Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with fractional Laplacian in the Schrödinger equation in R¹⁺¹. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schrödinger data in H^s1 and wave data in H^s2 ×H^s2?1 for 3/4?α < s₁ ≤ 0 and ?1/2 < s₂ < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α ≤ 1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s₁ = 0 and ?1/2 < s₂ < 1/2 by using the conservation law for the L² norm of u.     

Minimal <Emphasis Type="Italic">P</Emphasis>-symmetric period problem of first-order autonomous Hamiltonian systems

Let P ∈ Sp(2n) satisfying P ^k = I _2n . We consider the minimal P-symmetric period problem of the autonomous nonlinear Hamiltonian system \(\dot x\left( t \right) = JH'\left( {x\left( t \right)} \right)\). For some symplectic matrices P, we show that for any τ > 0, the above Hamiltonian system possesses a kτ periodic solution x with kτ being its period provided H satis Fies Rabinowitz's conditions on the minimal minimal P-symmetric period conjecture, together with that H is convex and H(Px) = H(x).     

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

On Dominions of the Rationals in Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a₁,..., a_n) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.     

A tournament approach to pattern avoiding matrices

We consider the following Turán-type problem: given a fixed tournament H, what is the least integer t = t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t = t(T _n ,H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several results on these problems, our main contributions are the following:

• Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n × n matrix with n(log n) ^O(1) entries equal to 1 contains the pattern M. We show that this conjecture is equivalent to the assertion that t(T _n ,H) = n(log n) ^O(1) if and only if H belongs to a certain (natural) family of tournaments.
• We propose an approach for determining if t(n,H) = n(log n) ^O(1). This approach combines expansion in sparse graphs, together with certain structural characterizations of H-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach–Tardos conjecture.

     

Goodness-of-fit Tests for Continuous-time Stationary Processes

The paper considers the following problem of hypotheses testing: based on a finite realization {X(t)}, 0 ≤ t ≤ T of a zero mean real-valued mean square continuous stationary Gaussian process X(t), t ? R, construct goodness-of-fit tests for testing a hypothesis H₀ that the hypothetical spectral density of the process X(t) has the specified form. We show that in the case where the hypothetical spectral density of X(t) does not depend on unknown parameters (the hypothesis H₀ is simple), then the suggested test statistic has a chi-square distribution. In the case where the hypothesis H₀ is composite, that is, the hypothetical spectral density of X(t) depends on an unknown p–dimensional vector parameter, we choose an appropriate estimator for unknown parameter and describe the limiting distribution of the test statistic, which is similar to that of obtained by Chernov and Lehman in the case of independent observations. The testing procedure works both for short- and long-memory models.     

Some Poisson structures and Lax equations associated with the Toeplitz lattice and the Schur lattice

The Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this paper, we unveil the origins of this Poisson structure and derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety H _n of GL _n (C), which we view as a real or complex Poisson–Lie group whose Poisson structure comes from a quadratic R-bracket on gl _n (C) for a fixed R-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on H _n , combined with the properties of the quadratic R-bracket allow us to give explicit formulas for the Lax equation. Then we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over R, we can construct a Poisson subvariety H _n ^τ of U _n which is itself a Poisson–Dirac subvariety of GL _n ^R (C). We then construct a Hamiltonian for the Poisson structure induced on H _n ^τ , corresponding to another system which derives from the Toeplitz lattice the modified Schur lattice. Thanks to the properties of Poisson–Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice.     

Finite groups in which Sylow normalizers have nilpotent Hall supplements

The normalizer of each Sylow subgroup of a finite group G has a nilpotent Hall supplement in G if and only if G is soluble and every tri-primary Hall subgroup H (if exists) of G satisfies either of the following two statements: (i) H has a nilpotent bi-primary Hall subgroup; (ii) Let π(H) = {p, q, r}. Then there exist Sylow p-, q-, r-subgroups H _p , H _q , and H _r of H such that H _q ? N _H (H _p ), H _r ? N _H (H _q ), and H _p ? N _H (H _r ).     

On an open problem of Guo-Skiba

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/A _G of G is called a G-boundary factor of A. For any Gboundary factor H/A _G of A, the subgroup (A ∩ H)/A _G of G/ A _G is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

Nonmatricial version of the Arveson-Wittstock extension principle,and its generalization

We consider the algebra ? = ?(H) of bounded operators in a Hilbert space H, ?-bimodules, and morphisms of these bimodules into the algebra ?(L ? H), where L is a Hilbert space. We study the problem of extension of a morphism defined on a sub-?-bimodule Y ? Z to Z. This problem is solved for Ruan bimodules.     

