Stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity

This paper is concerned with the stability of non-monotone traveling waves for a discrete diffusion equation with monostable convolution type nonlinearity. By using the anti-weighted energy method and nonlin-ear Halanay’s inequality, we prove that all noncritical traveling waves (waves with speeds c > c_*, c_* is minimal speed) are time-exponentially stable, when the initial perturbations around the waves are small. As a corollary of our stability result, we immediately obtain the uniqueness of the traveling waves.     

A maximum trip covering location problem with an alternative mode of transportation on tree networks and segments

In this paper the following facility location problem in a mixed planar-network space is considered: We assume that traveling along a given network is faster than traveling within the plane according to the Euclidean distance. A pair of points (A _i ,A _j ) is called covered if the time to access the network from A _i plus the time for traveling along the network plus the time for reaching A _j is lower than, or equal to, a given acceptance level related to the travel time without using the network. The objective is to find facilities (i.e. entry and exit points) on the network that maximize the number of covered pairs. We present a reformulation of the problem using convex covering sets and use this formulation to derive a finite dominating set and an algorithm for locating two facilities on a tree network. Moreover, we adapt a geometric branch and bound approach to the discrete nature of the problem and suggest a procedure for locating more than two facilities on a single line, which is evaluated numerically.     

Approximate Solution of Nonlinear Discrete Equations of Convolution Type

By the method of potential monotone operators we prove global theorems on existence, uniqueness, and ways to find a solution for different classes of nonlinear discrete equations of convolution type with kernels of special form both in weighted and in weightless real spaces ? _p . Using the property of potentiality of the operators under consideration, in the case of space ? ₂ and in the case of a weighted space ? _p (?) with a generic weight ?, we prove that a discrete equation of convolution type with an odd power nonlinearity has a unique solution and it (the main result) can be found by gradient method.     

Efficient long-time computations of time-domain boundary integrals for 2D and dissipative wave equation

Linear hyperbolic partial differential equations in a homogeneous medium, e.g., the wave equation describing the propagation and scattering of acoustic waves, can be reformulated as time-domain boundary integral equations. We propose an efficient implementation of a numerical discretization of such equations when the strong Huygens’ principle does not hold.For the numerical discretization, we make use of convolution quadrature in time and standard Galerkin boundary element method in space. The quadrature in time results in a discrete convolution of weights W_j with the boundary density evaluated at equally spaced time points. If the strong Huygens’ principle holds, W_j converge to 0 exponentially quickly for large enough j. If the strong Huygens’ principle does not hold, e.g., in even space dimensions or when some damping is present, the weights are never zero, thereby presenting a difficulty for efficient numerical computation.In this paper we prove that the kernels of the convolution weights approximate in a certain sense the time domain fundamental solution and that the same holds if both are differentiated in space. The tails of the fundamental solution being very smooth, this implies that the tails of the weights are smooth and can efficiently be interpolated. Further, we hint on the possibility to apply the fast and oblivious convolution quadrature algorithm of Schädle et al. to further reduce memory requirements for long-time computation. We discuss the efficient implementation of the whole numerical scheme and present numerical experiments.     

A compact finite difference scheme for the fourth-order time-fractional integro-differential equation with a weakly singular kernel

In this paper, a compact finite difference scheme is constructed and investigated for the fourth-order time-fractional integro-differential equation with a weakly singular kernel. In the temporal direction, the Caputo derivative term is treated by means of L₁ discrete formula and the Riemann–Liouville fractional integral term is discretized by the second-order convolution quadrature rule. A fully discrete compact difference scheme is constructed with the space discretization by the fourth-order compact approximation. The stability and convergence are obtained by the discrete energy method, the Cholesky decomposition and the reduced-order method. Numerical experiments are presented to verify the theoretical analysis.     

Convolution semigroups of states

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C ₀-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.     

Traveling waves for non-local delayed diffusion equations via auxiliary equations

In this paper, we study the existence of traveling wave solutions for a class of delayed non-local reaction-diffusion equations without quasi-monotonicity. The approach is based on the construction of two associated auxiliary reaction-diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space by using the traveling wavefronts of the auxiliary equations. Under monostable assumption, by using the Schauder's fixed point theorem, we then show that there exists a constant c_∗>0 such that for each c>c_∗, the equation under consideration admits a traveling wavefront solution with speed c, which is not necessary to be monotonic.     

