Positive invertibility of matrices and stability of Itô delay differential equations

We study the global exponential p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations of a special form using the theory of positively invertible matrices. To this end, we apply a method developed by N.V. Azbelev and his students for the stability analysis of deterministic functional-differential equations. We obtain sufficient conditions for the global exponential 2p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations in terms of the positive invertibility of a matrix constructed from the original system. We verify these conditions for specific equations.     

Regularity for a class of degenerate elliptic equations with discontinuous coefficients under natural growth

In this paper we are concerned with the regularity in Morrey spaces for weak solutions of a class of degenerate elliptic equations when the coefficient matrices satisfy certain VMO conditions in x uniformly with respect to u and the lower order terms satisfy a natural growth condition. Interior Hölder continuity of weak solutions is also derived with the improvement of the given data regularities.     

Partial differential equations driven by rough paths

We study a class of linear first and second order partial differential equations driven by weak geometric p-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving rough path. This allows a robust approach to stochastic partial differential equations. In particular, we may replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process. In the case of first order equations with Gaussian noise, we discuss the existence of a density with respect to the Lebesgue measure for the solution.     

Asymptotics and integrability properties for matrix coefficients of admissible representations of reductive p-adic groups

We obtain the necessary and sufficient conditions for the pth power integrability of a matrix coefficient of an admissible representation of a reductive p-adic group in terms of the exponents of the matrix coefficient.     

Fourier multipliers and periodic solutions of delay equations in Banach spaces

In this paper we characterize the existence and uniqueness of periodic solutions of inhomogeneous abstract delay equations and establish maximal regularity results for strong solutions. The conditions are obtained in terms of R-boundedness of linear operators determined by the equations and Lp- Fourier multipliers. Periodic mild solutions are also studied and characterized.     

Matrix Bernoulli equations. II

In this paper, we find sufficient conditions for the solvability by quadratures of J. Bernoulli’s equation defined over the set M ₂ of square matrices of order 2. We consider the cases when such equations are stated in terms of bases of a two-dimensional abelian algebra and a three-dimensional solvable Lie algebra over M ₂. We adduce an example of the third degree J. Bernoulli’s equation over a commutative algebra.     

Experimental and numerical study of noise effects in a FitzHugh–Nagumo system driven by a biharmonic signal

Using a nonlinear circuit ruled by the FitzHugh–Nagumo equations, we experimentally investigate the combined effect of noise and a biharmonic driving of respective high and low frequency F and f. Without noise, we show that the response of the circuit to the low frequency can be maximized for a critical amplitude B¹ of the high frequency via the effect of Vibrational Resonance (V.R.). We report that under certain conditions on the biharmonic stimulus, white noise can induce V.R. The effects of colored noise on V.R. are also discussed by considering an Ornstein–Uhlenbeck process. All experimental results are confirmed by numerical analysis of the system response.     

An application ofp-cyclic matrices,for solving periodic parabolic problems

Summary Finite-difference equations for the steady-state solution of a parabolic equation with periodic boundary conditions producep-cyclic matrices, wherep can be arbitrarily large. The periodic solution can be found by applying S.O.R. to the equations with thep-cyclic matrix, and this procedure is more economical than the standard step-by-step procedures for solving the parabolic equation. Bounds for the discretization error are found in terms of bounds for the known second derivatives of the boundary values.     

Admissibilities of linear estimator in a class of linear models with a multivariate t error variable

This paper discusses admissibilities of estimators in a class of linear models,which include the following common models:the univariate and multivariate linear models,the growth curve model,the extended growth curve model,the seemingly unrelated regression equations,the variance components model,and so on.It is proved that admissible estimators of functions of the regression coefficient β in the class of linear models with multivariate t error terms,called as Model II,are also ones in the case that error terms have multivariate normal distribution under a strictly convex loss function or a matrix loss function.It is also proved under Model II that the usual estimators of β are admissible for p 2 with a quadratic loss function,and are admissible for any p with a matrix loss function,where p is the dimension of β.     

On copositive matrices

A criterion for copositive matrices is given and for n = 3 the set of all copositive matrices is determined in terms of matrix elements. Copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations.     

A perturbed version of an inexact generalized Newton method for solving nonsmooth equations

In this paper, we present the combination of the inexact Newton method and the generalized Newton method for solving nonsmooth equations F(x)?=?0, characterizing the local convergence in terms of the perturbations and residuals. We assume that both iteration matrices taken from the B-differential and vectors F(x ^(k)) are perturbed at each step. Some results are motivated by the approach of C?tina? regarding to smooth equations. We study the conditions, which determine admissible magnitude of perturbations to preserve the convergence of method. Finally, the utility of these results is considered based on some variant of the perturbed inexact generalized Newton method for solving some general optimization problems.     

