Spanning trees on hypercubic lattices and nonorientable surfaces

We consider the problem of enumerating spanning trees on lattices. Closed-form expressions are obtained for the spanning tree generating function for a hypercubic lattice in d dimensions under free, periodic, and a combination of free and periodic boundary conditions. Results are also obtained for a simple quartic net embedded on two nonorientable surfaces, a Möbius strip and the Klein bottle. Our results are based on the use of a formula expressing the spanning tree generating function in terms of the eigenvalues of an associated tree matrix. An elementary derivation of this formula is given.     

Cutting the Same Fraction of Several Measures

We study some measure partition problems: Cut the same positive fraction of d+1 measures in ? ^d with a hyperplane or find a convex subset of ? ^d on which d+1 given measures have the same prescribed value. For both problems positive answers are given under some additional assumptions.     

Random perturbations of 2-dimensional hamiltonian flows

We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long time behavior corresponds to the behavior of the homogenized process - that is diffusion process with the constant diffusion matrix (effective diffusivity). We obtain the asymptotics of the effective diffusivity when the molecular diffusion tends to zero.     

The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds

In this paper, we shall estimate the growing speed for higher Sobolev norms of the solutions to Schrödinger equations on Riemannian manifolds (d?2), under some bilinear Strichartz estimate assumptions.     

Subdirect sums of P(P0)-matrices and totally nonnegative matrices

In this paper, the problem of when the sub-direct sum of two strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix is studied. In particular, it is shown that the subdirect sum of overlapping principal submatrices of strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix. It is also established that the 2-subdirect sum of two totally nonnegative matrices is a totally nonnegative matrix under some conditions. It is obtained that a partial totally nonnegative matrix, whose graph of the specified entries is a monotonically labeled 2-chordal graph, has a totally nonnegative completion. Finally, a positive answer to the question (IV) in Fallat and Johnson [Shaun M. Fallat, C.R. Johnson, J.R. Torregrosa, A.M. Urbano, P-matrix completions under weak symmetry assumptions, Linear Algebra Appl. 312 (2000) 73-91] is given for P₀-matrices.     

Global exponential stability and existence of periodic solutions in BAM networks with delays and reaction–diffusion terms

Both exponential stability and periodic solutions are considered for a class of bi-directional associative memory (BAM) neural networks with delays and reaction–diffusion terms by constructing suitable Lyapunov functional and some analysis techniques. The general sufficient conditions are given ensuring the global exponential stability and existence of periodic solutions of BAM neural networks with delays and reaction–diffusion terms. These presented conditions are in terms of system parameters and have important leading significance in the design and applications of globally exponentially stable and periodic oscillatory neural circuits for BAM with delays and reaction–diffusion terms.     

Qualitative analysis of a delayed free boundary problem for tumor growth under the effect of inhibitors

In this paper we study a delayed free boundary problem for the growth of tumors under the effect of inhibitors. The establishing of the model is based on the diffusion of nutrient and inhibitors, and mass conservation for the two processes proliferation and apoptosis. It is assumed that the process of proliferation is delayed compared to apoptosis. We mainly study the asymptotic behavior of the solution, and prove that under some assumptions, in the case where c₁ and c₂ are sufficiently small, the volume of the tumor cannot expand without limit; it will either disappear or evolve to a dormant state as t→∞.     

Dynamic behaviors of a delay differential equation model of plankton allelopathy

In this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results.     

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator H=(?Δ) ^m +B, m>0, in ? ^d which satisfies the following conditions: (i) the symbol of B is smooth in x, and (ii) the perturbation B has order less than 2m. Under these assumptions, we prove that the spectrum of H contains a half-line. This, in particular implies the Bethe-Sommerfeld conjecture for the Schrödinger operator with a periodic magnetic potential in all dimensions.     

Periodic solutions for Hamiltonian systems under strong constraining forces

We consider periodic solutions of Hamiltonian systems in Euclidean spaces whose motion is constrained to a submanifold M. We prove that under some nondegeneracy assumptions, periodic solutions persist when the constraint is replaced by a strong restoring potential.     

On Sobolev solutions of poisson equations in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> with a parameter

Smoothness with respect to a parameter is established under mild assumptions on the regularity of coefficients for Sobolev solutions of the Poisson equations in the whole ℝ ^d in the “ergodic case.” An assertion of this kind serves as one of the key tools in diffusion approximation and some other limit theorems. Bibliography: 12 titles.     

Asymptotic behavior of trajectories for some nonautonomous,almost periodic processes

In this paper, asymptotics are studied for some almost periodic processes on a complete metric space (X, d): (1) It is shown that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t → +∞. This property does not hold for general almost periodic contractive processes. (2) A compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. (3) The existence of almost periodic trajectories is studied for “affine” processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form $\text{du}\text{dt}\text{+ A (t) u(t) ? f(t)}$, where $\text{?(t)}$ is almost periodic: R → V uniformly convex Banach space and A(t) is a periodic, time-dependent, m-accretive operator in V.     

Breaking the rhythm on graphs

We study graph colorings avoiding periodic sequences with large number of blocks on paths. The main problem is to decide, for a given class of graphs F, if there are absolute constants t,k such that any graph from the class has a t-coloring with no k identical blocks in a row appearing on a path. The minimum t for which there is some k with this property is called the rhythm threshold of F, denoted by t(F). For instance, we show that the rhythm threshold of graphs of maximum degree at most d is between (d+1)/2 and d+1. We give several general conditions for finiteness of t(F), as well as some connections to existing chromatic parameters. The question whether the rhythm threshold is finite for planar graphs remains open.     

Exponential stability of delayed fuzzy cellular neural networks with diffusion

The exponential stability of delayed fuzzy cellular neural networks (FCNN) with diffusion is investigated. Exponential stability, significant for applications of neural networks, is obtained under conditions that are easily verified by a new approach. Earlier results on the exponential stability of FCNN with time-dependent delay, a special case of the model studied in this paper, are improved without using the time-varying term condition: dτ(t)/dt < μ.     

Nonnegative matrices having nonnegative Drazin pseudoinverses

Necessary and sufficient conditions for nonnegative matrices having nonnegative Drazin pseudoinverses are obtained. A decomposition theorem which characterizes the class of all nonnegative matrices with nonnegative Drazin pseudoinverses is proved, thus answering a question raised by several people. It is also shown that if a row (or column) stochastic matrix has a nonnegative Drazin pseudoinverse A(^d), then A(^d) is some power of A. These results extend known results for nonnegative group-monotone matrices.     

Tractability of Multivariate Integration for Periodic Functions

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε⁻¹ and d. Strong tractability means that we have only a polynomial dependence in ε⁻¹. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γ_j moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑^∞_j=1 γ_j<∞, whereas tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/ln d<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.     

TT-cross approximation for multidimensional arrays

As is well known, a rank-r matrix can be recovered from a cross of r linearly independent columns and rows, and an arbitrary matrix can be interpolated on the cross entries. Other entries by this cross or pseudo-skeleton approximation are given with errors depending on the closeness of the matrix to a rank-r matrix and as well on the choice of cross. In this paper we extend this construction to d-dimensional arrays (tensors) and suggest a new interpolation formula in which a d-dimensional array is interpolated on the entries of some TT-cross (tensor train-cross). The total number of entries and the complexity of our interpolation algorithm depend on d linearly, so the approach does not suffer from the curse of dimensionality.We also propose a TT-cross method for computation of d-dimensional integrals and apply it to some examples with dimensionality in the range from d=100 up to d=4000 and the relative accuracy of order 10^-10. In all constructions we capitalize on the new tensor decomposition in the form of tensor trains (TT-decomposition).     

The dynamics of a mutual interference age structured predator-prey model with time delay and impulsive perturbations on predators

In this paper, we introduce a mutual interference age structured predator-prey (natural enemy-pest) model with constant maturation time delay for the prey, and then propose a pest management strategy by constant periodic releasing for the predator. We show that there exists a global attractive pest-eradication periodic solution when the periodic releasing amount μ₁ and μ₂ are lager than some critical value. Further, to obtain a more effective pest control strategy, we give the conditions (involving the estimate of μ₁ and μ₂) in which the model is uniformly permanent and the pest population is under the economic threshold level. We believe that the results will provide reliable tactic basis for the practical pest management.     

Propagation of spiral waves in anisotropic media: from waves to stripes

The effects of anisotropic diffusion on the propagation of spiral waves in excitable media are studied numerically by means of time-linearized methods in very refined meshes and with very small time steps. It is shown that the anisotropy of the inhibitor's diffusivity tensor does not play as important role on wave propagation as that of the activator. It is also shown that the off-diagonal components of the activator's diffusivity tensor cause stretching of the activator's concentration along the principal directions of this tensor, and that a large difference between the diffusion coefficients in the x- and y-directions may result in the spiral wave annihilation and stripe formation.     

Cutoff for the Ising model on the lattice

Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L ¹ on a system of size n is O(logn). Whether in this regime there is cutoff, i.e. a sharp transition in the L ¹-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For ?² this carries all the way to the critical temperature. Specifically, for fixed d≥1, the continuous-time Glauber dynamics for the Ising model on (?/n?) ^d with periodic boundary conditions has cutoff at (d/2λ _∞)logn, where λ _∞ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating L ¹-mixing to L ²-mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e.g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.