Extended Kac-Akhiezer formulae and the Fredholm determinant of finite section Hilbert-Schmidt kernels

This paper deals with some results (known as Kac-Akhiezer formulae) on generalized Fredholm determinants for Hilbert-Schmidt operators onL ₂-spaces, available in the literature for convolution kernels on intervals. The Kac-Akhiezer formulae have been obtained for kernels, which are not necessarily of convolution nature and for domains in ? ⁿ .     

A first–passage time random walk distribution with five transition probabilities: a generalization of the shifted inverse trinomial

In this paper a univariate discrete distribution, denoted by GIT, is proposed as a generalization of the shifted inverse trinomial distribution, and is formulated as a first-passage time distribution of a modified random walk on the half-plane with five transition probabilities. In contrast, the inverse trinomial arises as a random walk on the real line with three transition probabilities. The probability mass function (pmf) is expressible in terms of the Gauss hypergeometric function and this offers computational advantage due to its recurrence formula. The descending factorial moment is also obtained. The GIT contains twenty-two possible distributions in total. Special cases include the binomial, negative binomial, shifted negative binomial, shifted inverse binomial or, equivalently, lost-games, and shifted inverse trinomial distributions. A subclass GIT_3,1 is a particular member of Kemp’s class of convolution of pseudo-binomial variables and its properties such as reproductivity, formulation, pmf, moments, index of dispersion, and approximations are studied in detail. Compound or generalized (stopped sum) distributions provide inflated models. The inflated GIT_3,1 extends Minkova’s inflated-parameter binomial and negative binomial. A bivariate model which has the GIT as a marginal distribution is also proposed.     

Hybrid and Hybrid Adaptive Petri Nets: On the computation of a Reachability Graph

Petri Nets (PNs) constitute a well known family of formalisms for the modelling and analysis of Discrete Event Dynamic Systems (DEDS). As general formalisms for DEDS, PNs suffer from the state explosion problem. A way to alleviate this difficulty is to relax the original discrete model and deal with a fully or partially continuous model. In Hybrid Petri Nets (HPNs), transitions can be either discrete or continuous, but not both. In Hybrid Adaptive Petri Nets (HAPNs), each transition commutes between discrete and continuous behaviour depending on a threshold: if its load is higher than its threshold, it behaves as continuous; otherwise, it behaves as discrete. This way, transitions adapt their behaviour dynamically to their load. This paper proposes a method to compute the Reachability Graph (RG) of HPNs and HAPNs.     

Global stability of traveling wavefronts for nonlocal reaction-diffusion equations with time delay

This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts(waves with speeds c c_*, where c = c~* is the minimal speed) is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x →-∞, but it can be allowed arbitrary large in other locations, which improves the results in [9, 18, 21].     

On Regularity for Measures in Multiplicative Free Convolution Semigroups

Given a probability measure  μ on the real line, there exists a semigroup μ _t with real parameter t > 1 which interpolates the discrete semigroup of measures μ _n obtained by iterating its free convolution. It was shown in (Math. Z. 248(4):665–674, 2004) that it is impossible that μ _t have no mass in an interval whose endpoints are atoms. We extend this result to semigroups related to multiplicative free convolution. The proofs use subordination results.     

Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the skew fields F=R,C,H. We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of K-biinvariant functions on G as special cases. The characters are given by the associated hypergeometric functions.     

Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension

In the present paper we present the tensor-product approximation of a multidimensional convolution transform discretized via a collocation-projection scheme on uniform or composite refined grids. Examples of convolving kernels are provided by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, and with x∈R^d. For piecewise constant elements on the uniform grid of size n^d, we prove quadratic convergence O(h²) in the mesh parameter h=1/n, and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to O(h³). A fast algorithm of complexity O(dR₁R₂nlogn) is described for tensor-product convolution on uniform/composite grids of size n^d, where R₁,R₂ are tensor ranks of convolving functions. We also present the tensor-product convolution scheme in the two-level Tucker canonical format and discuss the consequent rank reduction strategy. Finally, we give numerical illustrations confirming: (a) the approximation theory for convolution schemes of order O(h²) and O(h³); (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on an n×n×n grid in the range n≤16384.     

