A sharp upper bound on the signless Laplacian spectral radius of graphs

Let G be a simple connected graph of order n   with degree sequence _d1,_d2,…,_dn$d_1,d_2,\dots ,d_n$ in non-increasing order. The signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ of G   is the largest eigenvalue of its signless Laplacian matrix Q(G)$Q(G)$. In this paper, we give a sharp upper bound on the signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ in terms of _di$d_i$, which improves and generalizes some known results.     

War and peace in veto voting

Let I={_i1,…,_in}$I=\{i_1\text{,}\dots \text{,}{i}_n\}$ be a set of voters (players) and A={_a1,…,_ap}$A=\{a_1\text{,}\dots \text{,}{a}_p\}$ be a set of candidates (outcomes). Each voter i∈I$i\in I$ has a preference P_i over the candidates. We assume that P_i is a complete order on A  . The preference profile P={_Pi,i∈I}$P=\{P_i\text{,}i\in I\}$ is called a situation. A situation is called war if the set of all voters I is partitioned in two coalitions K₁ and K₂ such that all voters of K_i have the same preference, i=1,2,$i=1\text{,}2\text{,}$ and these two preferences are opposite. For a simple class of veto voting schemes we prove that the results of elections in all war situations uniquely define the results for all other (peace) situations. In other words, the results depend only on the veto (or effectivity) function. We give several other examples from game (and from graph) theory with the same property.     

Disjoint paths in tournaments

Given k   pairs of vertices (_si,_ti)$(s_i,t_i)$(1≤i≤k)$(1\le i\le k)$ of a digraph G, how can we test whether there exist k   vertex-disjoint directed paths from _si$s_i$ to _ti$t_i$ for 1≤i≤k$1\le i\le k$? This is NP-complete in general digraphs, even for k=2$k=2$ [2], but for k=2$k=2$ there is a polynomial-time algorithm when G is a tournament (or more generally, a semicomplete digraph), due to Bang-Jensen and Thomassen [1]. Here we prove that for all fixed k there is a polynomial-time algorithm to solve the problem when G is semicomplete.     

Invertible bimodule categories over the representation category of a Hopf algebra

For any finite-dimensional Hopf algebra H   we construct a group homomorphism BiGal(H)→BrPic(Rep(H))$\mathrm{BiGal}(H)\to \text{BrPic}(\mathrm{Rep}(H))$, from the group of equivalence classes of H  -biGalois objects to the group of equivalence classes of invertible exact Rep(H)$\mathrm{Rep}(H)$-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H=_Tq$H=T_q$ is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(_Tq)$\mathrm{Rep}(T_q)$-bimodule categories.     

On the least distance eigenvalue of a graph

Denote by D(G)=_(di,j)n×n$D(G)={(d_{i,j})}_{n\times n}$ the distance matrix of a connected graph G with n   vertices, where _dij$d_{ij}$ is equal to the distance between vertices _vi$v_i$ and _vj$v_j$ in G  . The least eigenvalue of D(G)$D(G)$ is called the least distance eigenvalue of G  , denoted by _λn${\lambda }_n$. In this paper, we determine all the graphs with _λn∈[−2.383,0]${\lambda }_n\in [-2.383,0]$.     

Capturing two elements in unavoidable minors of 3-connected binary matroids

Let M be a 3-connected binary matroid and let n   be an integer exceeding 2. Ding, Oporowski, Oxley, and Vertigan proved that there is an integer f(n)$f(n)$ so that if |E(M)|>f(n)$|E(M)|>f(n)$, then M has a minor isomorphic to one of the rank-n wheel, the rank-n   tipless binary spike, or the cycle or bond matroid of _K3,n$K_{3,n}$. This result was recently extended by Chun, Oxley, and Whittle to show that there is an integer g(n)$g(n)$ so that if |E(M)|>g(n)$|E(M)|>g(n)$ and x∈E(M)$x\in E(M)$, then x is an element of a minor of M isomorphic to one of the rank-n wheel, the rank-n   binary spike with a tip and a cotip, or the cycle or bond matroid of _K1,1,1,n$K_{1,1,1,n}$. In this paper, we prove that, for each i   in {2,3}$\{2,3\}$, there is an integer _hi(n)$h_i(n)$ so that if |E(M)|>_hi(n)$|E(M)|>h_i(n)$ and Z is an i-element rank-2 subset of M, then M has a minor from the last list whose ground set contains Z.     

