The rate of decay of stable periodic solutions for Duffing equation with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conditions

For a new class of g(t, x), the existence, uniqueness and stability of \({2\pi}\)-periodic solution of Duffing equation \({x'' + cx' + g(t, x) = h(t)}\) are presented. Moreover, the unique \({2\pi}\)-periodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes \({g_{x}^{\prime}(t, x) - c^2/4}\) with L ^p -norms \({(p \in [1, \infty])}\), and the classical criterion employs the \({L^{\infty}}\)-norm. The advantage is that we can deal with the case that \({g_{x}^{\prime}(t, x) - c^2/4}\) is beyond the optimal bounds of the \({L^{\infty}}\)-norm, because of the difference between the L ^p -norm and the \({L^{\infty}}\)-norm.     

<Emphasis Type="Italic">k</Emphasis>-Extreme Points in Symmetric Spaces of Measurable Operators

Let \({\mathcal{M}}\) be a semifinite von Neumann algebra with a faithful, normal, semifinite trace \({\tau}\) and E be a strongly symmetric Banach function space on \({[0,\tau({\bf 1}))}\) . We show that an operator x in the unit sphere of \({E(\mathcal{M}, \tau)}\) is k-extreme, \({k \in {\mathbb{N}}}\) , whenever its singular value function \({\mu(x)}\) is k-extreme and one of the following conditions hold (i) \({\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}\) or (ii) \({n(x)\mathcal{M}n(x^*) = 0}\) and \({|x| \geq \mu(\infty, x)s(x)}\) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever \({\mathcal{M}}\) is non-atomic. The global k-rotundity property follows, that is if \({\mathcal{M}}\) is non-atomic then E is k-rotund if and only if \(E(\mathcal{M}, \tau)\) is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement \({\mu(f)}\) is k-extreme and \({|f| \geq \mu(\infty,f)}\) . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit \({\Omega(g),\,g \in L_1 + L_{\infty}}\) , or \({\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}\) , if and only if \({\mu(f) = \mu(g)}\) and \({|f| \geq \mu(\infty, f)}\) . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.     

On Quantum Dynamics on <Emphasis Type="Italic">C</Emphasis>*-Algebras

We study three related extremal problems in the space H of functions analytic in the unit disk such that their boundary values on a part γ₁ of the unit circle Γ belong to the space \(L_{{\psi _1}}^\infty ({\gamma _1})\)of functions essentially bounded on γ₁ with weight ψ₁ and their boundary values on the set γ₀ = Γ γ₁ belong to the space \(L_{{\psi _0}}^\infty ({\gamma _0})\)with weight ψ₀. More exactly, on the class Q of functions from H such that the \(L_{{\psi _0}}^\infty ({\gamma _0})\)-norm of their boundary values on γ₀ does not exceed 1, we solve the problem of optimal recovery of an analytic function on a subset of the unit disk from its boundary values on γ₁ specified approximately with respect to the norm of \(L_{{\psi _1}}^\infty ({\gamma _1})\). We also study the problem of the optimal choice of the set γ₁ for a given fixed value of its measure. The problem of the best approximation of the operator of analytic continuation from a part of the boundary by bounded linear operators is investigated.     

Symbol <Emphasis Type="Italic">p</Emphasis>-algebras of prime degree and their <Emphasis Type="Italic">p</Emphasis>-central subspaces

We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is \({p+1}\). We do it by proving the following number theoretic fact: let \({\{s_1,\dots,s_{p+1}\}}\) be \({p+1}\) distinct nonzero elements in the additive group \({G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}\), then every nonzero element \({g \in G}\) can be expressed as \({d_1 s_1+\dots+d_{p+1} s_{p+1}}\) for some non-negative integers \({d_1,\dots,d_{p+1}}\) with \({d_1+\dots+d_{p+1}\leq p-1}\).     

Sharp weak bounds for <Emphasis Type="Italic">n</Emphasis>-dimensional fractional Hardy operators

We obtain the operator norms of the n-dimensional fractional Hardy operator H _α (0 < α < n) from weighted Lebesgue spaces \(L_{\left| x \right|^\rho }^p (\mathbb{R}^n )\) to weighted weak Lebesgue spaces \(L_{\left| x \right|^\beta }^{q,\infty } (\mathbb{R}^n )\).     

