Note on Periodic Complementary Sets of Binary Sequences

Denote by $PCS_p^n $ resp. $ACS_p^n $ thecollection consisting of ordered p-tuples of binary sequences(i.e., sequences whose elements are $ \pm 1$ ), each having length n, such that the sum of their periodic resp. aperiodicauto-correlation functions is a delta function. We fill many open cases inthe Bömer and Antweiler diagram [3] of the known cases where $PCS_p^n $ exist for $p \leqslant 12$ and $n \leqslant 50$ . In particular we show that $PCS_2^{34} $ exist, whileit is well known [1] that $ACS_2^{34} $ do not.     

New estimates of continuity in $$M/GI/1/ \infty$$ queues

The paper deals with an estimation of the total variation distance between stationary distributions of waiting time in two queueing systems with equal Poisson inputs and different distributions B and $\widetilde B$ of service time. Assuming equality of two first moments of B and $\widetilde B$ the continuity inequalities are derived in terms of difference pseudomoments of B and $\widetilde B$ . When in addition the third moments of B and $\widetilde B$ coincide then the constant involved in the corresponding inequality has the asymptotics ${\text{O}}\left[ {\left( {1 - \rho } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} } \right]$ in the heavy traffic limit $\rho \to 1$ .     

Local behavior and hitting probabilities of the \text{ Airy}_1 process

We obtain a formula for the $n$ -dimensional distributions of the $\text{ Airy}_1$ process in terms of a Fredholm determinant on $L^2(\mathbb{R })$ , as opposed to the standard formula which involves extended kernels, on $L^2(\{1,\dots ,n\}\times \mathbb{R })$ . The formula is analogous to an earlier formula of Prähofer and Spohn (J Stat Phys 108(5–6):1071–1106, 2002) for the $\text{ Airy}_2$ process. Using this formula we are able to prove that the $\text{ Airy}_1$ process is Hölder continuous with exponent $\frac{1}{2}$ —and that it fluctuates locally like a Brownian motion. We also explain how the same methods can be used to obtain the analogous results for the $\text{ Airy}_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the $\text{ Airy}_1$ process, analogous to that obtained in Corwin et al. (Commun Math Phys 2011, to appear) for the $\text{ Airy}_2$ process.     

The Gröbner ring conjecture in the lexicographic order case

We prove that a valuation domain $\mathbf{V}$ has Krull dimension $\le $ 1 if and only if, for any $n$ , fixing the lexicographic order as monomial order in $\mathbf{V}[X_1,\ldots ,X_n]$ , for every finitely generated ideal $I$ of $\mathbf{V}[X_1,\ldots ,X_n]$ , the ideal generated by the leading terms of the elements of $I$ is also finitely generated. This proves the Gröbner ring conjecture in the lexicographic order case. The proof we give is both simple and constructive. The same result is valid for Prüfer domains. As a “scoop”, contrary to the common idea that Gröbner bases can be computed exclusively on Noetherian ground, we prove that computing Gröbner bases over $\mathbf{R}[X_1,\ldots , X_n]$ , where $\mathbf{R}$ is a Prüfer domain, has nothing to do with Noetherianity, it is only related to the fact that the Krull dimension of $\mathbf{R}$ is $\le $ 1.     

An application of group expansion to the Anderson-Bernoulli model

We establish smoothness of the density of states for 1D lattice Schrödinger operators with potential taking values ${\pm\lambda}$ , for ${\lambda}$ in a class of small algebraic numbers and energy ${E \in\,) -2, 2(}$ suitably restricted away from ${\pm2}$ .     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Optimal Set of the Modulus of Continuity in the Sharp Jackson Inequality in the Space $$L_2 $$

To a function $f \in L_2 [ - \pi ,\pi ]$ and a compact set $Q \subset [ - \pi ,\pi ]$ we assign the supremum $\omega (f,Q) = \sup _{t \in Q} ||f( \cdot + t) - f( \cdot )||_{L_2 [ - \pi ,\pi ]} $ , which is an analog of the modulus of continuity. We denote by $K(n,Q)$ the least constant in Jackson's inequality between the best approximation of the function f by trigonometric polynomials of degree $n - 1$ in the space $L_2 [ - \pi ,\pi ]$ and the modulus of continuity $\omega (f,Q)$ . It follows from results due to Chernykh that $K(n,Q) \geqslant 1/\sqrt 2 $ and $K(n,[0,\pi /\pi ]) = 1/\sqrt 2 $ . On the strength of a result of Yudin, we show that if the measure of the set Q is less than $\pi /n$ , then $K(n,Q) >1/\sqrt 2 $ .     

