Global hypoellipticity and global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ) where Tⁿ is the n-dimensional torus and s?1. We prove that if P is s-globally hypoelliptic in Tⁿ then its transposed operator tP is s-globally solvable in Tⁿ, thus extending to the Gevrey classes the well-known analogous result in the corresponding C^∞ class.     

Global s-solvability, global s-hypoellipticity and Diophantine phenomena

In this paper we consider the problem of global Gevrey solvability for a class of sublaplacians on a toruswith coefficients in the Gevrey class Gs(TN). For this class of operators we show that global Gevrey solvability and global Gevrey hypoellipticity are both equivalent to the condition that the coefficients satisfy a Diophantine condition.     

An additive cohomological equation and typical behavior of Birkhoff sums over a translation of the multidimensional torus

For a periodic function f with a given decrease of the moduli of its Fourier coefficients, we analyze the solvability of the equation \(w(T_\alpha x) - w(x) = f(x) - \smallint _{\mathbb{T}^d } f(t) dt\) and the asymptotic behavior of the Birkhoff sums Σ _s=0 ^n?1 f(T _α ^s x) for almost every α. The results obtained are applied to the study of ergodic properties of a cylindrical cascade and of a special flow on the torus.     

On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? ^s , we construct Kronecker-like and van der Corput-like ergodic transformations T _1,Γ and T _2,Γ of [0, 1) ^s . We prove that for admissible lattices Γ, (T _ν,Γ ⁿ (x))_n≥0 is a low discrepancy sequence for all x ∈ [0, 1) ^s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) ^s , for almost all lattices Γ ∈ L _s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L _s ), the following asymptotic formula

$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$

holds with arbitrary small ? > 0, for all x ∈ [0, 1) ^s , and ν ∈ {1, 2}.

     

Some relations betweens-numbers of operators on Banach spaces

We considers-number sequences {s _n (T)} _n=1 ^∞ of linear and continuous operatorsT on Banach spaces and prove product formulas for the operator ideals ? _p,u ^(s) . Furthermore, we investigate the relationship between the eigenvalues of a Riesz operator and its Hilbert numbers.     

Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cⁿ, λ ∈ C, in a domain Ω ? Cⁿ which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.     

Localization of André-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K-theory at a prime p, and let T(n) be the telescope of a v_n-self map of a finite complex of type n. In this paper we study the K(n)_*-homology of Ω^∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PX be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map

s_n(X):L_T(n)P(X)→L_T(n)Σ^∞(Ω^∞X)₊     

On the stability of Voronoi cells

Let T be a given subset of ? ⁿ , whose elements are called sites, and let s??T. The Voronoi cell of s with respect to T consists of all points closer to s than to any other site. In many real applications, the position of some elements of T is uncertain due to either random external causes or to measurement errors. In this paper we analyze the effect on the Voronoi cell of small changes in s or in a given non-empty set P?T\{s}. Two types of perturbations of P are considered, one of them not increasing the cardinality of T. More in detail, the paper provides conditions for the corresponding Voronoi cell mappings to be closed, lower and upper semicontinuous. All the involved conditions are expressed in terms of the data.     

Gevrey global solvability of non-singular real first-order differential operators

In this article we deal with Gevrey global solvability of non-singular first-order operators defined on an n-dimensional s-Gevrey manifold, s > 1. As done by Duistermaat and Hörmander in the C ^∞ framework, we show that Gevrey global solvability is equivalent the existence of a global cross section.     

Matrix identities of the fast fourier transform

Let A_n(ω) be the nxn matrix A_n(ω)=(a_ij with a_ij=ω^ij, 0?i,j?n?1, ωⁿ=1. For n=rs we show

$\text{A}{}_{\mathrm{n}}\text{(w)P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{P}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{∵}\text{0}\text{s?1}\text{A}{}_{\mathrm{r}}\text{(w}{}^{\mathrm{s}}\text{)}\text{P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{{T}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{(w)}}\text{∵}\text{0}\text{r?1}\text{A}{}_{\mathrm{s}}\text{(w}{}^{\mathrm{r}}\text{)}$

=(A_r?I_s)T^s_r(I_r?A_s). When r and s are relatively prime this identity implies a wide class of identities of the form PA_n(ω)Q^T=A_r(ω^αs)?A_s(ω^βr). The matrices P^s_r, P^r_s, P, and Q are permutation matrices corresponding to the “data shuffling” required in a computer implementation of the FFT, and T^s_r is a diagonal matrix whose nonzeros are called “twiddle factors.” We establish these identities and discuss their algorithmic significance.     

