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.     

Weighted Berezin transform in the polydisc

For c>−1, let νc denote a weighted radial measure on C normalized so that νc(D)=1. If f is harmonic and integrable with respect to νc over the open unit disc D, then for every ψ∈Aut(D). Equivalently f is invariant under the weighted Berezin transform; Bcf=f. Conversely, does the invariance under the weighted Berezin transform imply the harmonicity of a function? In this paper, we prove that for any 1?p<∞ and c₁,c₂>−1, a function f∈Lp(D²,νc₁×νc₂) which is invariant under the weighted Berezin transform; Bc₁,c₂f=f needs not be 2-harmonic.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

Qualitative analysis of a time‐delayed free boundary problem for tumor growth under the action of external inhibitors

In this paper, a time‐delayed free boundary problem for tumor growth under the action of external inhibitors is studied. It is assumed that the process of proliferation is delayed compared with apoptosis. By L^p theory of parabolic equations, the Banach fixed point theorem and the continuation theorem, the existence and uniqueness of a global solution is proved. The asymptotic behavior of the solution is also studied. The proof uses the comparison principle and the iteration method. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability of travelling wave solutions of the active Josephson junction transmission line

The behavior of the Josephson line, which is a type of active pulse transmission line, is governed by a partial differential equation which is similar to the sine-Gordon equation. This equation has two solitary travelling wave solutions with different propagation speeds c₁ and c₂, 0 < c₁ < c₂, and a one-parameter family of spatially periodic travelling wave solutions whose propagation speeds range over the intervals (0, c₁) and (c₂, + ∞). First we prove the existence of these solutions. Second we consider the stability of these solutions by linearized stability analysis. It is shown that the slow solitary solution is stable in the sense of linearized stability and that the fast solitary solution is unstable. It is shown also that the periodic solution with the speed c, 0 < c < c₁, is stable in the sense of linearized stability and that the periodic solution with the speed c, c₂ < c < c₄, is unstable, where c₄ is a certain point in (c₂, + ∞).     

Run for your life. A note on the asymptotic speed of propagation of an epidemic

We study the large-time behaviour of the solution of a nonlinear integral equation of mixed Volterra-Fredholm type describing the spatio-temporal development of an epidemic. For this model it is known that there exists a minimal wave speed c₀ (i.e., travelling wave solutions with speed c exist if $\text{¦c¦ > c}{}_0$ and do not exist if $\text{¦c¦ < c}{}_0$). In this paper we show that c₀ is the asymptotic speed of propagation (i.e., for any c₁, c₂with 0 < c₁ < c₀ < c₂ the solution tends to zero uniformly in the region $\text{¦x¦ ? c}{}_2\text{t}$, whereas it is bounded away from zero uniformly in the region $\text{¦x¦ ? c}{}_1\text{t}$ for t sufficiently large).     

Reliability evaluation of multi-state systems under cost consideration

A more practical and desirable performance index of multi-state systems is the two-terminal reliability for level (d, c) (2TR_d,c), defined as the probability that d units of flow can be transmitted from the source node to the sink node with the total cost less than or equal to c. In this article, a simple algorithm is developed to calculate 2TR_d,c in terms of (d, c)-MPs. Two major advantages of the proposed algorithm include: (1) as of now, it is the only algorithm that searches for (d, c)-MPs without requiring all minimal paths (MPs) and the procedure of transforming feasible solutions; (2) it is more practical and efficient in solving (d, c)-MP problem in contrast to the best-known method. An example is provided to illustrate the generation of (d, c)-MPs by using the presented algorithm, and 2TR_d,c is thus evaluated. Furthermore, the computational experiments are conducted to verify the performance of the presented algorithm.     

The nullity and rank of linear combinations of idempotent matrices

Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004) 25-29] proved that the nonsingularity of P₁ + P₂, where P₁ and P₂ are idempotent matrices, is equivalent to the nonsingularity of any linear combinations c₁P₁ + c₂P₂, where c₁, c₂ ≠ 0 and c₁ + c₂ ≠ 0. In the present note this result is strengthened by showing that the nullity and rank of c₁P₁ + c₂P₂ are constant. Furthermore, a simple proof of the rank formula of Groß and Trenkler [J. Groß, G. Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999) 390-395] is obtained.     

Locally bounded coverings and factorial properties of graphs

For a graph property X, let X_n be the number of graphs with vertex set {1,…,n} having property X, also known as the speed of X. A property X is called factorial if X is hereditary (i.e. closed under taking induced subgraphs) and n^c₁n≤X_n≤n^c₂n for some constants c₁ and c₂. Hereditary properties with speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored. Only the properties with speeds up to the Bell number are well studied and well behaved. To better understand the behavior of factorial properties with faster speeds we introduce a structural tool called locally bounded coverings and show that a variety of graph properties can be described by means of this tool.     

