Two graph isomorphism polytopes

The convex hull ψ_n,n of certain (n!)² tensors was considered recently in connection with graph isomorphism. We consider the convex hull ψ_n of the n! diagonals among these tensors. We show: 1. The polytope ψ_n is a face of ψ_n,n. 2. Deciding if a graph G has a subgraph isomorphic to H reduces to optimization over ψ_n. 3. Optimization over ψ_n reduces to optimization over ψ_n,n. In particular, this implies that the subgraph isomorphism problem reduces to optimization over ψ_n,n.     

Compactly supported bidimensional wavelet bases with hexagonal symmetry

We study a class of subband coding schemes allowing perfect reconstruction for a bidimensional signal sampled on the hexagonal grid. From these schemes we construct biorthogonal wavelet bases ofL ²(R ²) which are compactly supported and such that the sets of generating functionsψ ₁,ψ ₂,ψ ₃ for the synthesis and \(\tilde \psi _1 , \tilde \psi _2 , \tilde \psi _3 ,\) for the analysis, as well as the scaling functions φ and \(\tilde \varphi \) , are globally invariant by a rotation of 2π/3. We focus on the particular case of linear splines and we discuss how to obtain a higher regularity. We finally present the possibilities of sharp angular frequency resolution provided by these new bases.     

Two soliton collision for nonlinear Schrödinger equations in dimension 1

We study the collision of two solitons for the nonlinear Schrödinger equation iψt=−ψxx+F(²|ψ|)ψ, F(ξ)=−2ξ+O(ξ²) as ξ→0, in the case where one soliton is small with respect to the other. We show that in general, the two soliton structure is not preserved after the collision: while the large soliton survives, the small one splits into two outgoing waves that for sufficiently long times can be controlled by the cubic NLS: iψt=−ψxx−2²|ψ|ψ.     

Sur une décomposition topologique des fonctions de Sobolev et sur la localisation de l'énergie dans les plasmas idéaux

Motivated by properties of “frozen topological invariants” in ideal plasmas theory, we consider the problem of an intrinsic characterization of the weak closure in H₀¹ (Ω) of all functions ψ₀ ∘ ϕ, with ϕ : Ω → Ω a diffeomorphism. We show that one may associate with ψ₀ a class W^weak (ψ₀) of functions which captures robust topological properties, of the level sets of {ψ₀} and, moreover, is closed under weak limits.     

Gauge Equivalence among Quantum Nonlinear Many Body Systems

Transformations performing on the dependent and/or the independent variables are an useful method used to classify PDE in class of equivalence. In this paper we consider a large class of U(1)-invariant nonlinear Schrödinger equations containing complex nonlinearities. The U(1) symmetry implies the existence of a continuity equation for the particle density ρ≡|ψ|² where the current j _ψ has, in general, a nonlinear structure. We introduce a nonlinear gauge transformation on the dependent variables ρ and j _ψ which changes the evolution equation in another one containing only a real nonlinearity and transforms the particle current j _ψ in the standard bilinear form. We extend the method to U(1)-invariant coupled nonlinear Schrödinger equations where the most general nonlinearity is taken into account through the sum of an Hermitian matrix and an anti-Hermitian matrix. By means of the nonlinear gauge transformation we change the nonlinear system in another one containing only a purely Hermitian nonlinearity. Finally, we consider nonlinear Schrödinger equations minimally coupled with an Abelian gauge field whose dynamics is governed, in the most general fashion, through the Maxwell-Chern-Simons equation. It is shown that the nonlinear transformation we are introducing can be applied, in this case, separately to the gauge field or to the matter field with the same final result. In conclusion, some relevant examples are presented to show the applicability of the method.     

Asymptotics of the basis functions of generalized Taylor series for the class H ρ,2

We study the basis functions φ _n,k and ψ _n,p of generalized Taylor series for the classH _ρ,2 and obtain asymptotic expansions of the functions φ _n,0 ^(l) and $\psi _{n,2 \cdot 4^n - 1}^{(l)} $ . We prove the existence of an asymptotics for the functions φ _n,k and ψ _n,p for k ≠ 0 and p ≠ 2·4 ⁿ ? 1. The first term of the asymptotic expansions of these functions is obtained.     

Symmetric multi-box splines for higher degree

We consider the space S _n =S _n (v ₀,…,v _n+r ) of compactly supported C ^n?1 piecewise polynomials on a mesh M of lines through ?² in directions v ₀,…,v _n+r . A sequence ψ=(ψ ₁,…,ψ _r ) of elements of S _n is called a multi-box spline if every element of S _n is a finite linear combination of shifts of (the components of) ψ. For the case n=2, 3 we give some examples for multi-box splines and show that they are not always stable. It is further shown that any C ^n?1 piecewise polynomial of degree n≥2 on M, is possibly a symmetric multi-box spline.     