Liouville-Green asymptotic approximation for a class of matrix differential equations and semi-discretized partial differential equations

A Liouville-Green (or WKB) asymptotic approximation theory is developed for the class of linear second-order matrix differential equations Y^″=[f(t)A+G(t)]Y on [a,+∞), where A and G(t) are matrices and f(t) is scalar. This includes the case of an “asymptotically constant” (not necessarily diagonalizable) coefficient A (when f(t)≡1). An explicit representation for a basis of the right-module of solutions is given, and precise computable bounds for the error terms are provided. The double asymptotic nature with respect to both t and some parameter entering the matrix coefficient is also shown. Several examples, some concerning semi-discretized wave and convection-diffusion equations, are given.     

On strong solutions for positive definite jump diffusions

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics.Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the αth positive semidefinite power of the process itself with 0.5<α<1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions.     

Multivariate Markov-switching ARMA processes with regularly varying noise

The tail behaviour of stationary R^d-valued Markov-switching ARMA (MS-ARMA) processes driven by a regularly varying noise is analysed. It is shown that under appropriate summability conditions the MS-ARMA process is again regularly varying as a sequence. Moreover, it is established that these summability conditions are satisfied if the sum of the norms of the autoregressive parameters is less than one for all possible values of the parameter chain, which leads to feasible sufficient conditions.Our results complement in particular those of Saporta [Tail of the stationary solution of the stochastic equation Y_n+1=a_nY_n+b_n with Markovian coefficients, Stochastic Process. Appl. 115 (2005) 1954-1978.] where regularly varying tails of one-dimensional MS-AR(1) processes coming from consecutive large values of the parameter chain were studied.     

On the Deligne-Simpson problem and its weak version

We consider the Deligne-Simpson problem (DSP) (respectively the weak DSP): Give necessary and sufficient conditions upon the choice of the p+1 conjugacy classes or so that there exist irreducible (p+1)-tuples (respectively (p+1)-tuples with trivial centralizers) of matrices A_j∈c_j with zero sum or of matrices M_j∈C_j whose product is I. The matrices A_j (respectively M_j) are interpreted as matrices-residua of Fuchsian linear systems (respectively as monodromy matrices of regular linear systems) of differential equations with complex time. In the paper we give sufficient conditions for solvability of the DSP in the case when one of the matrices is with distinct eigenvalues.     

A kind of nonnegative matrices and its application on the stability of discrete dynamical systems

In this paper we introduce a new kind of nonnegative matrices which is called (sp) matrices. We show that the zero solutions of a class of linear discrete dynamical systems are asymptotically stable if and only if the coefficient matrices are (sp) matrices. To determine that a matrix is (sp) matrix or not is very simple, we need only to verify that some elements of the coefficient matrices are zero or not. According to the result above, we obtain the conditions for the stability of several classes of discrete dynamical systems.     

Equations ax = c and xb = d in rings and rings with involution with applications to Hilbert space operators

This paper reviews the equations ax = c and xb = d from a new perspective by studying them in the setting of associative rings with or without involution. Results for rectangular matrices and operators between different Banach and Hilbert spaces are obtained by embedding the ‘rectangles’ into rings of square matrices or rings of operators acting on the same space. Necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian, skew-hermitian, reflexive, antireflexive, positive and real-positive solutions, and the general solutions are described in terms of the original elements or operators. New results are obtained, and many results existing in the literature are recovered and corrected.     

Non-convex sparse regularisation

We study the regularising properties of Tikhonov regularisation on the sequence space ?² with weighted, non-quadratic penalty term acting separately on the coefficients of a given sequence. We derive sufficient conditions for the penalty term that guarantee the well-posedness of the method, and investigate to which extent the same conditions are also necessary. A particular interest of this paper is the application to the solution of operator equations with sparsity constraints. Assuming a linear growth of the penalty term at zero, we prove the sparsity of all regularised solutions. Moreover, we derive a linear convergence rate under the assumptions of even faster growth at zero and a certain injectivity of the operator to be inverted. These results in particular cover non-convex ?p regularisation with 0<p<1.     

On some sets of permutation matrices

In [5], a class Σ of p×p circulant matrices was studied where p is a prime, and necessary and sufficient conditions were presented for the matrices in Σ to be commutative and to be closed with respect to matrix multiplication. Here we show that these properties also hold for n×n circulant matrices, where n is a positive integer, with an additional condition, namely, Σ contains an n-cycle.     

Process convergence of self-normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions

In this paper we show that the continuous version of the self-normalized process Y _n,p (t)?=?S _n (t)/V _n,p ?+?(nt???[nt])X _[nt]?+?1/V _n,p ,0?<?t?≤?1; p?>?0 where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=(\sum_{i=1}^{n}|X_i|^p)^{1/p}$ and X _i i.i.d. random variables belong to DA(α), has a non-trivial distribution iff p?=?α?=?2. The case for 2?>?p?>?α and p?≤?α?<?2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörg? et al. who showed Donsker’s theorem for Y _n,2(·), i.e., for p?=?2, holds iff α?=?2 and identified the limiting process as a standard Brownian motion in sup norm.