Loss probability properties in retrial queues

Under general conditions for c-server loss systems, B_c, the fraction of customers lost, is decreasing and convex. We study the shape of {B_c} for retrial queues. We show B_c+1>B_c is possible. For arbitrary arrivals and exponential service, we show {B_c} is decreasing, and report simulations where it is convex.     

Maximal periods of (Ehrhart) quasi-polynomials

A quasi-polynomial is a function defined of the form q(k)=cd(k)kd+cd−1(k)kd−1+?+c₀(k), where c₀,c₁,…,cd are periodic functions in k∈Z. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the cj(k) for Ehrhart quasi-polynomials. For generic polytopes, McMullen's bounds seem to be sharp, but sometimes smaller periods exist. We prove that the second leading coefficient of an Ehrhart quasi-polynomial always has maximal expected period and present a general theorem that yields maximal periods for the coefficients of certain quasi-polynomials. We present a construction for (Ehrhart) quasi-polynomials that exhibit maximal period behavior and use it to answer a question of Zaslavsky on convolutions of quasi-polynomials.     

Orbital stability of solitary waves for Kundu equation

In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c₃+s₂υ<0, while Guo and Wu (1995) only considered the case 2c₃+s₂υ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.     

Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h₀,h₁,…,h_c) is the Hilbert function of an artinian level module of embedding dimension two if and only if h_i−1−2h_i+h_i+1≤0 for all 0≤i≤c, where we assume that h₋₁=h_c+1=0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.     

Coloring some classes of mixed graphs

We consider the coloring problem for mixed graphs, that is, for graphs containing edges and arcs. A mixed coloring c is a coloring such that for every edge [x_i,x_j], c(x_i)≠c(x_j) and for every arc (x_p,x_q), c(x_p)<c(x_q). We will analyse the complexity status of this problem for some special classes of graphs.     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

The ball generated property in operator spaces

The Ball Generated Property (BGP) was introduced by Corson and Lindenstrauss and subsequently analysed in detail by Godefroy and Kalton. In this work, the (BGP) is studied in spaces of operators. It is shown that (BGP) is stable under c₀ and l_p-sums for 1 < p < ∞ and a characterization is provided for C(K, X)-spaces with (BGP). A similar characterization is obtained for L(X, C(K))-spaces. (BGP) is shown to be stable under injective tensor products.     

Rational Cherednik algebras and Hilbert schemes

Let H_c be the rational Cherednik algebra of type A_n-1 with spherical subalgebra U_c=eH_ce. Then U_c is filtered by order of differential operators, with associated graded ring where W is the nth symmetric group. We construct a filtered Z-algebra B such that, under mild conditions on c:• the category B-qgr of graded noetherian B-modules modulo torsion is equivalent to U_c-mod;• the associated graded Z-algebra has grB-lqgr?coh Hilb(n), the category of coherent sheaves on the Hilbert scheme of points in the plane.This can be regarded as saying that U_c simultaneously gives a non-commutative deformation of h⊕h^*/W and of its resolution of singularities Hilb(n)→h⊕h^*/W. As we show elsewhere, this result is a powerful tool for studying the representation theory of H_c and its relationship to Hilb(n).     

On the homeomorphism groups of manifolds and their universal coverings

Let H _c (M) stand for the path connected identity component of the group of all compactly supported homeomorphisms of a manifold M. It is shown that H _c (M) is perfect and simple under mild assumptions on M. Next, conjugation-invariant norms on H _c (M) are considered and the boundedness of H _c (M) and its subgroups is investigated. Finally, the structure of the universal covering group of H _c (M) is studied.     

Modeling of an insurance system and its large deviations analysis

We model an insurance system consisting of one insurance company and one reinsurance company as a stochastic process in R². The claim sizes {X_i} are an iid sequence with light tails. The interarrival times {τ_i} between claims are also iid and exponentially distributed. There is a fixed premium rate c₁ that the customers pay; c<c₁ of this rate goes to the reinsurance company. If a claim size is greater than R the reinsurance company pays for the claim. We study the bankruptcy of this system before it is able to handle N number of claims. It is assumed that each company has initial reserves that grow linearly in N and that the reinsurance company has a larger reserve than the insurance company. If c and c₁ are chosen appropriately, the probability of bankruptcy decays exponentially in N. We use large deviations (LD) analysis to compute the exponential decay rate and approximate the bankruptcy probability. We find that the LD analysis of the system decouples: the LD decay rate γ of the system is the minimum of the LD decay rates of the companies when they are considered independently and separately. An analytical and numerical study of γ as a function of (c,R) is carried out.