Superlocalization of waves and electrons and hopping conductivity on classical and superconductive fractals

We have found an unexpected paradoxical situation in the percolation transition: the superconductive behavior below and above the threshold. We have found also the two different density of states d_s=4/3 and d_v=1.05 and the inverse localization lengths for fractons with the scalar and vector interactions, respectively. In this concept the wave functions of electrons or waves on an incipient percolation cluster and fractal dilute structure exhibit superlocalization behavior of the form ψ(r)∝exp[−r^d_φ] with values of d_φ1=1.73 and d_φ2=2.4 for the former and the latter. Applications of these results for thermally activated hopping conductivity σ(t)∝exp[−(T₀/T)^β] between impurities on a random fractal structure give the values of β=2/5 for the scalar and β=1/2 (Mott's law) for the vector interactions, respectively. Band states are localized in classical and superlocalized in superconductive percolations.     

Decay and regularity in the inverse scattering problem

The regularity and decay properties for the potential q(x) in the Schrödinger equation ?ψ″ + qψ = k²ψ on the line are characterized in terms of the decay and regularity of the reflection coefficients R_± and their Fourier transforms.     

Commuting dual Toeplitz operators on the harmonic Dirichlet space

In this paper,we characterize the symbols for(semi-)commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space.We show that for φ,ψ∈W~(1,∞),S_φS_ψ=S_ψ Sφ on(D_h)~⊥ if and only if φ and ψ satisfy one of the following conditions:(1) Both φ and ψ are harmonic functions;(2) There exist complex constants α and β,not both 0,such that φ=αψ +β.     

Selection over classes of ordinals expanded by monadic predicates

A monadic formula ψ(Y) is a selector for a monadic formula φ(Y) in a structure M if ψ defines in M a unique subset P of the domain and this P also satisfies φ in M. If C is a class of structures and φ is a selector for ψ in every M∈C, we say that φ is a selector for φ over C.For a monadic formula φ(X,Y) and ordinals α≤ω₁ and δ<ω^ω, we decide whether there exists a monadic formula ψ(X,Y) such that for every P⊆αof order-type smaller thanδ, ψ(P,Y) selects φ(P,Y) in (α,<). If so, we construct such a ψ.We introduce a criterion for a class C of ordinals to have the property that every monadic formula φ has a selector over it. We deduce the existence of S⊆ω^ω such that in the structure (ω^ω,<,S) every formula has a selector.Given a monadic sentence π and a monadic formula φ(Y), we decide whether φ has a selector over the class of countable ordinals satisfying π, and if so, construct one for it.     

Scaling properties of functionals and existence of constrained minimizers

In this paper we develop a new method to prove the existence of minimizers for a class of constrained minimization problems on Hilbert spaces that are invariant under translations. Our method permits to exclude the dichotomy of the minimizing sequences for a large class of functionals. We introduce family of maps, called scaling paths, that permits to show the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to translations) is proved. We give an application to the energy functional I associated to the Schrödinger-Poisson equation in R³

iψt+Δψ−(|x|⁻¹?²|ψ|)ψ+|ψ|^p−2ψ=0     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

The pseudoweights of a space

Let X be a T_i-space, i ⩽ 2. We define the T_i-pseudoweight of X, ψ _i(X), to be the least weightof a coarser T_i topology on X. Reed and Zenor have shown that if X is a Moore space, and |X| ⩽ 2^ω, then ψ₁(X) = ω, but there is a Moore space, X, such that ψ₂(X) = w(X) = |X| = ω₁.Theorem 1: If X is metric, ψ₀(X) = log w(X), where log κ = min{λ:2^λ ⩾ κ}. Theorem 2: If X is compact and T₂, then ψ₁(X) = ψ₂(X) = w(X) (but it is possible to have ψ₀(X) = log w (X)< w(X)). Theorem 3: If X is a GO-space, then ψ₁(X) = ψ₂(X) (but it is possible to have ψ₀(X) =log ψ₁(X) < ψ₁(X) < w(X) even if X is a LOTS). Finally, Hart has shown that if X is an infinite LOTS, then w(X) = c (X) · ψ₁(X). Theorem 4: If X is an infinite LOTS, then w(X) =c(X) · ψ₀ (X).     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.     

On compactness of the difference of composition operators

Let φ and ψ be analytic self-maps of the unit disc, and denote by C_φ and C_ψ the induced composition operators. The compactness and weak compactness of the difference T=C_φ−C_ψ are studied on H^p spaces of the unit disc and L^p spaces of the unit circle. It is shown that the compactness of T on H^p is independent of p∈[1,∞). The compactness of T on L¹ and M (the space of complex measures) is characterized, and examples of φ and ψ are constructed such that T is compact on H¹ but non-compact on L¹. Other given results deal with L^∞, weakly compact counterparts of the previous results, and a conjecture of J.E. Shapiro.