Asymptotic analysis on the normalized <Emphasis Type="Italic">k</Emphasis>-error linear complexity of binary sequences

Linear complexity and k-error linear complexity are the important measures for sequences in stream ciphers. This paper discusses the asymptotic behavior of the normalized k-error linear complexity \({L_{n,k}(\underline{s})/n}\) of random binary sequences \({\underline{s}}\) , which is based on one of Niederreiter’s open problems. For k = n θ, where 0 ≤ θ ≤ 1/2 is a fixed ratio, the lower and upper bounds on accumulation points of \({L_{n,k}(\underline{s})/n}\) are derived, which holds with probability 1. On the other hand, for any fixed k it is shown that \({\lim_{n\rightarrow\infty} L_{n,k}(\underline{s})/n = 1/2}\) holds with probability 1. The asymptotic bounds on the expected value of normalized k-error linear complexity of binary sequences are also presented.     

Divergent solutions to the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-supercritical NLS equations

We investigate the nonlinear Schrödinger equation iu _t +Δu+|u| ^p?1 u = 0with 1+ 4/N < p < 1+ 4/N?2 (when N = 1, 2, 1 + 4/N < p < ∞) in energy space H ¹ and study the divergent property of infinite-variance and nonradial solutions. If \(M{\left( u \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( u \right) \prec M{\left( Q \right)^{\frac{{1 - {s_C}}}{{{s_C}}}}}E\left( Q \right)\) and \(\left\| {{u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}\left\| {\nabla {u_0}} \right\|_2^{\frac{{1 - {s_c}}}{{{s_c}}}}{\left\| {\nabla Q} \right\|_2}\), then either u(t) blows up in finite forward time or u(t) exists globally for positive time and there exists a time sequence t _n → +∞ such that \({\left\| {\nabla u\left( {{t_n}} \right)} \right\|_2} \to + \infty \). Here Q is the ground state solution of ?(1?s _c )Q+ΔQ+Q ^p?1 Q = 0. A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation obtained by Holmer to the general mass-supercritical and energy-subcritical case.     

<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>-Analogue of a linear transformation preserving log-convexity

In this paper, we establish the preserving log-convexity of linear transformation associated with p, q-analogue of Pascal triangle, i.e., if the sequence of nonnegative numbers {x_n}n is logconvex, then \({y_n} = {\sum\nolimits_{k = 0}^n {\left[ {\frac{n}{k}} \right]} _{pq}}{x_k}\) so is it for q ≠ p ≥ 1.     

On a two-phase free boundary condition for <Emphasis Type="Italic">p</Emphasis>-harmonic measures

Let \({\Omega^i\subset {\bf R}^n, i\in\{1,2\}}\) , be two (δ, r ₀)-Reifenberg flat domains, for some \({0 < \delta < \hat \delta}\) and r ₀ > 0, assume \({\Omega^1\cap\Omega^2=\emptyset}\) and that, for some \({w\in {\bf R}^n}\) and some 0 < r, \({w\in\partial\Omega^1\cap\partial\Omega^2, \partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . Let p, 1 < p < ∞, be given and let u ⁱ , \({i\in\{1,2\}}\) , denote a non-negative p-harmonic function in Ω ⁱ , assume that u ⁱ , \({i\in\{1,2\}}\), is continuous in \({\bar\Omega^i\cap B(w,2r) }\) and that u ⁱ  = 0 on \({\partial\Omega^i\cap B(w,2r)}\) . Extend u ⁱ to B(w, 2r) by defining \({u^i\equiv 0}\) on \({B(w,2r) {\setminus} \Omega^i}\). Then there exists a unique finite positive Borel measure μ ⁱ , \({i\in\{1,2\}}\) , on R ⁿ , with support in \({\partial\Omega^i\cap B(w,2r)}\) , such that if \({\phi \in C_0^\infty (B(w,2r))}\) , then

$\int\limits_{\mathbf R^n} \,|\nabla u^i|^{ p-2} \,\langle \nabla u^i, \,\nabla \phi \rangle \,dx =- \int\limits_{\mathbf R^n} \,\phi \,d \mu^i.$

Let \({\Delta(w,2r)=\partial\Omega^1\cap B(w,2r)=\partial\Omega^2\cap B(w,2r)}\) . The main result proved in this paper is the following. Assume that μ ² is absolutely continuous with respect to μ ¹ on Δ(w, 2r), d μ ² = kd μ ¹ for μ ¹-almost every point in Δ(w, 2r) and that \({\log k\in VMO(\Delta(w,r),\mu^1)}\) . Then there exists \({\tilde \delta = \tilde \delta(p,n) > 0}\) , \({\tilde \delta < \hat \delta}\) , such that if \({\delta\leq\tilde\delta}\) , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).

