much easier to compute the option value in the Fourier space than computing in terminal security price space. He computes the option value as an integral in the Fourier space, the integrand being some elementary functions and the characteristic functions of a wide range of Levy processes. The model also illustrates how the residue calculus leads to several variations of option formulas through the contour integrals. In this paper, we provide with, to a reasonable extent, some rigor into the mathematical background of Lewis’ model and validate his results for particular Levy processes. We also simply give the analyticity conditions for the characteristic function of the Carr–Geman–Madan–Yor model and a simple derivation of the characteristic function of Kou’s double exponential model.

We show that the phase tropical pair-of-pants is (ambient) isotopic to the complex pair-of-pants. This paper can serve as an addendum to the author's joint paper with Ruddat arXiv:2001.08267 where an isotopy between complex and ober-tropical pairs-of-pants was shown. Thus all three versions are isotopic., 10 pages, 8 figures. arXiv admin note: text overlap with arXiv:2001.08267

In the present paper, we obtain a Halanay inequality on time scales with unbounded coefficient for a dynamic problem, which extends a result of Wen et al. (J. Math. Anal. Appl., 347 (2008), 169–178.) to the inequality of integral type on time scales. Moreover, we list two dynamic problems to which the Halanay inequality obtained above can be applied and prove the zero solution of two delay dynamic problems are asymptotically stable. Moreover, it is worth mentioning that the Halanay inequality obtained in the present paper is more precise than the results in [3, 14, 17].

In this paper, we study the existence and multiplicity of solutions for a class of equations involving a nonlocal term and the biharmonic operator with critical exponent. By new techniques, multiplicity results are established together with variational method. It is worth noting that the method of this paper is obviously different from the existing literature.

In their remarkable paper, Fazli et al. (Int J Comput Math, 2019. https://doi.org/10.1080/00207160.2019.1658870 ) have recently established existence results for fractional Basset–Boussinesq–Oseen equation in an appropriate partially ordered Banach space. In this paper, we improve this study by proposing different approach that provides existence criterion for the addressed equation. The main assumptions are less restrictive and easily verifiable. Examples with graphical representations are constructed to demonstrate consistency to theoretical findings. Besides, we construct an iterative sequence that converges to the unique solution. We end the paper by a conclusion that demonstrates the advantage of our theorem compared to the previous results.

Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.

A very short proof of an unpublished result due to O. Gabber is given. More exactly, let R be a regular local ring containing a finite field k. Let G be a simply-connected reductive group scheme over k. It is proved that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. This is the mentioned unpublished result due to O. Gabber. In this paper, this result is derived from a purely geometric one, proved in another paper of the author and stated in the Introduction.

In a joint paper of the author with Alexei Stepanov, it was established that for any two comaximal ideals A and B of a commutative ring R, A + B = R, and any n ≥ 3 one has [E(n,R,A),E(n,R,B)] = E(n,R,AB). Alec Mason and Wilson Stothers constructed counterexamples demonstrating that the above equality may fail when A and B are not comaximal, even for such nice rings as ℤ [i]. The present note proves a rather striking result that the above equality and, consequently, also the stronger equality [GL(n,R,A), GL(n,R,B)] = E(n,R,AB) hold whenever R is a Dedekind ring of arithmetic type with infinite multiplicative group. The proof is based on elementary calculations in the spirit of the previous papers by Wilberd van der Kallen, Roozbeh Hazrat, Zuhong Zhang, Alexei Stepanov, and the author, and also on an explicit computation of the multirelative SK1 from the author’s paper of 1982, which, in its turn, relied on very deep arithmetical results by Jean-Pierre Serre and Leonid Vaserstein (as corrected by Armin Leutbecher and Bernhard Liehl). Bibliography: 50 titles.

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.

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.

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.

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.

This paper deals with abstract quasi-variational-like problems, which cover vector quasi-variational problems with set-valued mappings, for instance. The main objective of this paper is to study existence results for generalized solutions of quasi-variational-like problems extending the approach by Jadamba, Khan and Sama, see (Optim. Lett. 6, 1221–1231, 2012). It is well-known that a quasi-variational-like problem can conveniently be formulated as a set-valued fixed-point problem by using the so-called variational selection. Since the values of the corresponding set-valued mapping are non-convex in general and since the existing fixed-point theorems for non-convex mappings require very stringent conditions on the data of the problem, a few existence results have been obtained in the case where the underlying constraining set is bounded or even compact. In this paper, we focus on the relationship between solutions of a quasi-variational-like problem and generalized solutions of an optimization problem minimizing the difference between inputs and outputs of the variational selection. Doing so, we can solve the latter problem by the well-known Weierstras theorem with much less restrictive assumptions imposed on the given data. This approach allows to consider problems where the variational selection is non-convex and/or the underlying constraining set is unbounded.

