The maximal length of real trivectors of rank seven

Trivectors of rank seven over the complex number field have irreducible length less than or equal to four. Over the reals however this is not true. An example of a real trivector of rank seven and irreducible length five is presented. Then, in the notation of Busemann and Glassco, we have N(R, 7, 3) = 5 since for any field F we always have N(F ,7, 3) ≤ 5. This paper provides the first published example where N(F, n, r) ≠ N(K, n, r) for two different fields F and K.     

Linear circulant matricesover a finite field

Rings of polynomials R_N = Z_p[x]/x^N − 1 which are isomorphic to Z_P^N are studied, where p is prime and N is an integer. If I is an ideal in R_N, the code K whose vectors constitute the isomorphic image of I is a linear cyclic code. If I is a principle ideal and K contains only the trivial cycle 0 and one nontrivial cycle of maximal least period N, then the code words of K/ 0 obtained by removing the zero vector can be arranged in an order which constitutes a linear circulant matrix, C. The distribution of the elements of C is such that it forms the cyclic core of a generalized Hadamard matrix over the additive group of Z_Pp. A necessary condition that C = K/ 0 be linear circulant is that for each row vector v of C, the periodic infinite sequence a(v) produced by cycling the elements of v be period invariant under an arbitrary permutation of the elements of the first period. The necessary and sufficient condition that C be linear circulant is that the dual ideal generated by the parity check polynomial h(χ) of K be maximal (a nontrivial, prime ideal of R_N), with N = p^k − 1 and k = deg (h(χ)).     

Invariant record processes and applications to best choice modelling

Let X₁, X₂,…be identically distributed random variables from an unknown continuous distribution. Further let I_r(1), I_r(2),…be a sequence of indicator functions defined on X₁, X₂,…by I_r(k) = 0 if k < r, I_r(k) = 1 if X_k is a r-record AND = 0 otherwise. Suppose that we observe X₁, X₂,… at times T₁ < T₂ <… where the T_k's are realisations of some regular counting process (N(τ)) defined on the positive half-line. Having observed [0, τ], say, the problem is to predict the future behaviour of the counting processes (R_r(τ, s))_sτ = # r-records in [τ, s]. More specifically the objective of this paper is to show that these processes can be (inhomogeneous) Poisson processes even if (N(τ))_τ0 has dependent increments.

The strong link between optimal selection and optimal stopping of record sequences or record processes, perhaps not fully recognized so far, is pointed out in this paper. It is shown to lead to a unification of the treatment of problems which, at first sight, are rather different. Moreover the stopping of record processes in continuous time can lead to rigorous and elegant solutions in cases where dynamic programming is bound to fail. Several examples will be given to facilitate a comparison with other methods.     

Explicit N-fold Darboux transformation and multi-soliton solutions for the (1 + 1)-dimensional higher-order Broer–Kaup system

In this paper, we present a new N-fold Darboux transformations of the (1 + 1)-dimensional higher-order Broer–Kaup (HBK) system with the help of a gauge transformation of the spectral problem. As an application, new explicit (2N − 1)-soliton solutions of the (1 + 1)-dimensional HBK system are obtained. Both the N-fold Darboux transformation and (2N − 1)-soliton solutions can be written explicitly in terms of Vandermonde-like determinants which are remarkable compactness and transparency.     

An improved upper bound for Ramsey number N (3, 3, 3, 3; 2)

The Ramsey number N(3,3,3,3; 2) is the smallest integer n such that each 4-coloring by edges of the complete graph on n vertices contains monochromatic triangles. It is well known that 51 ≤ N(3, 3, 3, 3; 2) ≤ 65. Here we prove that N(3, 3, 3, 3; 2) ≤ 64.     

Harary's conjectures on integral sum graphs

Let N denote the set of positive integers and Z denote all integers. The (integral) sum graph of a finite subset S N(Z) is the graph (S, E) with uv ε E if and only if u + v ε S. A graph G is said to be an (integral) sum graph if it is isomorphic to the (integral) sum graph of some S N(Z). The (integral) sum number of a given graph G is the smallest number of isolated nodes which when added to G result in an (integral) sum graph.

We show that the integral sum number of a complete graph with n 4 nodes equals 2n − 3, which proves a conjecture of Harary. And we disprove another conjecture of Harary by showing that there are infinitely many trees which are not caterpillars but are integral sum graphs.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.     

Cases of equality for the spectral norm submultiplicativity of the hadamard product

In this note, we characterize those pairs of nonzero r-by-d complex matrices that satisfy N₂(A ∘ B) = N₂(A)N₂(B), in which N₂(·) is the spectral norm and · is the Hadamard product.     

Preconditioning for radial basis functions with domain decomposition methods

In our previous work, an effective preconditioning scheme that is based upon constructing least-squares approximation cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. The preconditioner costs O(N²) flops to set up and O(N) storage. The preconditioning technique is sufficiently general that it can be applied to different types of different operators. This was applied to the 2D multiquadric method, with c~1/√N on the Poisson test problem, the preconditioned GMRES converges in tens of iterations. In this paper, we combine the RBF methods and the ACBF preconditioning technique with the domain decomposition method (DDM). We studied different implementations of the ACBF-DDM scheme and provide numerical results for N > 10,000 nodes. We shall demonstrate that the efficiency of the ACBF-DDM scheme improves dramatically as successively finer partitions of the domain are considered.     