A linearized compact difference scheme for an one-dimensional parabolic inverse problem

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determining the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. The article is devoted to the following parabolic initial-boundary value problem with the control parameter: ∂u/∂t=∂²u/∂x²+p(t)u+?(x,t),0<x<1,0<t?T$\mathrm{\partial }u/\mathrm{\partial }t={\mathrm{\partial }}^{2}u/\mathrm{\partial }{x}^2+p(t)u+?(x\text{,}t)\text{,}0<x<1\text{,}0<t?T$ satisfying u(x,0)=f(x),0<x<1$u(x\text{,}0)=f(x)\text{,}0<x<1$; u(0,t)=_g0(t)$u(0\text{,}t)=g_0(t)$, u(1,t)=_g1(t)$u(1\text{,}t)=g_1(t)$, u(x^∗,t)=E(t),0?t?T$u(x^{\ast }\text{,}t)=E(t)\text{,}0?t?T$ where ?(x,t),f(x),_g0(t),_g1(t)$?(x\text{,}t)\text{,}f(x)\text{,}{g}_0(t)\text{,}{g}_1(t)$ and E(t)$E(t)$ are known functions, u(x,t)$u(x\text{,}t)$ and p(t)$p(t)$ are unknown functions. A linearized compact difference scheme is constructed. The discretization accuracy of the difference scheme is two order in time and four order in space. The solvability of the difference scheme is proved. Some numerical results and comparisons with the difference scheme given by Dehghan are presented. The numerical results show that the linearized difference scheme of this article improve the accuracy of the space and time direction and shorten computation time largely. The method in this article is also applicable to the two-dimensional inverse problem.     

Layered viscosity solutions of nonautonomous Hamilton–Jacobi equations: Semiconvexity and relations to characteristics

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ)$(H,\sigma )$ on a given domain Ω=(0,T)×Rⁿ$\Omega =(0,T)\times {\mathbb{R}}^n$. It is known that, if the Hamiltonian H=H(t,p)$H=H(t,p)$ is not a convex (or concave) function in p  , or H(⋅,p)$H(\cdot ,p)$ may change its sign on (0,T)$(0,T)$, then the Hopf-type formula does not define a viscosity solution on Ω  . Under some assumptions for H(t,p)$H(t,p)$ on the subdomains (_ti,_ti+1)×Rⁿ⊂Ω$(t_i,t_{i+1})\times {\mathbb{R}}^n\subset \Omega$, we are able to arrange “partial solutions” given by the Hopf-type formula to get a viscosity solution on Ω. Then we study the semiconvexity of the solution as well as its relations to characteristics.     

An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G   and 1<p<∞$1<p<\infty$ let _Ap(G)$A_p(G)$ be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G  , A(G)$A(G)$ (p=2$p=2$). It is shown that for any amenable group H  , a proper affine map α:Y⊂H→G$\alpha :Y\subset H\to G$ induces a p  -completely contractive algebra homomorphism _?α:_Ap(G)→_Ap(H)${?}_{\alpha }:A_p(G)\to A_p(H)$ by setting _?α(u)=u°α${?}_{\alpha }(u)=u°\alpha$ on Y   and _?α(u)=0${?}_{\alpha }(u)=0$ off of Y. Moreover, we show that if both G and H are amenable then any p  -completely contractive algebra homomorphism ?:_Ap(G)→_Ap(H)$?:A_p(G)\to A_p(H)$ is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2$p=2$) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.