Strongly quasi-Hamiltonian-connected semicomplete multipartite digraphs

A semicomplete multipartite or semicomplete c$c$-partite digraph D$D$ is a biorientation of a c$c$-partite graph. A semicomplete multipartite digraph D$D$ is called strongly quasi-Hamiltonian-connected, if for any two distinct vertices x$x$ and y$y$ of D$D$, there is a path P$P$ from x$x$ to y$y$ such that P$P$ contains at least one vertex from each partite set of D$D$.     

Cone-separation and star-shaped separability with applications

Let X$X$ be a (real) Banach space, A$A$ be a subset of X$X$ and x∉A$x\notin A$. We present cone-separation in terms of separation by a collection of linear functionals defined on X$X$ and obtain necessary and sufficient conditions for cone-separability A$A$ and x$x$. Also, we give characterizations for star-shaped separability. Finally, as an application of separability, we characterize best approximation problem by elements of star-shaped sets.     

Weak convergence of censored and reflected stable processes

It is shown that if a sequence of open n$n$-sets _Dk$D_k$ increases to an open n$n$-set D$D$ then reflected stable processes in _Dk$D_k$ converge weakly to the reflected stable process in D$D$ for every starting point x$x$ in D$D$. The same result holds for censored α$\alpha$-stable processes for every x$x$ in D$D$ if D$D$ and _Dk$D_k$ satisfy the uniform Hardy inequality. Using the method in the proof of the above results, we also prove the weak convergence of reflected Brownian motions in unbounded domains.     

On the complexity of the bondage and reinforcement problems

Let G=(V,E)$G=(V,E)$ be a graph. A subset D⊆V$D\subseteq V$ is a dominating set if every vertex not in D$D$ is adjacent to a vertex in D$D$. A dominating set D$D$ is called a total dominating set if every vertex in D$D$ is adjacent to a vertex in D$D$. The domination (resp. total domination) number of G$G$ is the smallest cardinality of a dominating (resp. total dominating) set of G$G$. The bondage (resp. total bondage) number of a nonempty graph G$G$ is the smallest number of edges whose removal from G$G$ results in a graph with larger domination (resp. total domination) number of G$G$. The reinforcement (resp. total reinforcement) number of G$G$ is the smallest number of edges whose addition to G$G$ results in a graph with smaller domination (resp. total domination) number. This paper shows that the decision problems for the bondage, total bondage, reinforcement and total reinforcement numbers are all NP-hard.     

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X$X$ in a bounded κ$\kappa$-fat open set; if u$u$ is a positive harmonic function with respect to X$X$ in a bounded κ$\kappa$-fat open set D$D$ and h$h$ is a positive harmonic function in D$D$ vanishing on D^c$D^c$, then the non-tangential limit of u/h$u/h$ exists almost everywhere with respect to the Martin-representing measure of h$h$.     

Disjoint cycles with different length in 4-arc-dominated digraphs

A d$d$-arc-dominated digraph is a digraph D$D$ of minimum out-degree d$d$ such that for every arc (x,y)$(x,y)$ of D$D$, there exists a vertex u$u$ of D$D$ of out-degree d$d$ such that (u,x)$(u,x)$ and (u,y)$(u,y)$ are arcs of D$D$. Henning and Yeo [Vertex disjoint cycles of different length in digraphs, SIAM J. Discrete Math. 26 (2012) 687–694] conjectured that a digraph with minimum out-degree at least four contains two vertex-disjoint cycles of different length. In this paper, we verify this conjecture for 4-arc-dominated digraphs.     

Constant-factor approximation of the domination number in sparse graphs

The k$k$-domination number   of a graph is the minimum size of a set D$D$ such that every vertex of G$G$ is at distance at most k$k$ from D$D$. We give a linear-time constant-factor algorithm for approximation of the k$k$-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, proper classes closed on topological minors and classes of graphs that can be drawn on a fixed surface with bounded number of crossings on each edge.     

Description of extended eigenvalues and extended eigenvectors of integration operators on the Wiener algebra

In the present paper we consider the Volterra integration operator V   on the Wiener algebra W(D)$W(\mathbb{D})$ of analytic functions on the unit disc D$\mathbb{D}$ of the complex plane C$\mathbb{C}$. A complex number λ$\lambda$ is called an extended eigenvalue of V if there exists a nonzero operator A   satisfying the equation AV=λVA$\mathit{AV}=\lambda \mathit{VA}$. We prove that the set of all extended eigenvalues of V   is precisely the set C?{0}$\mathbb{C}?\{0\}$, and describe in terms of Duhamel operators and composition operators the set of corresponding extended eigenvectors of V$V$. The similar result for some weighted shift operator on _?p${?}_p$ spaces is also obtained.     

A short proof of the Doob–Meyer theorem

Every submartingale S$S$ of class D$D$ has a unique Doob–Meyer decomposition S=M+A$S=M+A$, where M$M$ is a martingale and A$A$ is a predictable increasing process starting at 0.     

A note on distance-regular graphs with a small number of vertices compared to the valency

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2$\alpha >2$, there are finitely many distance-regular graphs Γ$\Gamma$ with valency k$k$, diameter D≥3$D\ge 3$ and v$v$ vertices satisfying v≤αk$v\le \alpha k$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3$k\ge 3$, diameter D≥3$D\ge 3$ and _c2≥εk$c_2\ge \epsilon k$ for a given 0<ε<1$0<\epsilon <1$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite).     

