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$D$.     

The domain of attraction and the stability region for stochastic partial differential equations with delays

In this paper, a stochastic partial differential equation with delays is considered. On the basis of the properties of nonnegative matrices, stochastic convolution and the inequality technique, sufficient conditions for determining the domain of p$p$th-moment attraction and the p$p$th-moment asymptotic stability region are obtained. An example is also discussed, to illustrate the efficiency of the results obtained.     

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.     

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear (n,δ)$(n,\delta )$-width and p$p$-average linear n$n$-width of diagonal matrix M$M$ are determined.     

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$h$.     

Nonlinear stability of general linear methods for neutral delay differential equations

This paper is concerned with the numerical solution of neutral delay differential equations (NDDEs). We focus on the stability of general linear methods with piecewise linear interpolation. The new concepts of GS(p)$GS\left(p\right)$-stability, GAS(p)$GAS\left(p\right)$-stability and weak GAS(p)$GAS\left(p\right)$-stability are introduced. These stability properties for (k,p,0)$(k,p,0)$-algebraically stable general linear methods (GLMs) are further investigated. Some extant results are unified.     

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.     

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.     

On prime ideals and radicals of polynomial rings and graded rings

We extend some known results on radicals and prime ideals from polynomial rings and Laurent polynomial rings to Z$\mathbb{Z}$-graded rings, i.e, rings graded by the additive group of integers. The main of them concerns the Brown–McCoy radical G$G$ and the radical S$S$, which for a given ring A$A$ is defined as the intersection of prime ideals I$I$ of A$A$ such that A/I$A/I$ is a ring with a large center. The studies are related to some open problems on the radicals G$G$ and S$S$ of polynomial rings and situated in the context of Koethe’s problem.     

Newton’s method and high-order algorithms for the nth root computation

Two modifications of Newton’s method to accelerate the convergence of the n$n$th root computation of a strictly positive real number are revisited. Both modifications lead to methods with prefixed order of convergence p∈N,p≥2$p\in \mathbb{N},p\ge 2$. We consider affine combinations of the two modified p$p$th-order methods which lead to a family of methods of order p$p$ with arbitrarily small asymptotic constants. Moreover the methods are of order p+1$p+1$ for some specific values of a parameter. Then we consider affine combinations of the three methods of order p+1$p+1$ to get methods of order p+1$p+1$ again with arbitrarily small asymptotic constants. The methods can be of order p+2$p+2$ with arbitrarily small asymptotic constants, and also of order p+3$p+3$ for some specific values of the parameters of the affine combination. It is shown that infinitely many p$p$th-order methods exist for the n$n$th root computation of a strictly positive real number for any p≥3$p\ge 3$.     

Modules satisfying the weak Nakayama property

Let R$R$ be a commutative ring with identity. We will say that an R$R$-module M$M$ satisfies the weak Nakayama property, if IM=M$IM=M$, where I$I$ is an ideal of R$R$, implies that for any x∈M$x\in M$ there exists a∈I$a\in I$ such that (a−1)x=0$(a-1)x=0$. In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R$R$ is a local ring, then R$R$ is a Max ring if and only if J(R)$J\left(R\right)$, the Jacobson radical of R$R$, is T$T$-nilpotent if and only if every R$R$-module satisfies the weak Nakayama property.     

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.     

Sensitivity of Boolean formulas

The sensitivity set of a Boolean function at a particular input is the set of input positions where changing that one bit changes the output. Analogously we define the sensitivity set of a Boolean formula in a conjunctive normal form at a particular truth assignment, it is the set of positions where changing that one bit of the truth assignment changes the evaluation of at least one of the conjunct in the formula. We consider Boolean formulas in a generalized conjunctive normal form. Given a set ?$?$ of Boolean functions, an ?$?$-constraint is an application of a function from ?$?$ to a tuple of literals built upon distinct variables, an ?$?$-formula is then a conjunction of ?$?$-constraints. In this framework, given a truth assignment I$I$ and a set of positions S$S$, we are able to enumerate all ?$?$-formulas that are satisfied by I$I$ and that have S$S$ as the sensitivity set at I$I$. We prove that this number depends on the cardinality of S$S$ only, and can be expressed according to the sensitivity of the Boolean functions in ?$?$.     

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.     

Products of conjugacy classes in finite and algebraic simple groups

We prove the Arad–Herzog conjecture for various families of finite simple groups — if A$A$ and B$B$ are nontrivial conjugacy classes, then AB$AB$ is not a conjugacy class. We also prove that if G$G$ is a finite simple group of Lie type and A$A$ and B$B$ are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB$AB$ is not a conjugacy class. We also prove a strong version of the Arad–Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product. A special case of this has been used by Prasad to prove a uniqueness result for Tits systems in quasi-reductive groups. Our final result is a generalization of the Baer–Suzuki theorem for p$p$-elements with p≥5$p\ge 5$.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.