     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

Elementary Graphs with Respect to <Emphasis Type="Italic">f</Emphasis>-Parity Factors

This note concerns the f-parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function \({f : V(G) \rightarrow \mathbb{N}}\), and our goal is to find a spanning subgraph F of G with deg _F ≤ f and minimizing the number of vertices x with \({\deg_F(x) \not\equiv f(x) \, {\rm mod} \, {2}}\) . First we prove a Gallai–Edmonds type structure theorem and some other known results on the f-parity subgraph problem, using an easy reduction to the matching problem. Then we use this reduction to investigate barriers and elementary graphs with respect to f-parity factors, where an elementary graph is a graph such that the union of f-parity factors form a connected spanning subgraph.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

On a hybrid bound for twisted <Emphasis Type="Italic">L</Emphasis>-values

Let f ∈S _k (M, ψ) be a newform, and let χ be a primitive character of conductor q. We express \({L(\frac{1}{2}+it,f\otimes\chi)}\) as a short combination of bilinear forms involving Kloosterman fractions. Using this we establish the convexity breaking bound \({L\left(\tfrac{1}{2}+it,f\otimes\chi\right)\ll_{f,\varepsilon} [q(1+|t|)]^{\frac{1}{2}-\frac{1}{118}+\varepsilon}}\) for any ε > 0.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

A Note on Derivations on the Algebra of Operators in Hilbert <Emphasis Type="Italic">C</Emphasis>*-Modules

Let \({\mathfrak{M}}\) be a Hilbert C*-module on a C*-algebra \({\mathfrak{A}}\) and let \({End_\mathfrak{A}(\mathfrak{M})}\) be the algebra of all operators on \({\mathfrak{M}}\). In this paper, first the continuity of \({\mathfrak{A}}\)-module homomorphism derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) is investigated. We give some sufficient conditions on which every derivation on \({End_\mathfrak{A}(\mathfrak{M})}\) is inner. Next, we study approximately innerness of derivations on \({End_\mathfrak{A}(\mathfrak{M})}\) for a σ-unital C*-algebra \({\mathfrak{A}}\) and full Hilbert \({\mathfrak{A}}\)-module \({\mathfrak{M}}\). Finally, we show that every bounded linear mapping on \({End_\mathfrak{A}(\mathfrak{M})}\) which behave like a derivation when acting on pairs of elements with unit product, is a Jordan derivation.     

On the Dunford Property (<Emphasis Type="Italic">C</Emphasis>) for Bounded Linear Operators <Emphasis Type="Italic">RS</Emphasis> and <Emphasis Type="Italic">SR</Emphasis>

In this paper we show that if \({S\in L(X,Y)}\) and \({R\in L(Y,X),}\) X and Y complex Banach spaces, then the products RS and SR share the Dunford property (C). We also study property (C) for R, S, RS and \({SR \in L(X)}\) in the case that R and S satisfies the operator equations RSR = R ² and SRS = S ².     

Small weight codewords in the dual code of points and hyperplanes in PG(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">q</Emphasis> even

The main theorem of this article gives a classification of the codewords in \({C^{\bot}_{n-1}(n,q)}\) , the dual code of points and hyperplanes in PG(n, q), q even, with weight smaller than \({q+\sqrt[3]{q^{2}}+1}\). In the proof, we rely on the classification of the small blocking sets in PG(2, q), q even.     

Around <Emphasis Type="Italic">q</Emphasis>-Appell polynomial sequences

First we show that the quadratic decomposition of the Appell polynomials with respect to the q-divided difference operator is supplied by two other Appell sequences with respect to a new operator \(\mathcal{M}_{q;q^{-\varepsilon}}\), where ε represents a complex parameter different from any negative even integer number. While seeking all the orthogonal polynomial sequences invariant under the action of \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\) (the \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\)-Appell), only the Wall q-polynomials with parameter q ^ε/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.     

A geometric approach to alternating <Emphasis Type="Italic">k</Emphasis>-linear forms

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.