Research Problem

We define an arithmetic invariant for the congruence subgroups Γ₀(N), denoted by

b(Γ₀(N)) and pose the problem of finding good asymptotic upper bonds for b(Γ₀(N)) as N approaches X especially when N is prime or the product of two (not necessarily distinct) primes.     

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large reaction term: The almost periodic framework

Homogenization of a stochastic nonlinear reaction–diffusion equation with a large nonlinear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic nonlinear convection–diffusion equation which we explicitly derive in terms of appropriate functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable generalized concept of $\Sigma$-convergence for stochastic processes.     

Multiple bifurcations and periodic “bubbling” in a delay population model

In this paper, the flip bifurcation and periodic doubling bifurcations of a discrete population model without delay influence is firstly studied and the phenomenon of Feigenbaum’s cascade of periodic doublings is also observed. Secondly, we explored the Neimark–Sacker bifurcation in the delay population model (two-dimension discrete dynamical systems) and the unique stable closed invariant curve which bifurcates from the nontrivial fixed point. Finally, a computer-assisted study for the delay population model is also delved into. Our computer simulation shows that the introduction of delay effect in a nonlinear difference equation derived from the logistic map leads to much richer dynamic behavior, such as stable node → stable focus → an lower-dimensional closed invariant curve (quasi-periodic solution, limit cycle) or/and stable periodic solutions → chaotic attractor by cascading bubbles (the combination of potential period doubling and reverse period-doubling) and the sudden change between two different attractors, etc.     

Application of the Averaging Method to the Investigation of Nonlinear Wave Processes in Elastic Systems with Circular Symmetry

We apply asymptotic methods of nonlinear mechanics (the Bogolyubov–Mitropol'skii averaging method) to the construction of approximate solutions of a system of nonlinear equations describing wave processes in elastic systems with circular symmetry. As an example, we study the dynamics of interaction of two flexural waves that propagate in a cylindrical shell under the conditions of free oscillations and periodic excitation.     

Large deviation principle for a stochastic Allen–Cahn equation

The Allen–Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction–diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen–Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber (Stoch Partial Differ Equ Anal Comput 1(1):175–203, 2013). We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder continuous in time, which extends results by Budhiraja et al. (Ann Probab 36(4):1390–1420, 2008). From this result and a continuity argument we deduce a large deviation principle for the Allen–Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.     

Nonlinear dynamics and solitons in the presence of rapidly varying periodic perturbations

Problems of nonlinear dynamics and soliton propagation in the presence of rapidly varying periodic perturbations are considered applying a rigorous analytical approach based on asymptotic expansions. The method we develop allows derivation of an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the parameter ω⁻¹, ω being the frequency of the rapidly varying (direct or parametric) driving force. The general approach is demonstrated on several examples of different physical nature, including chaos suppression in the parametrically driven Duffing oscillator, dynamics of the sine-Gordon kinks in the presence of rapidly varying direct or parametric driving force, propagation of envelope (nonlinear Schrödinger) solitons in optical fibres with periodic amplification, stability of solitons on rapidly varying spatial periodic potential, and so on.     

Dynamics of transport under random fluxes on the boundary

The impact of boundary noise on the dynamical evolution of the scalar transport equation in shear flows is studied, taking off from earlier studies in shear-flow dispersion in internal waves, a mechanism for horizontal mixing in the ocean. In particular, we model a gravity current evolving under an assumed shear-flow. The transport equation is deterministic, with a noise term at the inlet boundary. This was motivated by observed seasonal fluctuations in some known sources of salty, dense water in the oceans, like the Red Sea overflow, as well as observed thermal and saline anomalies in the thermohaline circulation.The noises used were: Wiener white, Wiener colored, Lévy white, and Lévy colored noise. Lévy processes form a more general class of processes which are generally non-Gaussian in distribution, and may have infinitely many jumps in finite time. They have been used to model pollutant point-sources, the flight time of particles in vortices, and linear and nonlinear anomalous diffusion.The major finding was that white noises (Wiener and Lévy ) and colored Wiener noise all have the effect of impeding the diffusion process, by as much as 33%. However, colored Lévy noise (non-Gaussian, time-correlated) does not have this effect on diffusion. This would suggest that time-correlation is more important in distinguishing noises than the distribution of the process that produced the noise. This also explains why Lévy colored noise showed great sensitivity to the stability parameter α, while Lévy white noise is unaffected by its stability parameter.     

The L��vy�CKhintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Ito??s construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy?CKhintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein?CKantorovich metrics. This is a non-trivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.     

一类具有非线性扩散和时滞的捕食系统的持续性与周期解

研究了一类具有非线性扩散和Beddington-Deangelis功能性反应,且同时具有连续时滞和离散时滞的非自治两食饵一捕食者系统,证明了在适当条件下该系统是一致持久的,并且得到了系统正周期解全局渐近稳定的充分条件.最后,给出一个例子以说明得到的结果.     

On the maximum density of 0–1 matrices with no forbidden rectangles

This note provides bounds for the maximal number of ones allowed in an N × N 0–1 matrix, N=2ⁿ, in which there are no ‘forbidden rectangles’ of a special type.