On the deficiency index of the vector-valued Sturm–Liouville operator

Let R₊:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R₊ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R₊ and the elements of the matrix functions P^?1, Q, and R are measurable on R₊ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n²(R₊) by formal expressions of the form l[f] = ?(P(f' ? Rf))' ? R*P(f' ? Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = ?(P₀f')' + i((Q₀f)' + Q₀f') + P'₁f, where everywhere the derivatives are understood in the sense of distributions and P₀, Q₀, and P₁ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P₀^?1 exists and ∥P0∥, ∥P₀^?1∥, ∥P₀^?1∥∥P₁∥², ∥P₀^?1∥∥Q₀∥² ∈ L_loc¹(R₊). Themain goal in this paper is to study of the deficiency index of the minimal operator L₀ generated by expression l[f] in L_n²(R₊) in terms of the matrix functions P, Q, and R (P₀, Q₀, and P₁). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where x_k, k = 1, 2,..., is an increasing sequence of positive numbers, with lim_k→+∞x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.     

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

Boolean Lattices: Ramsey Properties and Embeddings

A subposet Q ^′ of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q ^′ such that x ≤ y in P iff f(x) ≤ f(y) in Q. For posets P, P ^′, let the poset Ramsey number R(P, P ^′) be the smallest N such that no matter how the elements of the Boolean lattice Q _N are colored red and blue, there is a copy of P with all red elements or a copy of P ^′ with all blue elements. We provide some general bounds on R(P, P ^′) and focus on the situation when P and P ^′ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of Q _n in Q _N and for a multicolor version of a poset Ramsey number.     

Coercivity results of a modified <Emphasis Type="Italic">Q</Emphasis><Subscript>1</Subscript>-finite volume element scheme for anisotropic diffusion problems

In this paper, we study a so-called modified Q ₁-finite volume element scheme that is obtained by employing the trapezoidal rule to approximate the line integrals in the classical Q ₁-finite volume element method. A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness matrix. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme on arbitrary convex quadrilateral meshes. When the diffusion tensor is an identity matrix, this sufficient condition reduces to a geometric one, covering some standard meshes, such as the traditional h ^1+γ -parallelogram meshes and some trapezoidal meshes. More interesting is that, this sufficient condition has explicit expression, by which one can easily judge on any diffusion tensor and any mesh with any mesh size h > 0. The H ¹ error estimate of the modified Q ₁-finite volume element scheme is obtained without the traditional h ^1+γ -parallelogram assumption. Some numerical experiments are carried out to validate the theoretical analysis.     

On idempotent <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated to a von Neumann algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let L_p(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ L_p(M, τ), then A² ∈ L_p(M, τ). Let n be a positive integer, n > 2, and A = Aⁿ ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μ_t(A) belong to the set {0} ∪ [‖A^n?2‖^?1, ‖A‖] for all t > 0; 2) either μ_t(A) ≥ 1 for all t > 0 or there is a t₀ > 0 such that μ_t(A) = 0 for all t > t₀. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z² = 0, ZP = 0, and PZ = Z. Here, if Q ∈ L_p(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L₁(M, τ) and if A = A³ and A ? A² ∈ M, then τ(A) ∈ R.     

High accuracy analysis of the lowest order <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H¹-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h²)/O(h² + τ²) in H¹-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Pointwise estimates of solutions for the multi-dimensional bipolar Euler–Poisson system

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. By making a new analysis on Green’s functions for the Euler system with damping and the Euler–Poisson system with damping, we obtain the pointwise estimates of the solution for the multi-dimensions bipolar Euler–Poisson system. As a by-product, we extend decay rates of the densities \({\rho_i(i=1,2)}\) in the usual L²-norm to the L^p-norm with \({p\geq1}\) and the time-decay rates of the momentums m_i(i = 1,2) in the L²-norm to the L^p-norm with p > 1 and all of the decay rates here are optimal.     

Invariance of the generalized oscillator under a linear transformation of the related system of orthogonal polynomials

We consider the families of polynomials P = { P _n (x)} _n=0 ^∞ and Q = { Q _n (x)} _n=0 ^∞ orthogonal on the real line with respect to the respective probability measures μ and ν. We assume that { Q _n (x)} _n=0 ^∞ and {P _n (x)} _n=0 ^∞ are connected by linear relations. In the case k = 2, we describe all pairs (P,Q) for which the algebras A _P and A _Q of generalized oscillators generated by { Qn(x)} _n=0 ^∞ and { Pn(x)} _n=0 ^∞ coincide. We construct generalized oscillators corresponding to pairs (P,Q) for arbitrary k ≥ 1.     

New a posteriori L∞(L2) and L2(L2)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems

We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in L^∞(J; L²Ω)-norm and L²(J; L²Ω)-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.     

On the Ramsey number of the triangle and the cube

The Ramsey number r(K ₃,Q _n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K _N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd?s conjectured that r(K ₃,Q _n )=2 ⁿ⁺¹?1 for every n∈?, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K ₃,Q _n )?7000·2 ⁿ . Here we show that r(K ₃,Q _n )=(1+o(1))2 ⁿ⁺¹ as n→∞.     

Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P _k (A). We show that P₁(A) is a subset of P _k (A) for all k. If T in P₁(A), then Tⁿ lies in P₁(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P _k (A), then UTV lies in P _k (A) for all natural k. Let A be unital, then 1) if an element T in P₁(A) is right invertible, then any right inverse element T^?1 lies in P₁(A); 2) for ßßIßß = 1 the class P₁(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P₁(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P₁(A) coincides with the set of all paranormal operators on a Hilbert space H.     

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T], when the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L _∞(0, T; L ₂(Ω))-norm and to ?u in the \(L_{\infty }(0, \textit {T}; \mathbf {L}_{2}({\Omega }))\)-norm are proven to converge with the rate h ^k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h ^k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.     

Normal number constructions for Cantor series with slowly growing bases

Let Q = (q_n)_n=1^∞ be a sequence of bases with q_i ≥ 2. In the case when the q_i are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose Q-Cantor series expansion is both Q-normal and Q-distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of Q, and from this construction we can provide computable constructions of numbers with atypical normality properties.     

Ramsey numbers of cubes versus cliques

The cube graph Q _n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2 ⁿ vertices. The Ramsey number r(Q _n ;K _s ) is the minimum N such that every graph of order N contains the cube graph Q _n or an independent set of order s. In 1983, Burr and Erd?s asked whether the simple lower bound r(Q _n ;K _s )≥(s?1)(2 ⁿ ?1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.     

Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method

Superconvergence and recovery type a posteriori error estimators are analyzed for Pian and Sumihara's 4-node hybrid stress quadrilateral finite element method for linear elasticity problems. Superconvergence of order O(h~(1+min){α,1}) is established for both the displacement approximation in H~1-norm and the stress approximation in L~2-norm under a mesh assumption, where α 0 is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. Recovery type approximations for the displacement gradients and the stress tensor are constructed, and a posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.     

Maximum Distance Between Slater Orders and Copeland Orders of Tournaments

Given a tournament T?=?(X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this paper is to compare the results provided by these two methods: to which extent can they lead to different orders? We consider three cases: T is any tournament, T is strongly connected, T has only one Slater order. For each one of these three cases, we specify the maximum of the symmetric difference distance between Slater orders and Copeland orders. More precisely, thanks to a result dealing with arc-disjoint circuits in circular tournaments, we show that this maximum is equal to n(n???1)/2 if T is any tournament on an odd number n of vertices, to (n ²???3n?+?2)/2 if T is any tournament on an even number n of vertices, to n(n???1)/2 if T is strongly connected with an odd number n of vertices, to (n ²???3n???2)/2 if T is strongly connected with an even number n of vertices greater than or equal to 8, to (n ²???5n?+?6)/2 if T has an odd number n of vertices and only one Slater order, to (n ²???5n?+?8)/2 if T has an even number n of vertices and only one Slater order.     

Compactification of the space of branched coverings of the two-dimensional sphere

For a closed oriented surface Σ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let X_Σ,n be the set of isomorphism classes of orientation-preserving n-fold branched coverings Σ → S ² of the two-dimensional sphere. We complete X_Σ,n with the isomorphism classes of mappings that cover the sphere by the degenerations of Σ. In the case Σ = S ², the topology that we define on the obtained completion \({\overline X _{\Sigma ,n}}\) coincides on \({X_{{s^2},n}}\) with the topology induced by the space of coefficients of rational functions P/Q, where P and Q are homogeneous polynomials of degree n on ?P¹ ≌ S ². We prove that \({\overline X _{\Sigma ,n}}\) coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H(Σ, n) ? X _Σ,n consisting of isomorphism classes of branched coverings with all critical values being simple.     

A partition theorem for a large dense linear order

Let κ be a cardinal which is measurable after generically adding ?_κ+ω many Cohen subsets to κ, and let ?_κ = (Q, ≤ _Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t _m ⁺ such that the set [Q] ^m of m-tuples from Q can be partitioned into classes 〈C _i : i < t _m ⁺ }〉 such that for any coloring a class C _i in fewer than κ colors, there is a copy ?* of ?_κ such that [?*] ^m ? C _i is monochromatic. It follows that \(\mathbb{Q}_\kappa \to (\mathbb{Q}_\kappa )_{ < \kappa /t_m^ + }^m \), that is, for any coloring of [?_κ] ^m with fewer than κ colors there is a copy Q′ ? Q of ?_κ such that [Q′] ^m has at most t _m ⁺ colors. On the other hand, we show that there are colorings of ?_κ such that if Q′ ? Q is any copy of ?_κ then C _i ? [Q′] ≠ ø; for all i < t _m ⁺ , and hence \(\mathbb{Q}_\kappa \nrightarrow [\mathbb{Q}_\kappa ]_{t_m^ + }^m \).We characterize t _m ⁺ as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t ₂ ⁺ = 2 and for m > 2 we have t _m ⁺ > t _m , the m-th tangent number.The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ^+ω-strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah’s theorem, is not directly implied by any large cardinal assumption.