Decomposition of Minkowski–Rådström–Hörmander Space to the Direct Sum of Symmetric and Asymmetric Subspaces

Any abstract convex cone S with a uniformity satisfying the law of cancellation can be embedded in a topological vector space $\widetilde{S}$ (Urbański, Bull Acad Pol Sci, Sér Sci Math Astron Phys 24:709–715, 1976). We introduce a notion of a cone symmetry and decompose in Theorem 2.12 a quotient vector space $\widetilde{S}$ into a topological direct sum of its symmetric subspace $\widetilde{S}_s$ and asymmetric subspace $\widetilde{S}_a$ . In Theorem 2.19 we prove a similar decomposition for a normed space $\widetilde{S}$ . In section 3 we apply decomposition to Minkowski–Rådström–Hörmander (MRH) space with three best known norms and four symmetries. In section 4 we obtain a continuous selection from a MRH space over ?² to the family of pairs of nonempty compact convex subsets of ?².     

A Note on Variable Exponent Hörmander Spaces

In this paper we introduce the variable exponent Hörmander spaces and we study some of their properties. In particular, it is shown that ${{(\mathcal{B}_{p_{(\cdot)}}^{c}(\Omega))^{\prime}}}$ is isomorphic to ${{\mathcal{B}^{loc}_{\widetilde{p^\prime(\cdot)}(\Omega)}}}$ (Ω open set in ${{\mathbb{R}^n, p? > 1}}$ and the Hardy–Littlewood maximal operator M is bounded in ${L_p(\cdot))}$ extending a Hörmander’s result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hörmander spaces are given.     

Expansion in SL 2 {(\mathbb{R})} and monotone expanders

This work presents an explicit construction of a family of monotone expanders, which are bi-partite expander graphs whose edge-set is defined by (partial) monotone functions. The family is (roughly) defined by the Möbius action of SL ₂ ${\mathbb{R}}$ on the interval [0,1]. A key part of the proof is a product-growth theorem for certain subsets of SL ₂ ${\mathbb{R}}$ . This extends recent results on finite/compact groups to the non-compact scenario. No other proof-of-existence for monotone expanders is known.     

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ $\mathbb{T}^d $ , where $\mathbb{T}^d $ is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ?^d, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ $\mathbb{T}^d $ if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H₊(k), k ∈ $\mathbb{T}^d $ , corresponding to a two-particle system with repulsive interaction.     

Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation

In this paper we study the long time behavior of a discrete approximation in time and space of the cubic nonlinear Schrödinger equation on the real line. More precisely, we consider a symplectic time splitting integrator applied to a discrete nonlinear Schrödinger equation with additional Dirichlet boundary conditions on a large interval. We give conditions ensuring the existence of a numerical ground state which is close in energy norm to the continuous ground state. Such result is valid under a CFL condition of the form $\tau h^{-2}\le C$ where $\tau $ and $h$ denote the time and space step size respectively. Furthermore we prove that if the initial datum is symmetric and close to the continuous ground state $\eta $ then the associated numerical solution remains close to the orbit of $\eta ,\Gamma =\cup _\alpha \{e^{i\alpha }\eta \}$ , for very long times.     

Destruction Of Lagrangian Torus For Positive Definite Hamiltonian Systems

For an integrable Hamiltonian ${H_0=\frac{1}{2} \sum_{i=1}^dy_i^2}$ ${(d \geq 2)}$ , we show that any Lagrangian torus with a given unique rotation vector can be destructed by arbitrarily ${C^{2d-\delta}}$ -small perturbations. In contrast with it, it has been shown that KAM torus with constant type frequency persists under ${C^{2d+\delta}}$ -small perturbations by Pöschel (Comm Pure Appl Math 35:653–696, 1982).     

