On Σ1-definable Functions Provably Total in I ∏

We prove the following theorem: Let φ(x) be a formula in the language of the theory PA^? of discretely ordered commutative rings with unit of the form ?yφ′(x,y) with φ′ and let ∈ Δ₀ and let fφ: ? → ? such that f_φ(x) = y iff φ′(x,y) & (?z < y) φ′(x,z). If I ∏ ∈(?x ≥ 0), φ then there exists a natural number K such that I ∏ ? ?y?x(x > y ? ?φ(x) < x^K). Here I ∏₁^? denotes the theory PA^? plus the scheme of induction for formulas φ(x) of the form ?yφ′(x,y) (with φ′) with φ′ ∈ Δ₀.     

Iteratively reweighted least squares minimization for sparse recovery

Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ?^N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ^?1(y) is typically an (N ? m)—dimensional hyperplane; in addition, x is then equal to the element in Φ^?1(y) of minimal ??₁‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ^?1(y) with smallest ??₂(w)‐norm. If x⁽ⁿ⁾ is the solution at iteration step n, then the new weight w⁽ⁿ⁾ is defined by w := [|x|² + ε]^?1/2, i = 1, …, N, for a decreasing sequence of adaptively defined ε_n; this updated weight is then used to obtain x^{(n + 1)} and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x⁽ⁿ⁾ converges for all y, regardless of whether Φ^?1(y) contains a sparse vector. If there is a sparse vector in Φ^?1(y), then the limit is this sparse vector, and when x⁽ⁿ⁾ is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w = [|x|² + ε]^?1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc.     

For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution

We consider linear equations y = Φx where y is a given vector in ?ⁿ and Φ is a given n × m matrix with n < m ≤ τn, and we wish to solve for x ∈ ?^m. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We prove the existence of ρ = ρ(τ) > 0 so that for large n and for all Φ's except a negligible fraction, the following property holds: For every y having a representation y = Φx₀ by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, the solution x₁ of the ??₁‐minimization problem is unique and equal to x₀. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost‐spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices. © 2006 Wiley Periodicals, Inc.     

For most large underdetermined systems of equations,the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution

We consider inexact linear equations y ≈ Φx where y is a given vector in ?ⁿ, Φ is a given n × m matrix, and we wish to find x_0,? as sparse as possible while obeying ‖y ? Φx_0,?‖₂ ≤ ?. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the ??₁‐minimization problem is convex and is considered tractable. We show that for most Φ, if the optimally sparse approximation x_0,? is sufficiently sparse, then the solution x_1,? of the ??₁‐minimization problem is a good approximation to x_0,?. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We study the underdetermined case where m ～ τn and τ > 1, and prove the existence of ρ = ρ(τ) > 0 and C = C(ρ, τ) so that for large n and for all Φ's except a negligible fraction, the following approximate sparse solution property of Φ holds: for every y having an approximation ‖y ? Φx₀‖₂ ≤ ? by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, This has two implications. First, for most Φ, whenever the combinatorial optimization result x_0,? would be very sparse, x_1,? is a good approximation to x_0,?. Second, suppose we are given noisy data obeying y = Φx₀ + z where the unknown x₀ is known to be sparse and the noise ‖z‖₂ ≤ ?. For most Φ, noise‐tolerant ??₁‐minimization will stably recover x₀ from y in the presence of noise z. We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost‐spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices. © 2006 Wiley Periodicals, Inc.     

