Integers with dense divisors

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are y-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?y. Let D(x,y,z) be the number of positive integers not exceeding x whose divisors are y-dense and whose prime divisors are bigger than z, and let , and . We show that is equivalent, in a large region, to a function d(u,v) which satisfies a difference-differential equation. Using that equation we find that d(u,v)?(1−u/v)/(u+1) for v?3+ε. Finally, we show that d(u,v)=e^−γd(u)+O(1/v), where γ is Euler's constant and d(u)∼x⁻¹D(x,y,1), for fixed u. This leads to a new estimate for d(u).     

Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation

We discuss the existence of global or periodic solutions to the nonlinear wave equation with the boundary condition , where Ω is a bounded domain in R^N,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting.     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

A maximum principle for evolution Hamilton-Jacobi equations on Riemannian manifolds

We establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form ut+F(t,dxu)=0, u(0,x)=u₀(x), where is a bounded uniformly continuous function, M is a Riemannian manifold, and . This yields uniqueness of the viscosity solutions of such Hamilton-Jacobi equations.     

Bandwidth of the strong product of two connected graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)-f(y)|:xy∈E(G)} taken over all proper numberings f of G. The strong product of two graphs G and H, written as G(S_P)H, is the graph with vertex set V(G)×V(H) and with (u₁,v₁) adjacent to (u₂,v₂) if one of the following holds: (a) u₁ and v₁ are adjacent to u₂ and v₂ in G and H, respectively, (b) u₁ is adjacent to u₂ in G and v₁=v₂, or (c) u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the strong product of two connected graphs. Let G be a connected graph. We denote the diameter of G by D(G). Let d be a positive integer and let x,y be two vertices of G. Let denote the set of vertices v so that the distance between x and v in G is at most d. We define δ_d(G) as the minimum value of over all vertices x of G. Let denote the set of vertices z such that the distance between x and z in G is at most d-1 and z is adjacent to y. We denote the larger of and by . We define η(G)=1 if G is complete and η(G) as the minimum of over all pair of vertices x,y of G otherwise. Let G and H be two connected graphs. Among other results, we prove that if δ_D(H)(G)?B(G)D(H)+1 and B(H)=⌈(|V(H)|+η(H)-2)/D(H)⌉, then B(G(S_P)H)=B(G)|V(H)|+B(H). Moreover, we show that this result determines the bandwidth of the strong product of some classes of graphs. Furthermore, we study the bandwidth of the strong product of power of paths with complete bipartite graphs.     

Existence of entire explosive positive radial solutions for a class of quasilinear elliptic systems

We show the existence of entire explosive positive radial solutions for quasilinear elliptic systems div(|∇u|^m−2∇u)=p(|x|)g(v), div(|∇v|ⁿ⁻²∇v)=q(|x|)f(u) on , where f and g are positive and non-decreasing functions on (0,∞) satisfying the Keller-Osserman condition.     

Isomorphisms between unital C-algebras

It is shown that every almost linear bijection of a unital C^∗-algebra A onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C^∗-algebra A of real rank zero onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C^∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C^∗-algebra isomorphisms between unital C^∗-algebras.     

Dynamical systems method (DSM) for selfadjoint operators

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation Av=f consists of solving the Cauchy problem , u(0)=u₀, where Φ is a suitable operator, and proving that (i) ∃u(t)∀t>0, (ii) ∃u(∞), and (iii) A(u(∞))=f. It is proved that if equation Av=f is solvable and u solves the problem , u(0)=u₀, where a>0 is a parameter and u₀ is arbitrary, then lima→0limt→∞u(t,a)=y, where y is the unique minimal-norm solution of the equation Av=f. Stable solution of the equation Av=f is constructed when the data are noisy, i.e., fδ is given in place of f, ‖fδ−f‖?δ. The case when a=a(t)>0, , a(t)↘0 as t→∞ is considered. It is proved that in this case limt→∞u(t)=y and if fδ is given in place of f, then limt→∞u(tδ)=y, where tδ is properly chosen.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

On the optimal control problem for Schrödinger equation with complex potential

The existence and uniqueness for the solution of the problem of determining the v(x,t) potential in the Schrödinger equation from the measured final data ψ(x,T)=y(x) is investigated. For the objective functional , it is proven that the problem has at least one solution for α?0, and has a unique solution for α>0. The necessary condition for solvability the problem is stated as the variational principle.     

Hyper- and reverse-Wiener indices of F-sums of graphs

The Wiener index W(G)=∑_{u,v}⊂V(G)d(u,v), the hyper-Wiener index and the reverse-Wiener index , where d(u,v) is the distance of two vertices u,v in G, d²(u,v)=d(u,v)², n=|V(G)| and D is the diameter of G. In [M. Eliasi, B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math. 157 (2009) 794-803], Eliasi and Taeri introduced the F-sums of two connected graphs. In this paper, we determine the hyper- and reverse-Wiener indices of the F-sum graphs and, subject to some condition, we present some exact expressions of the reverse-Wiener indices of the F-sum graphs.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

The utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term

The authors study the problem , and u(0,t)=u(1,t)=ψ(t), where ψ(t)=u₀ for t2k<t<t2k+1 and ψ(t)=0 for , with t₀=0 and the sequence tk is determined by the equations , for , and , for k=2,4,6,… and where 0<m<M. Note that the switching points , are unknown. Existence and uniqueness are demonstrated. Theoretical estimates of the tk and tk+1−tk are obtained and numerical verifications of the estimates are presented. The case of ux(0,t)=ux(1,t)=ψ(t) is also considered and analyzed.     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

A necessary and sufficient condition for the existence of large solutions to sublinear elliptic systems

We prove that the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) where 0<α?1, 0<β?1, and p and q are nonnegative continuous functions has a nonnegative entire radial solution satisfying lim|x|→∞u(x)=lim|x|→∞v(x)=∞ if and only if the functions p and q satisfy

     

A unique solution to a nonlinear elliptic equation

The purpose of this paper is to prove the existence of a unique, classical solution to the nonlinear elliptic partial differential equation −∇⋅(a(u(x))∇u(x))=f(x) under periodic boundary conditions, where u(x₀)=u₀ at x₀∈Ω, with Ω=TN, the N-dimensional torus, and N=2,3. The function a is assumed to be smooth, and a(u(x))>0 for , where G⊂R is a bounded interval. We prove that if the functions f and a satisfy certain conditions, then a unique classical solution u exists. The range of the solution u is a subset of a specified interval . Applications of this work include stationary heat/diffusion problems with a source/sink, where the value of the solution is known at a spatial location x₀.     

On controlled extensions of functions

For any numerical function we give sufficient conditions for resolving the controlled extension problem for a closed subset A of a normal space X. Namely, if the functions , and satisfy the equality E(f(a),g(a))=h(a), for every a∈A, then we are interested to find the extensions f? and ? of f and g, respectively, such that , for every x∈X. We generalize earlier results concerning E(u,v)=u·v by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.