A Convolution Operator Related to the Generalized Mehler–Fock and Kontorovich–Lebedev Transforms

In this paper we study a generalization of an index integral involving the product of modified Bessel functions and associated Legendre functions. It is applied to a convolution construction associated with this integral, which is related to the classical Kontorovich–Lebedev and generalized Mehler–Fock transforms. Mapping properties and norm estimates in weighted L _p -spaces, 1 ≤ p ≤ 2, are investigated. An application to a class of convolution integral equations is considered. Necessary and sufficient conditions are found for the solvability of these equations in L ₂.     

Fuzzy control of original UPOs of unknown discrete chaotic systems

This paper presents a fuzzy algorithm for controlling original unstable periodic orbits of unknown discrete chaotic systems. In the modeling phase, only input–output data pairs provided from the true system are required. The fuzzy model is developed using Gaussian membership functions and consequent functions where the Levenberg–Marquardt computational algorithm is employed for the model parameters calculation. In the controller design phase, the L₂-stability criterion is used, which forms the basis of the main design principle. Simulation results are given to illustrate the effectiveness and control performance of the proposed method.     

Spatially explicit ecological models: a spatial convolution approach

Spatial structure tends to have a stabilizing influence on predator–prey interactions in which the local model predicts extinction of the system. This result is well supported by laboratory observations of simple systems. Here, we use a spatially explicit version of the Nicholson–Bailey model having Moran–Ricker host reproduction to repeat and extend some of these results. Our model is a discrete spatial convolution model analogous to the integrodifference equations (IDEs) used by other authors. We show a spatial rescue effect which prevents extinction of the system by reducing the size (standard deviation) of the dispersal pdf. We also show that very favorable habitat (K=∞) and marginal habitat (K=1.0), when mixed randomly together in an explicit map, are highly stabilizing whereas either kind of habitat alone will cause extinction. The marginal habitat in this situation has host densities below parasite replacement level and thus constitutes a host refuge (although not a complete one) from the parasite. When a host–parasitoid model having spiral wave dynamics in two-dimensional space was extended to one- and three-dimensional space, we observed analogous dynamics, i.e., traveling waves of evasion and pursuit in one dimension and ‘spiral-like’ structures in a three-dimensional spatial volume. We illustrate an approach to analysis of spatial convolution models via the frequency response of the system transfer function. In spatial convolution format, local interaction and dispersal are conveniently isolated from one another, and this allows us to vary these components independently and thus to study their effects on the dynamics of the total system. We show two examples of nonrandom dispersal pdf’s – a bimodal form representing two dispersal types in the population and a ‘ripple’ pdf representing a repulsive process.     

Weighted L p -algebras on groups

The space L _p (G), 1 > p < ∞, on a locally compact group G is known to be closed under convolution only if G is compact. However, the weighted spaces L _p (G, w) are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra L _p (G,w) on an Abelian group G is described. It is shown that all algebras of this type are semisimple.     

On phase separation points for one-dimensional models

In the paper, the one-dimensional model with nearest-neighbor interactions I _n , n ∈ Z, and the spin values ±1 is considered. It is known that, under some conditions on parameters of I _n , a phase transition occurs for this model. We define the notion of a phase separation point between two phases. We prove that the expectation value of this point is zero and its mean-square fluctuation is bounded by a constant C(β) which tends to ¼ as β → ∞, where β = 1/T and T is the temperature.     

Multidimensional stability of traveling fronts in monostable reaction-diffusion equations with complex perturbations

This paper studies the multidimensional stability of traveling fronts in monostable reaction-difusion equations,including Ginzburg-Landau equations and Fisher-KPP equations.Eckmann and Wayne(1994)showed a one-dimensional stability result of traveling fronts with speeds c c(the critical speed)under complex perturbations.In the present work,we prove that these traveling fronts are also asymptotically stable subject to complex perturbations in multiple space dimensions(n=2,3),employing weighted energy methods.