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

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

Insolvency risk measures play important role in the theory and practice of risk management. In this paper, we provide a numerical procedure to compute vectors of their exact values and prove for them new upper and/or lower bounds which are shown to be attainable. More precisely, we investigate a general insolvency risk measure for a regime-switching Sparre Andersen model in which the distributions of claims and/or wait times are driven by a Markov chain. The measure is defined as an arbitrary increasing function of the conditional expected harm of the deficit at ruin, given the initial state of the Markov chain. A vector-valued operator L, generated by the regime-switching process, is introduced and investigated. We show a close connection between the iterations of L and the risk measure in a finite horizon. The approach assumed in the paper enables to treat in a unified way several discrete and continuous time risk models as well as a variety of important vector-valued insolvency risk measures.

This paper investigates, using a simulation program, a retrial queuing system with a single server which is subject to random breakdowns. The number of sources of calls is finite, and collisions can take place. We assume that the failure of the server blocks the system’s operation such that newly arriving customers cannot enter the system, contrary to an earlier paper where the failure does not affect the arrivals. All the random variables included in the model construction are assumed to be independent of each other, and all times are exponentially distributed except for the service time, which is gamma distributed. The novelty of this analysis is the inspection of the blocking effect on the performance measures using different distributions. Various figures represent the impact of the squared coefficient of the variation of the service time on the main performance measures such as the mean and variance of the number of customers in the system, the mean and variance of the response time, the mean and variance of the time a customer spends in the service, and the mean and variance of the sojourn time in the orbit.

This paper continues a series of studies dealing with noninvasive preoperative methods for localizing eloquent areas of the human brain. The inverse problem of magnetoencephalography (MEG) is illposed and difficult for both analytical and numerical solutions. An analytical formula is derived for the solution of the forward problem that computes the magnetic field on the surface of the head from the known location and orientation of a current dipole in the low-frequency approximation in the spherical model. In addition, the paper considers the question of stability of solutions of the inverse problem of MEG to the effect of noise. The solution is unstable to the effect of noise on its angular component, but the deviation from the true solution is much less than the noise variance.

A graph Γ is called a bi-circulant if it admits a cyclic group as a group of automorphisms acting semiregularly on the vertices of Γ with two orbits. The characterization of tetravalent edgetransitive bi-circulants was given in several recent papers. In this paper, a classification is given of connected tetravalent vertex-transitive bi-circulants of order twice an odd integer.

The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.

This is a follow-up of the paper J. Fernandez-Bonder, A. Ritorto and A. Salort, H-convergence result for nonlocal elliptic-type problems via Tartar’s method, SIAM J. Math. Anal., 49 (2017), pp. 2387–2408, where the classical concept of H-convergence was extended to fractional $$p$$ -Laplace type operators. In this short paper we provide an explicit characterization of this notion by demonstrating that the weak- $$*$$ convergence of the coefficients is an equivalent condition for H-convergence of the sequence of nonlocal operators. This result takes advantage of nonlocality and is in stark contrast to the local $$p$$ -Laplacian case.

In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation, $${\rm{i}}u_t-u_{xx}+{\rm{i}}(|u|^4u)_x=0, \;\; x\in\mathbb{T}.$$ The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established. The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem. We mention that in the present paper the mean value of u does not need to be zero, but small enough, which is different from the assumption (1.7) in Geng-Wu [J. Math. Phys., 53, 102702 (2012)].

In this paper, we have pointed out that the proof of Theorem 11 in the recent paper (Lafhim in Positivity, 2019. https://doi.org/10.1007/s11117-019-00685-1 ) is erroneous. Using techniques from variational analysis, we propose other proofs to detect necessary optimality conditions in terms of Karush–Kuhn–Tucker multipliers. Our main results are given in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first order necessary optimality conditions are then given in the smooth setting while using the generalized differentiation calculus of Mordukhovich.