Hopf subalgebras and tensor powers of generalized permutation modules

By means of a certain module V$V$ and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R$R$ of a finite-dimensional Hopf algebra H$H$ is finite. The module V$V$ is the counit representation induced from R$R$ to H$H$, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V$V$ is either semisimple with R^∗$R^{\ast }$ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R$R$, then the depth of R$R$ in H$H$ is finite. One assigns a nonnegative integer depth to V$V$, or any other H$H$-module, by comparing the truncated tensor algebras of V$V$ in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character.     

Gradient skewness tensors and local illumination detection for images

In this paper, we propose the definition of D$D$-eigenvalue for an arbitrary order tensor related with a second-order tensor D$D$, and introduce the gradient skewness tensor which involves a three-order tensor derived from the skewness statistic of gradient images. As we happen to find out that the skewness of oriented gradients can measure the directional characteristic of illumination in an image, the local illumination detection problem for an image can be abstracted as solving the largest D$D$-eigenvalue of gradient skewness tensors. We discuss the properties of D$D$-eigenvalues, and especially for gradient skewness tensors we provide the calculation method of its D$D$-eigenvalues and D$D$-characteristic polynomial. Some numerical experiments show its effective application in illumination detection. Our method also presents excellent results in a class of image authenticity verification problems, which is to distinguish artificial “flat” objects in a photograph.     

Quandle cocycles from invariant theory

Let G$G$ be a group. Any G$G$-module M$M$ has an algebraic structure called a G$G$-family of Alexander quandles. Given a 2-cocycle of a cohomology associated with this G$G$-family, topological invariants of (handlebody) knots in the 3-sphere are defined. We develop a simple algorithm to algebraically construct n$n$-cocycles of this G$G$-family from G$G$-invariant group n$n$-cocycles of the abelian group M$M$. We present many examples of 2-cocycles of these G$G$-families using facts from (modular) invariant theory.     

Chern–Simons classes for a superconnection

In this note we define the Chern–Simons classes of a flat superconnection, D+L$D+L$, on a complex Z/2Z$\mathbb{Z}/2\mathbb{Z}$-graded vector bundle E$E$ on a manifold such that D$D$ preserves the grading and L$L$ is an odd endomorphism of E$E$. As an application, we obtain a definition of Chern–Simons classes of a (not necessarily flat) morphism between flat vector bundles on a smooth manifold. An application of Reznikov's theorem shows the triviality of these classes when the manifold is a compact Kähler manifold or a smooth complex quasi-projective variety in degrees >1$>1$.     

Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation

We develop a notion of nonlinear expectation–G$G$-expectation–generated by a nonlinear heat equation with infinitesimal generator G$G$. We first study multi-dimensional G$G$-normal distributions. With this nonlinear distribution we can introduce our G$G$-expectation under which the canonical process is a multi-dimensional G$G$-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô’s type with respect to our G$G$-Brownian motion, and derive the related Itô’s formula. We have also obtained the existence and uniqueness of stochastic differential equations under our G$G$-expectation.     

Fixed point theorem for generalized Φ -pseudocontractive mappings

Let E$E$ be a real Banach space, C$C$ be a nonempty closed convex subset of E$E$ and T:C→C$T:C\to C$ be a continuous generalized Φ$\Phi$-pseudocontractive mapping. It is proved that T$T$ has a unique fixed point in C$C$.     

Tournaments associated with multigraphs and a theorem of Hakimi

A tournament of order n$n$ is usually considered as an orientation of the complete graph _Kn$K_n$. In this note, we consider a more general definition of a tournament that we call aC$C$-tournament, where C$C$ is the adjacency matrix of a multigraph G$G$, and a C$C$-tournament is an orientation of G$G$. The score vector of a C$C$-tournament is the vector of outdegrees of its vertices. In 1965 Hakimi obtained necessary and sufficient conditions for the existence of a C$C$-tournament with a prescribed score vector R$R$ and gave an algorithm to construct such a C$C$-tournament which required, however, some backtracking. We give a simpler and more transparent proof of Hakimi’s theorem, and then provide a direct construction of such a C$C$-tournament which works even for weighted graphs.     

Continuity properties of the ball hull mapping

The ball hull mapping  β$\beta$ associates with each closed bounded convex set K$K$ in a Banach space its ball hull β(K)$\beta \left(K\right)$, defined as the intersection of all closed balls containing K$K$. We are concerned in this paper with continuity and Lipschitz continuity (with respect to the Hausdorff metric) of the ball hull mapping. It is proved that β$\beta$ is a Lipschitz map in finite dimensional polyhedral spaces. Both properties, finite dimension and polyhedral norm, are necessary for this result. Characterizing the ball hull mapping by means ofH$H$-convexity we show, with the help of a remarkable example from combinatorial geometry, that there exist norms with noncontinuous β$\beta$ map, even in finite dimensional spaces. Using this surprising result, we then show that there are infinite dimensional polyhedral spaces (in the usual sense of Klee) for which the map β$\beta$ is not continuous. A property known as ball stability implies that β$\beta$ has Lipschitz constant one. We prove that every Banach space of dimension greater than two can be renormed so that there is an intersection of closed balls for which none of its parallel bodies is an intersection of closed balls, thus lacking ball stability.     

Every point is critical

We show that, for any compact Alexandrov surface S$S$ (without boundary) and any point y$y$ in S$S$, there exists a point x$x$ in S$S$ for which y$y$ is a critical point. Moreover, we prove that uniqueness characterizes the surfaces homeomorphic to the sphere among smooth orientable surfaces.