Some remarks on essential self‐adjointness and ultracontractivity of a class of singular elliptic operators

We study the properties of essential self‐adjointness on C^∞_c (ℝ^N ) and semigroup ultracontractivity of a class of singular second order elliptic operators defined in L² (ℝ^N , σ^{–a –N} (x) dx) with Dirichlet boundary conditions, where a, b ∈ ℝ and σ: ℝ^N → (0, ∞) is a C^∞‐function satisfying c^‐1(1 + |x |) ≤ σ (x) ≤ c (1 + |x |) (x ∈ ℝN). We also obtain sharp short time upper and lower diagonal bounds on the heat kernel of e –^Ht. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Hankel operators on the Fock space

In this paper we study generalized Hankel operators ofthe form : ?²(|z |²) → L²(|z |²). Here, (f):= (Id–P_l )( ^kf) and Pl is the projection onto A_l ²(?, |z |²):= cl(span{ ^m zⁿ | m, n ∈ N, m ≤ l }). The investigations in this article extend the ones in [11] and [6], where the special cases l = 0 and l = 1 are considered, respectively. The main result is that the operators are not bounded for l < k – 1. The proof relies on a combinatoric argument and a generalization to general conjugate holomorphic L² symbols, generalizing arguments from [6], seems possible and is planned for future work (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Obstacle Problems with Linear Growth: Hölder Regularity for the Dual Solution

For a strictly convex integrand f : ℝⁿ → ℝ with linear growth we discuss the variational problem among mappings u : ℝⁿ ⊃ Ω → ℝ of Sobolev class W¹₁ with zero trace satisfying in addition u ≥ ψ for a given function ψ such that ψ|_∂Ω < 0. We introduce a natural dual problem which admits a unique maximizer σ. In further sections the smoothness of σ is investigated using a special J-minimizing sequence with limit u* ∈ C^1,α (Ω) for which the duality relation holds.     

Almost Sure Convergence in Extreme Value Theory

Let X₁, …, X_n be independent random variables with common distribution function F. Define and let G(x) be one of the extreme-value distributions. Assume F ∈ D(G), i.e., there exist a_n> 0 and b_n ∈ ? such that . Let l(?∞,x)(·) denote the indicator function of the set (?∞,x) and S(G) =: {x : 0 < G(x) < 1}. Obviously, 1(?∞,x)((M_n?b_n)/a_n) does not converge almost surely for any x ∈ S(G). But we shall prove .     

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.

     

Time asymptotics for the polyharmonic wave equation in waveguides

Let Ω denote an unbounded domain in ?ⁿ having the form Ω=?^l×D with bounded cross‐section D??^n?l, and let m∈? be fixed. This article considers solutions u to the scalar wave equation ?u(t,x) +(?Δ)^mu(t,x) = f(x)e^?iωt satisfying the homogeneous Dirichlet boundary condition. The asymptotic behaviour of u as t→∞ is investigated. Depending on the choice of f ,ω and Ω, two cases occur: Either u shows resonance, which means that ∣u(t,x)∣→∞ as t→∞ for almost every x ∈ Ω, or u satisfies the principle of limiting amplitude. Furthermore, the resolvent of the spatial operators and the validity of the principle of limiting absorption are studied. Copyright © 2003 John Wiley & Sons, Ltd.     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.     

A note on a third‐order multi‐point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third‐order multi‐point boundary value problem at resonance where f: [0, 1] × R³ → R is a continuous function, 0 < ξ₁ < ??? < ξ_m < 1, α_i ∈ R, i = 1, …, m, m ≥ 1 and 0 < η₁ < η₂ < ??? < η_n < 1, β_j ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′^′′) is equal to 2. Since all the existence results for third‐order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.     

Modal stabilization of a certain class of uniformly controllable systems

The system $\dot x = A( \cdot )x + b( \cdot )u,$ where A(·) ∈ ? ^n×n and b(·) ∈ ? ^n×1, is considered. The elements of the matrix A(·) and the column b(·) are bounded by nonanticipating functionals of an arbitrary nature that satisfy the condition $\mathop {\inf }\limits_{( \cdot )} A^{n - 1} ( \cdot )b( \cdot ),...,A( \cdot )b( \cdot ),b( \cdot )| > 0$ . From a given constant spectrum contained in the left half-plane, a feedback u = (s(·), x) is constructed, the coefficients of which are expressed in terms of A(·) and b(·). Conditions for the closed system to be globally exponentially stable are found. A similar result is obtained for the system $x(k + 1) = A(k)x(k) + b(k)u(k)$ .     

