A new approach to fixed point theorems on G-metric spaces

In this paper, we introduce the metric d^G$d^G$ on a G  -metric space (X,G)$(X,G)$ and use this notion to show that many contraction conditions for maps on the G  -metric space (X,G)$(X,G)$ reduce to certain contraction conditions for maps on the metric space (X,d^G)$(X,d^G)$. As applications, the proofs of many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ may be simplified, and many fixed point theorems for maps on the G  -metric space (X,G)$(X,G)$ are direct consequences of preceding results for maps on the metric space (X,d^G)$(X,d^G)$.     

On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form _ut+Au(t)=f(u(t),t)$u_t+Au(t)=f(u(t),t)$, u(1)=φ$u(1)=\phi$, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f   in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0$\beta >0$) is well-posed and that its solution _Uβ(t)$U_{\beta }(t)$ converges on [0,1]$[0,1]$ to the exact solution u(t)$u(t)$ as β→0⁺$\beta \to 0^{+}$. These results extend some earlier works on the nonlinear backward problem.     

Cheeger-harmonic functions in metric measure spaces revisited

Let (X,d,μ)$(X,d,\mu )$ be a complete metric measure space, with μ   a locally doubling measure, that supports a local weak L²$L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on (X,d,μ)$(X,d,\mu )$. Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.     

The reduced expressions in a Coxeter system with a strictly complete Coxeter graph

Let (W,S)$(W,S)$ be a Coxeter system with a strictly complete Coxeter graph. The present paper concerns the set Red(z)$\mathrm{Red}(z)$ of all reduced expressions for any z∈W$z\in W$. By associating each bc-expression to a certain symbol, we describe the set Red(z)$\mathrm{Red}(z)$ and compute its cardinal |Red(z)|$|\mathrm{Red}(z)|$ in terms of symbols. An explicit formula for |Red(z)|$|\mathrm{Red}(z)|$ is deduced, where the Fibonacci numbers play a crucial role.     

Reduced limit for semilinear boundary value problems with measure data

We study boundary value problems for semilinear elliptic equations of the form −Δu+g°u=μ$-\mathrm{\Delta }u+g°u=\mu$ in a smooth bounded domain Ω⊂R^N$\Omega \subset R^N$. Let {_μn}$\{{\mu }_n\}$ and {_νn}$\{{\nu }_n\}$ be sequences of measure in Ω and ∂Ω   respectively. Assume that there exists a solution _un$u_n$ with data (_μn,_νn)$({\mu }_n,{\nu }_n)$, i.e., _un$u_n$ satisfies the equation with μ=_μn$\mu ={\mu }_n$ and has boundary trace _νn${\nu }_n$. Further assume that the sequences of measures converge in a weak sense to μ and ν   respectively while {_un}$\{u_n\}$ converges to u   in L¹(Ω)$L^1(\Omega )$. In general u   is not a solution of the boundary value problem with data (μ,ν)$(\mu ,\nu )$. However there exists a pair of measures (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ such that u   is a solution of the boundary value problem with this data. The pair (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ is called the reduced limit of the sequence {(_μn,_νn)}$\{({\mu }_n,{\nu }_n)\}$. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence. A closely related problem was studied by Marcus and Ponce [3].     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (= width) _ε0${\epsilon }_0$, the same happens for the solution u(t,⋅)$u(t,\cdot )$ for a certain radius ε(t)$\epsilon (t)$, as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t)$\epsilon (t)$ as t grows.     

Laguerre functions on symmetric cones and recursion relations in the real case

In this article we derive differential recursion relations for the Laguerre functions on the cone Ω$\Omega$ of positive definite real matrices. The highest weight representations of the group Sp(n,R)$\mathrm{Sp}(n,\mathbb{R})$ play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω+iSym(n,R)$\Omega +i\mathrm{Sym}(n,\mathbb{R})$. We then use the Laplace transform to carry the Lie algebra action over to L²(Ω,d_μν)$L^2(\Omega ,\mathrm{d}{\mu }_{\nu })$. The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R).$\mathfrak{sl}(2,\mathbb{R}).$     

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p)$G(\mathbf{p})$ is globally rigid in Euclidean space R^d${\mathbb{R}}^d$, where G is a finite graph and p is a configuration of points corresponding to the vertices of G  . G(p)$G(\mathbf{p})$ is globally rigid in  R^d${\mathbb{R}}^d$ if for any other configuration q for G   such that the edge lengths of G(q)$G(\mathbf{q})$ are the same as the corresponding edge lengths of G(p)$G(\mathbf{p})$, then p is congruent to q. A framework G(p)$G(\mathbf{p})$ is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.     

Orthogonally additive polynomials on Fourier algebras

We show that an n-homogeneous polynomial P   on the Fourier algebra A(G)$A(G)$ of a locally compact group G   can be represented in the form P(f)=〈T,fⁿ〉$P(f)=〈T,f^n〉$(f∈A(G))$(f\in A(G))$ for some T   in the group von Neumann algebra VN(G)$\mathit{VN}(G)$ of G if and only if it is orthogonally additive and completely bounded.     

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.