Blaschke products and proper holomorphic mappings

Suppose f: $\mathbb{D} \to V$ is a proper holomorphic map of the unit disk $\mathbb{D} \subset \mathbb{C}$ onto a subset $V \subset \mathbb{D}$ of degree d>0. We show that f is conjugate to either an affine map or a degree d Blaschke product. As an application we give a unified treatment of theorems of Böttcher and Schröder coordinates.     

On the Durrmeyer-type modification of some discrete approximation operators

In [10], for continuous functionsf from the domain of certain discrete operatorsL _n the inequalities are proved concerning the modulus of continuity ofL _nf. Here we present analogues of the results obtained for the Durrmeyer-type modification $\tilde L_n $ ofL _n. Moreover, we give the estimates of the rate of convergence of $\tilde L_n f$ in Hölder-type norms     

About “bulky” links generated by generalized Möbius–listing bodies GML_2^n

In this paper, we consider the “bulky knots” and “bulky links,” which appear after cutting of a Generalized Möbius–Listing $ GML_2^n $ body (with the radial cross section a convex plane 2-symmetric figure with two vertices) along a different Generalized Möbius–Listing surfaces $ GML_2^n $ situated in it. The aim of this report is to investigate the number and geometric structure of the independent objects that appear after such a cutting process of $ GML_2^n $ bodies. In most cases we are able to count the indices of the resulting mathematical objects according to the known classification for the standard knots and links.     

Weighted Composition Operators from the Minimal Möbius Invariant Space into the Bloch Space

Let ${{\varphi}}$ be an analytic self-map of the open unit disk ${{\mathbb{D}}}$ in the complex plane ${{\mathbb{C}, H(\mathbb{D})}}$ the space of complex-valued analytic functions on ${{\mathbb{D}}}$ , and let u be a fixed function in ${{H(\mathbb{D})}}$ . The weighted composition operator is defined by $$(uC_{\varphi}f)(z) = u(z)f({\varphi}(z)), \quad z \in \mathbb{D}, f \in H(\mathbb{D}).$$ In this paper, we study the boundedness and the compactness of the weighted composition operators from the minimal Möbius invariant space into the Bloch space and the little Bloch space.     

Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions

Let J and ${{\mathfrak{J}}}$ be operators on a Hilbert space ${{\mathcal{H}}}$ which are both self-adjoint and unitary and satisfy ${J{\mathfrak{J}}=-{\mathfrak{J}}J}$ . We consider an operator function ${{\mathfrak{A}}}$ on [0, 1] of the form ${{\mathfrak{A}}(t)={\mathfrak{S}}+{\mathfrak{B}}(t)}$ , ${t \in [0, 1]}$ , where ${\mathfrak{S}}$ is a closed densely defined Hamiltonian ( ${={\mathfrak{J}}}$ -skew-self-adjoint) operator on ${{\mathcal{H}}}$ with ${i {\mathbb{R}} \subset \rho ({\mathfrak{S}})}$ and ${{\mathfrak{B}}}$ is a function on [0, 1] whose values are bounded operators on ${{\mathcal{H}}}$ and which is continuous in the uniform operator topology. We assume that for each ${t \in [0,1] \,{\mathfrak{A}}(t)}$ is a closed densely defined nonnegative (=J-accretive) Hamiltonian operator with ${i {\mathbb{R}} \subset \rho({\mathfrak{A}}(t))}$ . In this paper we give sufficient conditions on ${{\mathfrak{S}}}$ under which ${{\mathfrak{A}}}$ is conditionally reducible, which means that, with respect to a natural decomposition of ${{\mathcal{H}}}$ , ${{\mathfrak{A}}}$ is diagonalizable in a 2×2 block operator matrix function such that the spectra of the two operator functions on the diagonal are contained in the right and left open half planes of the complex plane. The sufficient conditions involve bounds on the resolvent of ${{\mathfrak{S}}}$ and interpolation of Hilbert spaces.     

On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? ⁿ , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H ¹(B ₁ ⁺ ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives u_x are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where H_n?p(σ)=0,p>2. It is shown for continuous solutions u from H¹(B₁/⁺) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives u_x are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles.