Mean first-passage time of a bistable kinetic model driven by two different kinds of coloured noises

The transient properties of a bistable system are investigated when both the multiplicative noise and the coupling between additive and multiplicative noises are coloured with different correlation time τ₁ and τ₂. The mean first-passage time (MFPT) is obtained. The numerical computations show that the intensities of additive and multiplicative noises and the coupling strength λ affected the MFPT in the same way, while τ₁ and τ₂ play different roles in the MFPT. The increase of τ₁ can reduce the escape rate, while increase of τ₂ can enhance the escape rate.     

Stochastic stability and state shifts for a time-delayed cancer growth system subjected to correlated multiplicative and additive noises

In the present paper, we investigate the stationary probability distribution(SPD) and the mean treatment time of a time-delayed cancer growth system induced by cross-correlated intrinsic and extrinsic noises. Our main results show that the resonant-like phenomenon of the mean first-passage time (MFPT) appears in the tumor cell growth model due to the interaction of all kinds of noises and time delay. Due to the existence of the resonant-like peak value, by increasing the intensity of multiplicative noise and time delay, it is possible to restrain effectively the development of the cancer cells and enhance the stability of the system. During the process of controlling the diffusion of the tumor cells, it contributes to inhibiting the development of cancer by increasing the cross-correlated noise strength and weakening the additive noise intensity and time delay. Meanwhile, the proper multiplicative noise intensity is conducive to the process of inhibition. Conversely, in the process of exterminating cancer cells of a large density, it can exert positive effects on eliminating the tumor cells by increasing noises intensities and the value of time delay.     

Characterization of edge‐colored complete graphs with properly colored Hamilton paths

An edge‐colored graph H is properly colored if no two adjacent edges of H have the same color. In 1997, J. Bang‐Jensen and G. Gutin conjectured that an edge‐colored complete graph G has a properly colored Hamilton path if and only if G has a spanning subgraph consisting of a properly colored path C₀ and a (possibly empty) collection of properly colored cycles C₁,C₂,…, C_d such that provided . We prove this conjecture. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 333–346, 2006     

Factorization Representations in the Boundary Crossing Problems for Random Walks on a Markov Chain

Let τ be some stopping time for a random walk S _n defined on transitions of a finite Markov chain and let τ(t) be the first passage time across the level t which occurs after τ. We prove a theorem that establishes a connection between the dual Laplace-Stieltjes transforms of the joint distributions of (τ, S _τ) and (τ(t), S _τ(t)). This result applies to the study of the number of crossings of a strip by sample paths of a random walk.Original Russian Text Copyright © 2005 Lotov V. I. and Orlova N. G.The authors were partially supported by the Russian Foundation for Basic Research (Grant 05-01-00810) and the Grant Council of the President of the Russian Federation (Grant NSh-2139.2003.1).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 833–840, July–August, 2005.     

Extensions of p-adic vector measures

For R being a separating algebra of subsets of a set X, E a complete Hausdorff non-Archimedean locally convex space and m: R → E a bounded finitely additive measure, it is shown that:

a If m is σ-additive and strongly additive, then m has a unique σ-additive extension m^σ on the σ-algebra R^σ generated by R.

b If m is strongly additive and τ-additive, then m has a unique τ-additive extension m^τ on the α-algebra R^bo of all τ_R-Borel sets, where τ_R is the topology having R as a basis.

Also, some other results concerning such measures are given.     

Complexity results for scheduling tasks with discrete starting times

Suppose that n independent tasks are to be scheduled without preemption on a set of identical parallel processors. Each task T_i requires a given execution time τ_i and it may be started for execution on any processor at any of its prescribed starting times s_i1, s_i2, …, s_iki, with k_i ≤ k for some fixed integer k. We first prove that the problem of finding a feasible schedule on a single processor is NP-complete in the strong sense even when τ_i ε {τ, τ′} and k_i ≤ 3 for 1 ≤ i ≤ n. The same problem is, however, shown to be solvable in O(n log n) time, provided s_iki − s_i1 < τ_i for 1 ≤ i ≤ n. We then show that the problem of finding a feasible schedule on an arbitrary number of processors is strongly NP-complete even when τ_i ε {τ, τ′}, k_i = 2 and s_i2 − s_i1 = δ < τ_i for 1 ≤ i ≤ n. Finally a special case with k_i = 2 and s_i2 − s_i1 = 1, 1 ≤ i ≤ n, of the above multiprocessor scheduling problem is shown to be solvable in polynomial time.     

Additivity of Lie multiplicative maps on triangular algebras

Let 𝒜 and ? be unital algebras over a commutative ring ?, and ? be a (𝒜,??)-bimodule, which is faithful as a left 𝒜-module and also as a right ?-module. Let 𝒰?=?Tri(𝒜,??,??) be the triangular algebra and 𝒱 any algebra over ?. Assume that Φ?:?𝒰?→?𝒱 is a Lie multiplicative isomorphism, that is, Φ satisfies Φ(ST???TS)?=?Φ(S)Φ(T)???Φ(T)Φ(S) for all S, T?∈?𝒰. Then Φ(S?+?T)?=?Φ(S)?+?Φ(T)?+?Z _S,T for all S, T?∈?𝒰, where Z _S,T is an element in the centre 𝒵(𝒱) of 𝒱 depending on S and T.     

On Disjoint Sums in the Lattice of Linear Topologies

Let M be a vector space over a skew-field equipped with the discrete topology, (M) be the lattice of all linear topologies on M ordered by inclusion,and τ_*, τ₀, τ₁ ∈ (M). We write τ₁ = τ_* ⊔ τ₀ or say that τ₁ is a disjoint sum of τ_* and τ₀ if τ₁ = inf{τ₀, τ_*} and sup{τ₀, τ_*} is the discrete topology. Given τ₁, τ₀ ∈ (M), we say that τ₀ is a disjoint summand of τ₁ if τ₁ = τ_* ⊔ τ₀ for a certain τ_* ∈ (M). Some necessary and some sufficient conditions are proved for τ₀ to be a disjoint summand of τ₁.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 3–18, 2003.     

Impact of two types of time delays and cross-correlated multiplicative and additive noises on stability and stochastic resonance for a metapopulation system

In this paper, we investigate the stability and the shift between the extinction state and the stable one of a large density and the stochastic resonance (SR) for a metapopulation system subjected to two types of time delay terms, cross-correlation noises and multiplicative signal. By using the fast descent method and the method of small delay approximation, the expressions of the effective potential function and the signal-to-noise ratio (SNR) are obtained. We denote by Q the intensity of the multiplicative noise, and M the intensity of the additive noise, θ and τ the two time delay terms introduced into the metapopulation system. Our main results show some facts that time delay θ and the strength of correlation noise λ can restrain the development of the metapopulation, while the other term of time delay τ can accelerate the expansion of the population from the extinction state to the large stable one. We discover that it is possible to enhance the signal-to-noise ratio by adjusting the intensities of the multiplicative, additive noises and the time delays of the stochastic metapopulation system     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

The complexity of finding generalized paths in tournaments

Given a tournament with n vertices, we consider the number of comparisons needed, in the worst case, to find a permutation υ₁υ₂…υ_n of the vertices, such that the results of the games υ₁υ₂, υ₂υ₃,…, υ_n−1υ_n match a prescribed pattern. If the pattern requires all arcs to go forwrd, i.e., υ₁ → υ₂, υ₂ → υ₃,…, υ_n−1 → υ_n, and the tournament is transitive, then this is essentially the problem of sorting a linearly ordered set. It is well known that the number of comparisons required in this case is at least cn lg n, and we make the observation that O(n lg n) comparisons suffice to find such a path in any (not necessarily transitive) tournament. On the other hand, the pattern requiring the arcs to alternate backward-forward-backward, etc., admits an algorithm for which O(n) comparisons always suffice. Our main result is the somewhat surprising fact that for various other patterns the complexity (number of comparisons) of finding paths matching the pattern can be cn lg^αn for any α between 0 and 1. Thus there is a veritable spectrum of complexities, depending on the prescribed pattern of the desired path. Similar problems on complexities of algorithms for finding Hamiltonian cycles in graphs and directed graphs have been considered by various authors, [2, pp. 142, 148, 149; 4].     

Signed permutations and the four color theorem

Let σ₁,σ₂ be two permutations in the symmetric group S_n. Among the many sequences of elementary transpositions τ₁,…,τ_r transforming σ₁ into σ₂=τ_rτ₁σ₁, some of them may be signable, a property introduced in this paper. We show that the four color theorem in graph theory is equivalent to the statement that, for any n≥2 and any σ₁,σ₂S_n, there exists at least one signable sequence of elementary transpositions from σ₁ to σ₂. This algebraic reformulation rests on a former geometric one in terms of signed diagonal flips, together with a codification of the triangulations of a convex polygon on n+2 vertices by permutations in S_n.     

The heat flow in nonlinear Hodge theory

We study the nonlinear Hodge system dω=0 and δ(ρ(|ω|²)ω)=0 for an exterior form ω on a compact oriented Riemannian manifold M, where ρ(Q) is a given positive function. The solutions are called ρ-harmonic forms. They are the stationary points on cohomology classes of the functional with e′(Q)=ρ(Q)/2. The ρ-codifferential of a form ω is defined as δ_ρω=ρ⁻¹δ(ρω) with ρ=ρ(|ω|²).We evolve a given closed form ω₀ by the nonlinear heat flow system for a time-dependent exterior form ω(x,t) on M. This system is the differential of the normalized gradient flow for E(ω) with ω=ω₀+du. Under a technical assumption on the function 2ρ′(Q)Q/ρ(Q), we show that the nonlinear heat flow system , with initial condition ω(·,0)=ω₀, has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ω₀. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative.     

A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

Average CaseL∞-Approximation in the Presence of Gaussian Noise

We consider the average caseL_∞-approximation of functions fromC^r([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ²>0. We show that the n th minimal average error is of ordern^{−(2r+1)/(4r+4)} ln^1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forL_q-approximation withq<∞ andσ0, and forL_∞-approximation withσ=0.     

One‐Dimensional Birth‐Death Process and Delbrück‐Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions

As a mathematical theory for the stochastic, nonlinear dynamics of individuals within a population, Delbrück‐Gillespie process (DGP) is a birth–death system with state‐dependent rates which contain the system size V as a natural parameter. For large V , it is intimately related to an autonomous, nonlinear ODE as well as a diffusion process. For nonlinear dynamical systems with multiple attractors, the quasi‐stationary and stationary behavior of such a birth–death process can be understood in terms of a separation of time scales by a T*～e^αV (α > 0) : a relatively fast, intra‐basin diffusion for t?T* and a much slower inter‐basin Markov jump process for t?T* . In this paper for one‐dimensional systems, we study both stationary behavior (t=∞ ) in terms of invariant distribution , and finite time dynamics in terms of the mean first passsage time (MFPT) . We obtain an asymptotic expression of MFPT in terms of the “stochastic potential”. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the for a DGP. When n₁ and n₂ belong to two different basins of attraction, the MFPT yields the T*(V) in terms of Φ (x, V) ≈φ₀(x) + (1/V)φ₁(x) . For systems with saddle‐node bifurcations and catastrophe, discontinuous “phase transition” emerges, which can be characterized by Φ (x, V) in the limit of . In terms of timescale separation, the relation between deterministic local nonlinear bifurcations, and stochastic global phase transition is discussed. The one‐dimensional theory is a pedagogic first step toward a general theory of DGP.     

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ₀>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes

where σ=σ_n=(s₁,s₂,…,s_n) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d_max=2∑_ν=1ⁿs_ν+2n−1 if and only if

The proof of the uniqueness of the extremal nodes τ₁,τ₂,…,τ_n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ_ν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.     

Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series

We study the rate of uniform approximation to continuous functions ƒ(x, y), 2π-periodic in each variable, in Lipschitz classes Lip(α, β) and in Zygmund classes Z(α, β), 0 < α, β 1, by Cesàro means σ_mn^γδ(ƒ) of positive orders of the rectangular partial sums of double Fourier series. The rate of uniform approximation to the conjugate functions ^1,0, ^0,1 and ^1,1 by the corresponding Cesàro means is also discussed in detail. The difference between the classes Lip(α, β) and Z(α, β), similar to the one-dimensional case, appears again when max(α, β) = 1. (Compare Theorems 2 and 3 with Theorems 4 and 5.) One surprising result is the following: The uniform approximation rate by σ_mn^{γδ 1,0} to ^1,0 is worse in general than that by σ_mn^{γδ 1,1} to ^1,1 for ƒ ε Lip(1, 1). In fact, the appearance of an extra factor [log(n + 2)]² in the former case is unavoidable (see Theorem 6). All approximation rates we obtain, with one exception, are shown to be exact. Two conjectures are also included.     

Numerical Schottky Uniformizations

A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut₊(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut₊(S,τ)| ≥ 12(g−1). We say that (S,τ) is maximally symmetric if |Aut₊(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.Partially supported by Projects Fondecyt 1030252 1030373 and UTFSM 12.03.21     

Nonnegative solutions of a nonlinear recurrence

Orthonormal polynomials with weight ¦τ¦ exp(−τ⁴) have leading coefficients with recurrence properties which motivate the more general equations ξ_m(ξ_{m − 1} + ξ_m + ξ_{m + 1}) = γ_m², M = 1, 2,…, where ξ_o is a fixed nonnegative value and γ₁, γ₂,… are positive constants. For this broader problem, the existence of a nonnegative solution is proved and criteria are found for its uniqueness. Then, for the motivating problem, an asymptotic expansion of its unique nonnegative solution is obtained and a fast computational algorithm, with error estimates, is given.