Wiener–Hopf equation whose kernel is a probability distribution

We prove the existence of a solution of an inhomogeneous generalized Wiener–Hopf equation whose kernel is a probability distribution on R generating a random walk drifting to +∞, while the inhomogeneous term f of the equation belongs to the space L ₁(0,∞) or L _∞(0,∞). We establish the asymptotic properties of the solution of this equation under various assumptions about the inhomogeneity f.     

Local Wiener–Hopf factorization and indices over arbitrary fields

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].     

Canonical Wiener–Hopf and spectral factorization of large-degree matrix polynomials

We present a novel Newton method for canonical Wiener–Hopf and spectral factorization of matrix polynomials. The initial vector results from solving a block Toeplitz-like system, and the Jacobi matrix governing the Newton iteration has nice structural and numerical properties. The local quadratic convergence of the method is proved and was tested numerically. For scalar polynomials of degree 20000, a superfast version of the method implemented on a laptop typically reqired about half a minute to produce an initial vector and then performed the Newton iteration within one second. In the matrix case, the method worked reproachless on a laptop with 8 Gigabyte RAM if the degree of the polynomial times the squared matrix dimension did not exceed 20000. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extremum Problem for the Wiener–Hopf Equation

The extremum problem for the Wiener–Hopf equation obtained by replacing the condition u(x) = 0, x < 0, by the condition of the minimum of the quadratic functional of the function u(x)exp(–x), – < x < , is solved in closed form.     

On the Inhomogeneous Conservative Wiener–Hopf Equation

We prove the existence of a solution to the inhomogeneous Wiener–Hopf equation whose kernel is a probability distribution generating a random walk drifting to ?∞. Asymptotic properties of a solution are found depending on the corresponding properties of the free term and the kernel of the equation.     

Wiener–Hopf Equations Technique for Quasimonotone Variational Inequalities

In this paper, we use the Wiener–Hopf equations technique to suggest and analyze new iterative methods for solving general quasimonotone variational inequalities. These new methods differ from previous known methods for solving variational inequalities.     

Kendall random walk,Williamson transform,and the corresponding Wiener–Hopf factorization

We give some properties of hitting times and an analogue of the Wiener–Hopf factorization for the Kendall random walk. We also show that the Williamson transform is the best tool for problems connected with the Kendall convolution.     

Hopf bifurcation and bistability of a nutrient–phytoplankton–zooplankton model

In this paper we consider a nutrient–phytoplankton–zooplankton model in aquatic environment and study its global dynamics. The existence and stability of equilibria are analyzed. It is shown that the system is permanent as long as the coexisting equilibrium exists. The discontinuous Hopf and classical Hopf bifurcations of the model are analytically verified. It is shown that phytoplankton bloom may occur even if the input rate of nutrient is low. Numerical simulations reveal the existence of saddle-node bifurcation of nonhyperbolic periodic orbit and subcritical discontinuous Hopf bifurcation, which presents a bistable phenomenon (a stable equilibrium and a stable limit cycle).     

On $$\mathbb {R}$$-Linear Problem and Truncated Wiener–Hopf Equation

Siberian Advances in Mathematics - We consider the $$\mathbb {R}$$-linear problem (also known as the Markushevich problem and the generalized Riemann boundary value problem) and the convolution...     

A Note on Wiener–Hopf Factorization for Markov Additive Processes

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.     

On the Volterra Factorization of the Wiener–Hopf Integral Operator

Mathematical Notes - The problem of the factorization of the Wiener–Hopf integral operator in the form of the product of the upper and lower Volterra operators is considered. Conditions for...     

Representation theory of finite groups in Wiener–Hopf factorization problem

We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.     

A Motzkin–Straus Type Result for 3-Uniform Hypergraphs

Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in [8]. They showed that if G is a 2-graph in which a largest clique has order l then ${\lambda(G)=\lambda(K^{(2)}_l),}$ where λ(G) denotes the Lagrangian of G. It is interesting to study a generalization of the Motzkin–Straus Theorem to hypergraphs. In this note, we give a Motzkin–Straus type result. We show that if m and l are positive integers satisfying ${{l-1 \choose 3} \le m \le {l-1 \choose 3} + {l-2 \choose 2}}$ and G is a 3-uniform graph with m edges and G contains a ${K_{l-1}^{(3)}}$ , a clique of order l?1, then ${\lambda(G) = \lambda(K_{l-1}^{(3)})}$ . Furthermore, the upper bound ${{l-1 \choose 3} + {l-2 \choose 2}}$ is the best possible.     

On a Result of Pandey and Bhargava

Let S denote the class of functions f(z) which are analytic in the unit disc D={z:|z|<1} and normalized by the conditions f(0)=0=f′(0)-1. Let K(μ)and S (m,M ) be the subclasses of S and satisfying the conditions     

Exponential functional of Lévy processes: Generalized Weierstrass products and Wiener–Hopf factorization

In this note, we state a representation of the Mellin transform of the exponential functional of Lévy processes in terms of generalized Weierstrass products. As by-product, we obtain a multiplicative Wiener–Hopf factorization generalizing previous results obtained by Patie and Savov (2012) [14] as well as smoothness properties of its distribution.     

Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.