On weakly <Emphasis Type="Italic">S</Emphasis>Φ-supplemented subgroups of finite groups

Let G be a finite group. We say that a subgroup H of G is weakly SΦ-supplemented in G if G has a subgroup T such that G = HT and H∩T ≤ Φ(H)H_sG, where H_sG is the subgroup of H generated by all those subgroups of H that are s-permutable in G. In this paper, we investigate the influence of weakly SΦ-supplemented subgroups on the structure of finite groups. Some new characterizations of p-nilpotency and supersolubility of finite groups are obtained.     

On control of the prime spectrum of a finite simple group

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H ₁, …, H _m of some special form such that π(H ₁)∪…∪π(H _m ) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections H _i so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H ₁,…,H _m } controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.     

On the problem of determining the parameters of a layered elastic medium and an impulse source

Considering the linear system of elasticity equations describing the wave propagation in the half-space ? ₊ ³ = {x ∈ ?³ | x ₃ > 0} we address the problem of determining the density and elastic parameters which are piecewise constant functions of x ₃. The shape is unknown of a point-like impulse source that excites elastic oscillations in the half-space. We show that under certain assumptions on the source shape and the parameters of the elastic medium the displacements of the boundary points of the half-space for some finite time interval (0, T) uniquely determine the normalized density (with respect to the first layer) and the elastic Lamé parameters for x ₃ ∈ [0, H], where H = H(T). We give an algorithmic procedure for constructing the required parameters.     

A condition for the solvability of finite groups

A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H ₁ B is a proper subgroup of G for every maximal subgroup H ₁ of H. We investigate the influence of ?-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite groups.     

EXACT BOUNDARY CONTROLLABILITY OF 1-D NONLINEAR SCHRODINGER EQUATION

In this paper, the boundary control problem of a distributed parameter system described by the Schr(o)dinger equation posed on finite interval α≤ x ≤β:{iyt yxx |y|2y = 0,y(α,t) = h1(t),y(β,t) = h2(t) for t ＞ 0 (S)is considered. It is shown that by choosing appropriate control inputs (hj), (j = 1,2) one can always guide the system (S) from a given initial state ψ∈ Hs(α,β),(s ∈ R) to a terminal state ψ∈ Hs(α,β), in the time period [0, T]. The exact boundary controllability is obtained by considering a related initial value control problem of Schr(o)dinger equation posed on the whole line R. The discovered smoothing properties of Schr(o)dinger equation have played important roles in our approach; this may be the first step to prove the results on boundary controllability of (semi-linear) nonlinear Schr(o)dinger equation.     

Gabor (super)frames with Hermite functions

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H _n . Let h = (H ₀, H ₁, . . . , H _n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices \({\Lambda \subseteq \mathbb{R} ^2}\) such that the Gabor system \({\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}\) is a frame for \({L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}\). As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor frames.     

A Note on Generalized Lagrangians of Non-uniform Hypergraphs

Set \(A\subset {\mathbb N}\) is less than \(B\subset {\mathbb N}\) in the colex ordering if m a x(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of \({\mathbb N}^{(r)}\) in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus implies that this conjecture is true for r=2. This conjecture seems to be challenging even for r=3. For a hypergraph H=(V,E), the set T(H)={|e|:e∈E} is called the edge type of H. In this paper, we study non-uniform hypergraphs and define L(H) a generalized Lagrangian of a non-uniform hypergraph H in which edges of different types have different weights. We study the following two questions: 1. Let H be a hypergraph with m edges and edge type T. Let C _m,T denote the hypergraph with edge type T and m edges formed by taking the first m sets with cardinality in T in the colex ordering. Does L(H)≤L(C _m,T ) hold? If T={r}, then this question is the question by Frankl and Füredi. 2. Given a hypergraph H, find a minimum subhypergraph G of H such that L(G) = L(H). A result of Motzkin and Straus gave a complete answer to both questions if H is a graph. In this paper, we give a complete answer to both questions for {1,2}-hypergraphs. Regarding the first question, we give a result for {1,r ₁,r ₂,…,r _l }-hypergraph. We also show the connection between the generalized Lagrangian of {1,r ₁,r ₂,? ,r _l }-hypergraphs and {r ₁,r ₂,? ,r _l }-hypergraphs concerning the second question.     

Automorphism group of Green ring of Sweedler Hopf algebra

Let H ₂ be Sweedler’s 4-dimensional Hopf algebra and r(H ₂) be the corresponding Green ring of H ₂. In this paper, we investigate the automorphism groups of Green ring r(H ₂) and Green algebra F(H ₂) = r(H ₂)?_? F, where F is a field, whose characteristics is not equal to 2. We prove that the automorphism group of r(H ₂) is isomorphic to K ₄, where K ₄ is the Klein group, and the automorphism group of F(H ₂) is the semidirect product of ?₂ and G, where G = F {1/2} with multiplication given by a · b = 1? a ? b + 2ab.