Representation and approximation of solutions to semilinear Volterra equations with delay

In this paper we use a theorem of Crandall and Pazy to provide the product integral representation of the nonlinear evolution operator associated with solutions to the semilinear Volterra equation: x(?)(t) = W(t, τ) ?(0) + ∝_τ^tW(t, s)F(s, x_s(?)) ds.Here the kernel W(t, s) is a linear evolution operator on a Banach space X; I is an interval of the form [?r, 0] or (?∞, 0] and F is a nonlinear mapping of R × C(I, X) into X. The abstract theory is applied to examples of partial functional differential equations.     

New Partial Difference Sets in Ztp2 and a Related Problem about Galois Rings

We generalize a construction of partial difference sets (PDS) by Chen, Ray-Chaudhuri, and Xiang through a study of the Teichmüller sets of the Galois rings. Let R=GR(p², t) be the Galois ring of characteristic p² and rank t with Teichmüller set T and let π:R→R/pR be the natural homomorphism. We give a construction of PDS in R with the parameters ν=p^2t, k=r(p^t−1), λ=p^t+r²−3r, μ=r²−r, where r=lp^{t−s(p, t)}, 1≤l≤p^{s(p, t)}, and s(p, t) is the largest dimension of a GF(p)-subspace W⊂R/pR such that π⁻¹(W)∩T generates a subgroup of R of rank <t. We prove that s(p, t) is the largest dimension of a GF(p)-subspace W of GF(p^t) such that dim W^p<t, where W^p is the GF(p)-space generated by {∏^p_i=1w_i∣w_i∈W, 1≤i≤p}. We determine the values of s(p, t) completely and solve a general problem about dim_EW^r for an E-vector space W in a finite extension of a finite field E. The PDS constructed here contain the family constructed by Chen, Ray-Chaudhuri, and Xiang and have a wider range of parameters.     

Differentiation formulas for stochastic integrals in the plane

For a one-parameter process of the form X_t=X₀+∫^t₀φ_sdW_s+∫^t₀ψ_sds, where W is a Wiener process and ∫φdW is a stochastic integral, a twice continuously differentiable function f(X_t) is again expressible as the sum of a stochastic integral and an ordinary integral via the Ito differentiation formula. In this paper we present a generalization for the stochastic integrals associated with a two-parameter Wiener process.Let {W₂, z∈R²₊} be a Wiener process with a two-dimensional parameter. Ertwhile, we have defined stochastic integrals ∫ φdWand ∫ψdWdW, as well as mixed integrals ∫h dz dW and ∫gdW dz. Now let X_z be a two-parameter process defined by the sum of these four integrals and an ordinary Lebesgue integral. The objective of this paper is to represent a suitably differentiable function f(X_z) as such a sum once again. In the process we will also derive the (basically one-dimensional) differentiation formulas of f(X_z) on increasing paths in $\text{R}{}^2{}_{\mathrm{+}}$.     

Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

The spaces Cp(X): decomposition into a countable union of bounded subspaces and completeness properties

A Tychonoff space X has to be finite if C_p(X) is σ-countably compact [23]. However, this is not true if only σ-pseudocompactness of C_p(X) is assumed. It is proved that C_p(X) is σ-pseudocompact iff X is pseudocompact and b-discrete. The technique developed yields an example showing that the theorem of Grothendieck [7] cannot be extended over the class of pseudocompact spaces. Some generalizations of the results of Lutzer and McCoy [9] are obtained. We establish also that ∏{C_p(X_t):tϵT} is a Baire space in case C_p(X_t) is Baire for each t ∈ T.     

Asymptotic behavior of the convex hull of a stationary Gaussian process?

Let X = {X(t), t ?? T} be a stationary centered Gaussian process with values in ? ^d , where the parameter set T equals ? or ?+. Let ?? _t = Cov(X ₀ ,X _t ) be the covariance function of X, and (??,?, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W _t = conv{X _u , u ?? T ?? [0, t]} as t ?? +?? and show that under the condition ??t ?? 0, t????, the rescaled convex hull (2 ln t) ^?1/2 W _t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ? associated to the covariance matrix ?? ₀. The asymptotic behavior of the mathematical expectations E f(W _t ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171?C179, 2011].     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

On the Hamilton-Jacobi equation and infimal convolution in the framework of Sobolev-functions

We study the regularity properties of the Hamilton-Jacobi flow equation and infimal convolution in the case where the initial datum function is continuous and lies in a given Sobolev-space W ^1,p (? ⁿ ). We prove that under suitable assumptions it holds for solutions w(x, t) that D _x w(·, t) → Du(·) in L ^p (? ⁿ ) as t → 0. Moreover, we construct examples showing that our results are essentially optimal.     

The fractal dimensions of the level sets of the generalized iterated Brownian motion

Let{W1(t), t∈R+} and {W2(t), t∈R+} be two independent Brownian motions with W1(0) = W2(0) = 0. {H (t) = W1(|W2(t)|), t ∈R+} is called a generalized iterated Brownian motion. In this paper, the Hausdorff dimension and packing dimension of the level sets {t ∈[0, T ], H(t) = x} are established for any 0 < T ≤ 1.     

Boundary noncrossings of additive Wiener fields∗

Let {W _i (t), t ∈ ?₊}, i = 1, 2, be two Wiener processes, and let W ₃ = {W ₃(t), t ∈ ? ₊ ² } be a two-parameter Brownian sheet, all three processes being mutually independent. We derive upper and lower bounds for the boundary noncrossing probability P _f = P{W ₁(t ₁) + W ₂(t ₂) + W ₃(t) + f(t) ≤ u(t), t ∈ ? ₊ ² }, where f, u : ? ₊ ² → ? are two general measurable functions. We further show that, for large trend functions γf > 0, asymptotically, as γ → ∞, P _γf is equivalent to \( {P}_{\gamma}\underset{\bar{\mkern6mu}}{{}_f} \) , where \( \underset{\bar{\mkern6mu}}{f} \) is the projection of f onto some closed convex set of the reproducing kernel Hilbert space of the field W(t) = W ₁(t ₁) + W ₂(t ₂) + W ₃(t). It turns out that our approach is also applicable for the additive Brownian pillow.     

On the simpleness of zeros of Stokes multipliers

The aim of this paper is to discuss the simpleness of zeros of Stokes multipliers associated with the differential equation -Φ^″(X)+W(X)Φ(X)=0, where W(X)=X^m+a₁X^m-1+?+a_m is a real monic polynomial. We show that, under a suitable hypothesis on the coefficients a_k, all the zeros of the Stokes multipliers are simple.     

Distribution and characteristic functions for correlated complex Wishart matrices

Let A(t) be a complex Wishart process defined in terms of the M×N complex Gaussian matrix X(t) by A(t)=X(t)X(t)^H. The covariance matrix of the columns of X(t) is Σ. If X(t), the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t₁, t₂, where t₁<t₂. In particular, we derive the following results: the joint density of the elements of A(t₁), A(t₂), the joint density of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂), the characteristic function of the elements of A(t₁), A(t₂), the characteristic function of the eigenvalues of Σ^-1A(t₁),Σ^-1A(t₂). In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

The Lindelöf Number of Functional Spaces on Monolithic Compacta

Let X be a compactum, τ be an infinite cardinal, and t(X) ≤ τ. In this case, l(C_p(X)) ≤ 2^τ. If X is τ-monolitliic, then l(C_p(X)) ≤ τ⁺. In addition, if X is zero-dimensional and there are no τ⁺-Aronszajn trees, then l(C_p(X)) ≤ τ.     

Global solutions of first order linear systems of ordinary differential equations with distributional coefficients

With the help of our distributional product we define four types of new solutions for first order linear systems of ordinary differential equations with distributional coefficients. These solutions are defined within a convenient space of distributions and they are consistent with the classical ones. For example, it is shown that, in a certain sense, all the solutions of X₁′=(1+δ)X₁−X₂, X₂′=(2+δ′)X₁+4X₂+δ″ have the form X₁(t)=c₁(e^2t−2e^3t)−14e^3t−δ(t), X₂(t)=c₁(4e^3t−e^2t−δ(t))+28e^3t−18δ(t)+δ′(t), where c₁ is an arbitrary constant and δ is the Dirac measure concentrated at zero. In the spirit of our preceding papers (which concern ordinary and partial differential equations) and under certain conditions we also prove existence and uniqueness results for the Cauchy problem.     

On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W ₀(x) ∪ W ₁(x), where W ₀(x) is Noetherian and W ₁(x) consists of neighbourhoods of x, then X is metacompact.     

Atomic decomposition and boundedness criterion of operators on multi-parameter hardy spaces of homogeneous type

The main purpose of this paper is to derive a new ( p, q)-atomic decomposition on the multi-parameter Hardy space Hp (X1 × X2 ) for 0 p0 p ≤ 1 for some p0 and all 1 q ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both Lq (X1 × X2 ) (for 1 q ∞) and Hardy space Hp (X1 × X2 ) (for 0 p ≤ 1). As an application, we prove that an operator T1, which is bounded on Lq (X1 × X2 ) for some 1 q ∞, is bounded from Hp (X1 × X2 ) to Lp (X1 × X2 ) if and only if T is bounded uniformly on all (p, q)-product atoms in Lp (X1 × X2 ). The similar boundedness criterion from Hp (X1 × X2 ) to Hp (X1 × X2 ) is also obtained.     

Invariance principle for integral type functionals of square-integrable martingales

Let X_n = {X_n(t): 0 ⩽ t ⩽1}, n ⩾ 0, be a sequence of square-integrable martingales. The main aim of this paper is to give sufficient conditions under which ∫^·₀f_n (A_n(t), X_n(t)) dX_n(t) converges weakly in D[0, 1] to ∫^·₀f₀(A₀(t), X₀(t)) dX₀ (t) as n → ∞, where {A_n, n ⩾ 0} is some sequence of increasing processes corresponding to the sequence {X_n, n ⩾ 0}.     

On differential inclusions with prescribed attainable sets

In this article, the inverse problem of the differential inclusion theory is studied. For a given ε>0 and a continuous set valued map t→W(t), t∈[t₀,θ], where W(t)⊂Rⁿ is compact and convex for every t∈[t₀,θ], it is required to define differential inclusion so that the Hausdorff distance between the attainable set of the differential inclusion at the time moment t with initial set (t₀,W(t₀)) and W(t) would be less than ε for every t∈[t₀,θ].