H-global well-posedness for semilinear wave equations

We consider the Cauchy problem for semilinear wave equations in with n?3. Making use of Bourgain's method in conjunction with the endpoint Strichartz estimates of Keel and Tao, we establish the H^s-global well-posedness with s<1 of the Cauchy problem for the semilinear wave equation. In doing so a number of nonlinear a priori estimates is established in the framework of Besov spaces. Our method can be easily applied to the case with n=3 to recover the result of Kenig-Ponce-Vega.     

Continuity properties of the averaging projection onto the set of Hankel matrices

Some continuity properties of the averaging projection $\text{P}$ onto the set of Hankel matrices are investigated. It is proved that this projection is of weak type (1, 1) which means that for any nuclear operator T the s-numbers of $\text{P}$T satisfy S_n($\text{P}$T) ? const(n + 1). As a consequence it is obtained that $\text{P}$ maps the Matsaev ideal $\text{G}$_ω = {T:∑_n?0S_n(T)(2n + 1)^?1 < ∞} into the set of compact operators.     

Separately radial and radial Toeplitz operators on the projective space and representation theory

We consider separately radial (with corresponding group T ⁿ ) and radial (with corresponding group U(n)) symbols on the projective space P ⁿ (C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between T ⁿ and U(n).     

Solvability of the Dirichlet problem for second-order elliptic equations

In our preceding papers, we obtained necessary and sufficient conditions for the existence of an (n?1)-dimensionally continuous solution of the Dirichlet problem in a bounded domain Q ? ? _n under natural restrictions imposed on the coefficients of the general second-order elliptic equation, but these conditions were formulated in terms of an auxiliary operator equation in a special Hilbert space and are difficult to verify. We here obtain necessary and sufficient conditions for the problem solvability in terms of the initial problem for a somewhat narrower class of right-hand sides of the equation and also prove that the obtained conditions become the solvability conditions in the space W ₂ ¹ (Q) under the additional requirement that the boundary function belongs to the space W ₂ ^1/2 (?Q).     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

The problem of determining the one-dimensional kernel of the electroviscoelasticity equation

We consider the problem of finding the kernel K(t), for t ∈ [0, T], in the integrodifferential system of electroviscoelasticity. We assume that the coefficients depend only on one spatial variable. Replacing the inverse problem with an equivalent system of integral equations, we apply the contraction mapping principle in the space of continuous functions with weighted norms. We prove a global unique solvability theorem and obtain a stability estimate for the solution to the inverse problem.     

Solvability of Newton equations in smoothing-type algorithms for the SOCCP

In this paper, we first investigate the invertibility of a class of matrices. Based on the obtained results, we then discuss the solvability of Newton equations appearing in the smoothing-type algorithm for solving the second-order cone complementarity problem (SOCCP). A condition ensuring the solvability of such a system of Newton equations is given. In addition, our results also show that the assumption that the Jacobian matrix of the function involved in the SOCCP is a P₀-matrix is not enough for ensuring the solvability of such a system of Newton equations, which is different from the one of smoothing-type algorithms for solving many traditional optimization problems in ℜⁿ.     

PCR algorithm for parallel computing the solution of the general restricted linear equations

In this paper we present a parallel algorithm for parallel computing the solution of the general restricted linear equations Ax=b,x∈T, where T is a subspace of ? ⁿ and b∈AT. By this algorithm the solution x=A _T,S ⁽²⁾ b is obtained in n(log?₂ m+log?₂(n?s+1)+7)+log?₂ m+1 steps with P=mn processors when m≥2(n?1) and with P=2n(n?1) processors otherwise.