Dynamics of a three‐dimensional systems of rational difference equations

We investigate in this paper the solutions and the periodicity of the following rational systems of difference equations of three‐dimensional with initial conditions x_?2,x_?1,x₀,y_?2,y_?1,y₀,z_?2,z_?1andz₀ are nonzero real numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Weighted Norm Inequalities with General Weights for the Commutator of Calderón

In this paper, by a sharp function estimate and an idea of Lerner, the authors establish someweighted estimates for the m-multilinear integral operator which is bounded from L1(Rn)×···×L1 (Rn)to L1/m,∞ (Rn),, and the associated kernel K(x; y1, . . . , ym)) enjoys a regularity on the variable x. As anapplication, weighted estimates with general weights are given for the commutator of Calderón.     

Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise sensitivity. Consider a source of truly random bits x ∈ {0, 1}ⁿ, and k parties, who have noisy version of the source bits yⁱ ∈ {0, 1}ⁿ, when for all i and j, it holds that P [y = x_j] = 1 ? ?, independently for all i and j. That is, each party sees each bit correctly with probability 1 ? ?, and incorrectly (flipped) with probability ?, independently for all bits and all parties. The parties, who cannot communicate, wish to agree beforehand on balanced functions f_i: {0, 1}ⁿ → {0, 1} such that P [f₁(y¹) = … = f_k(y^k)] is maximized. In other words, each party wants to toss a fair coin so that the probability that all parties have the same coin is maximized. The function f_i may be thought of as an error correcting procedure for the source x. When k = 2,3, no error correction is possible, as the optimal protocol is given by f_i(yⁱ) = y. On the other hand, for large values of k, better protocols exist. We study general properties of the optimal protocols and the asymptotic behavior of the problem with respect to k, n, and ?. Our analysis uses tools from probability, discrete Fourier analysis, convexity, and discrete symmetrization. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Robust stabilization of some class of uncertain systems

Consider an uncertain system $$ \dot x = A( \cdot )x + B( \cdot )u, $$ where A(·) ∈ ? ^{n × n} , B(·) ∈ ? ^{n × m} , and the elements of matrices A(·) and B(·) are arbitrary functionals. It is assumed that all elements are uniformly bounded, and that the first r elements counted from above and situated on a certain fixed upper superdiagonal are alternating. It is also assumed that m = n ? r, and that a matrix formed by the last m rows of matrix B(·) is nonsingular. The control u = S(·)x is synthesized, and conditions on the admissible matrix B(·) ensuring the global asymptotical stability of the system are obtained. We consider the case when modulation of the components of the vector u is realized by means of synchronous amplitude-frequency pulse modulators of the first kind. A lower estimate for the pulse frequency under which the pulse system is globally asymptotically stable is obtained.     

Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

In this paper a system of differential equations y′ ? A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A₁+ A ₀ +λ _?1A_?1(·,λ) and A ₁,A ₀, A _{? 1} are n × n matrix-functions. The main assumptions: A ₁ is absolutely continuous on the interval [a, b], A ₀ and A _{- 1}(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ₁(x),…,φ_n (x) of the characteristic equation det (φ E — A ₁) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ? C on which the inequalities Re(λφ₁(x)) ≤ … ≤ Re (λφ_n(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φ_j(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A ₁ (x), A _o (x) and A _{- 1} (x).     

Complete systems of recursive integrals and Taylor series for solutions of Sturm–Liouville equations

Given a regular nonvanishing complex valued solution y₀ of the equation , x ∈ (a,b), assume that it is n times differentiable at a point x₀ ∈ [a,b]. We present explicit formulas for calculating the first n derivatives at x₀ for any solution of the equation . That is, a map transforming the Taylor expansion of y₀ into the Taylor expansion of u is constructed. The result is obtained with the aid of the representation for solutions of the Sturm‐Liouville equation in terms of spectral parameter power series. Copyright © 2012 John Wiley & Sons, Ltd.