Hecke operators and an identity for the dedekind sums

If h, k ∈ Z, k > 0, the Dedekind sum is given by

$\text{s(h,k) =}\text{∑}\text{μ=1}\text{k}\text{μ}\text{k}\text{hμ}\text{k}$

, with

$\text{((x)) = x ? [x] ?}\text{1}\text{2}\text{, x?Z}$

,

$\text{=0 , x∈Z}$

. The Hecke operators T_n for the full modular group SL(2, Z) are applied to log η(τ) to derive the identities (n ∈ Z⁺)

$\text{∑ ∑ s(ah+bk,dk) = σ(n)s(h,k)}$

,

$\text{ad=n b(}\text{mod}\text{d)}$

$\text{d>0}$

where (h, k) = 1, k > 0 and σ(n) is the sum of the positive divisors of n. Petersson had earlier proved (1) under the additional assumption k ≡ 0, h ≡ 1 (mod n). Dedekind himself proved (1) when n is prime.     

On the existence of periodic motions of the excited inverted pendulum by elementary methods

Using purely elementary methods, necessary and sufficient conditions are given for the existence of 2T-periodic and 4T-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating with period 2T. The equation of the motion is of the form

$$\ddot{\theta}-\frac{1}{l}(g+a(t)) \theta=0,$$

where l, g are constants and

$$a(t) := \begin{cases} A &\text{if } 2kT\leq t < (2k+1)T,\\ -A &\text{if } (2k+1)T\leq t < (2k+2)T,\end{cases}\quad (k=0,1,\dots);$$

A, T are positive constants. The exact stability zones for the upper equilibrium are presented.

     

L boundedness of commutator operator associated with schrödinger operators on heisenberg group

Let L = −ΔHⁿ + V be a Schrödinger operator on Heisenberg group Hⁿ, where ΔHⁿ is the sublaplacian and the nonnegative potential V belongs to the reverse Hölder class B_Q/2, where Q is the homogeneous dimension of Hⁿ  . Let T₁=(−ΔHn+V)⁻¹V,T₂=(−ΔHn+V)^−1/V₂1/2$T_1={(-\Delta H^n+V)}^{-1}V,T_2={(-\Delta H^n+V)}^{-1/2_{V}1/2}$, and T₃=(−Δ_Hn+V)^−1/2_∇Hn$T_3={(-{\Delta }_{H^n}+V)}^{-1/2}{\nabla }_{H^n}$, then we verify that [b, T_i], i = 1,2,3 are bounded on some L^p(Hⁿ), where b ∈ BMO(Hⁿ). Note that the kernel of T_i, i = 1,2,3 has no smoothness.     

On the uniform Kreiss resolvent condition

Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition

$\parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M}{{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots ,$

implies the uniform Kreiss resolvent condition

$\left\| {\sum\limits_{k = 0}^n {\frac{{T^k }}{{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L}{{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots .$

We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T ^m for each integer m ? 2.

     

Coverings of random ellipsoids,and invertibility of matrices with i.i.d. heavy-tailed entries

Let A = (a_ij) be an n × n random matrix with i.i.d. entries such that Ea₁₁ = 0 and Ea ₁₁ ² = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B ₂ ⁿ of cardinality at most exp(δn) such that with probability very close to one we have

$$A\left( {B_2^n} \right)\subset\mathop \cup \limits_{y \in A\left( \mathcal{N} \right)} \left( {y + L\sqrt n B_2^n} \right)$$

. In fact, a stronger statement holds true. As an application, we show that for some L' > 0 and u ∈ [0, 1) depending only on the distribution law of a11, the smallest singular value sn of the matrix A satisfies

$$\mathbb{P}\left\{ {{s_n}\left( A \right) \leq \varepsilon {n^{ - 1/2}}} \right\} \leq L'\varepsilon + {u^n}$$

for all ε > 0. The latter result generalizes a theorem of Rudelson and Vershynin which was proved for random matrices with subgaussian entries.

     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

On an inverse spectral problem for a quadratic Jacobi matrix pencil

Given two monic polynomials P2n and P2n−2 of degree 2n and 2n−2 (n?2) with complex coefficients and with disjoint zero sets. We give necessary and sufficient conditions on these polynomials such that there exist two n×n Jacobi matrices B and C for which

     

Circulant partial Hadamard matrices

A question arising in stream cypher cryptanalysis is reframed and generalized in the setting of Hadamard matrices as follows: For given n, what is the maximum value of k   for which there exists a k×n$k\times n$(±1)$(\pm 1)$-matrix A   such that AA^T=n_Ik$A{A}^{\mathsf{T}}=n{I}_k$, with each row after the first obtained by a cyclic shift of its predecessor by one position? For obvious reasons we call such matrices circulant partial Hadamard matrices. Further, what is the maximum value of k subject to the condition that the row sums are equal to r?     

On the solvability of boundary value problems for nonlinear differential systems

We obtain in a sense optimal tests for the solvability of the nonlinear boundary value problem

$$\frac{{dx}}{{dt}} = f(t,x),x(a) = h(x,(b)),$$

where the function f: [a, b] × ? ⁿ → ? ⁿ belongs to the Carathéodory class and the function h: ? ⁿ → ? ⁿ is continuous.

     

Numerical algorithm for solving quadratic matrix equations of a certain class

A relationship is found between the solutions to the quadratic matrix equation X ^T DX + AX + X ^T B + C = 0, where all the matrix coefficients are n × n matrices, and the neutral subspaces of the 2n × 2n matrix \(M = \left( \begin{gathered} CA \\ BD \\ \end{gathered} \right) \) . This relationship is used to design an algorithm for solving matrix equations of the indicated type. Numerical results obtained with the help of the proposed algorithm are presented.     

Dynamic control for non-linear systems with delay

We consider some class of non-linear systems of the form

$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$

where A(·) ∈ ? ^{n × n} , A _i (·) ∈ ? ^{n × n} , b(·) ∈ ? ⁿ , whose coefficients are arbitrary uniformly bounded functionals.

A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation

$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$

where ρ(·) ∈ ?¹, m(·) ∈ ? ⁿ , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.

     

Oscillation of third order differential equation with damping term

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term

$$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$

We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ? 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>) boundedness for Calderón commutator with rough variable kernel

For b ∈ Lip(?ⁿ); the Calderón commutator with variable kernel is defined by

$$[b,{T_1}]f(x) = p.v.\int_{{R n}} {\frac{{\Omega (x,x - y)}}{{|x - y{| {n + 1}}}}(b(x) - b(y))f(y)dy} $$

In this paper, we establish the L²(?ⁿ) boundedness for [b, T₁] with Ω(x, z′) ∈ L^∞(?ⁿ)-L^q(S^n-1) (q > 2(n-1)=n) satisfying certain cancellation conditions. Moreover, the exponent q > 2(n-1)/n is optimal. Our main result improves a previous result of Calderón.

     

Estimates for n-widths of the Hardy-type operators (Addendum to “Improved estimates for the approximation numbers of the Hardy-type operators”)

Consider the Hardy-type operator T : L^p(a,b)→L^p(a,b),-∞a<b∞, which is defined by

It is shown that

where ρ_n(T) stands for any of the following: the Kolmogorov n-width, the Gel’fand n-width, the Bernstein n-width or the nth approximation number of T.     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Properties of two-dimensional sets with small sumset

We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B⊆R² in terms of the minimum number h₁(A,B) of parallel lines covering each of A and B. We show that, if h₁(A,B)?s and |A|?|B|?2s²−3s+2, then

     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.