Stability of IMEX Runge–Kutta methods for delay differential equations

Stability of IMEX (implicit–explicit) Runge–Kutta methods applied to delay differential equations (DDEs) is studied on the basis of the scalar test equation du/dt=λu(t)+μu(t-τ)$\mathrm{d}u/\mathrm{d}t=\lambda u(t)+\mu u(t-\tau )$, where τ$\tau$ is a constant delay and λ,μ$\lambda ,\mu$ are complex parameters. More specifically, P-stability regions of the methods are defined and analyzed in the same way as in the case of the standard Runge–Kutta methods. A new IMEX method which possesses a superior stability property for DDEs is proposed. Some numerical examples which confirm the results of our analysis are presented.     

Finite Element implementation for the EULER-EYTELWEIN-problem and further use in FEM-simulation of common nautical knots

The computation of ropes and cable constructions considering contact as well as friction is often required in engineering practice. That includes mutual frictional contact between two and more ropes and frictional contact between ropes and solid –mostly assumed as rigid– bodies. The latter occurs especially in the mechanical model of a rope wound around a cylinder. So the main topic of this contribution is the development and FE-implementation of a "Segment- To-Analytical-Surface (STAS)"-contact-element which can be used to describe the mechanical model of a rope wound around a cylinder. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of nonnegative symmetric matrices with given spectrum

Let σ = (λ₁, … , λ_n) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ₁, a diagonal entry c and let τ = (μ₁, … , μ_m) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ₁. We show how to construct a nonnegative symmetric matrix C with the spectrum

(λ₁+max{0,μ₁-c},λ₂,…,λ_n,μ₂,…,μ_m).     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

On the critical dimension of a fourth order elliptic problem with negative exponent

We study the regularity of the extremal solution of the semilinear biharmonic equation on a ball B⊂RN, under Navier boundary conditions u=Δu=0 on ∂B, where λ>0 is a parameter, while τ>0, β>0 are fixed constants. It is known that there exists λ^∗ such that for λ>λ^∗ there is no solution while for λ<λ^∗ there is a branch of minimal solutions. Our main result asserts that the extremal solution u^∗ is regular (supBu^∗<1) for N?8 and β,τ>0 and it is singular (supBu^∗=1) for N?9, β>0, and τ>0 with small. Our proof for the singularity of extremal solutions in dimensions N?9 is based on certain improved Hardy-Rellich inequalities.     

Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Let LGn denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2n. For each strict partition λ=(λ₁,…,λk) with λ₁?n there is a Schubert variety X(λ). Let T denote a maximal torus of the symplectic group acting on LGn. Consider the T-equivariant cohomology of LGn and the T-equivariant fundamental class σ(λ) of X(λ). The main result of the present paper is an explicit formula for the restriction of the class σ(λ) to any torus fixed point. The formula is written in terms of factorial analogue of the Schur Q-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for LGn. As another consequence of the main result, we obtained a presentation of the ring .     

Permutation resolutions for Specht modules

For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M ^μ for the symmetric group \(\mathfrak{S}_{r}\) with Young subgroup stabilizers \(\mathfrak{S}_{\mu}\). The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S ^λ . We prove the exactness in special cases.     

Weakly nonlinear Schrödinger equation with random initial data

It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ _t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈? ^d and t∈?. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ _t (x) has a limit as λ→0 for t=λ ^?2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.     

Bifurcation formulae derived from center manifold theory

Applicable formulae for the parameters μ₂, β₂, τ₂, μ₄, β₄ and τ₄ of N-dimensional Hopf bifurcation theory are presented. The center manifold theorem is used to reduce the system from N dimensions to 2 dimensions. Approximate solution of the system in Poincaré normal form provides the formulae. The formulae are explicit so that the parameters may be computed directly from partial derivatives of the system in real canonical form. The formula for μ₂ is shown to be identical to that of I. D. Hsu and N. D. Kazarinoff. Formulae for the parameters μ₁, μ₂, μ₃, τ₁, τ₂, τ₃, β₂, β₃ and β₄ in the “tangency” case Re λ₁′(v_c) = 0, Re λ₁′'(v_c) ≠ 0 are also presented.     

Sums of commutators in non-commutative Banach function spaces

Let M be a II_∞-factor and denote by τ its normal faithful semi-finite trace. For any rearrangement invariant Köthe function space X on [0,+∞[, let X(M,τ) be the associated non-commutative Banach function space. This paper is concerned with ideals in M of the form I_X(M,τ)=M∩X(M,τ) that are contained in L^p(M,τ) for some p>0. It is proved that an element T in I_X(M,τ) is a finite sum of commutators of the form [A,B] with A∈I_X(M,τ) and B∈M if and only if the function belongs to X, where ν_T is the Brown spectral measure of T and t→λ_t(T) is the non-increasing rearrangement of the function λ→|λ| with respect to ν_T. This extends to general Banach function spaces a result obtained by Kalton for quasi-Banach ideals of compact operators and implies that the Dixmier's trace of a quasi-nilpotent element in L^1,∞(M,τ) is always zero.     

A counting problem in ergodic theory and extrapolation for one-sided weights

The purpose of this paper is to prove that, given a dynamical system (X,M,μ, τ) and 0 < q < 1, the Lorentz spaces L^1,q(μ) satisfy the so-called Return Times Property for the Tail, contrary to what happens in the case q = 1. In fact, we consider a more general case than in previous papers since we work with a σ-finite measure μ and a transformation τ which is only Cesàro bounded. The proof uses the extrapolation theory of Rubio de Francia for one-sided weights. These results are of independent interest and can be applied to many other situations.     

Algebraic numbers and density modulo 1

Let λ₁,μ₁ and λ₂,μ₂ be two pairs of rationally independent real algebraic numbers of degree 2, with absolute values greater than 1, such that the absolute values of their conjugates are also greater than 1. Under some additional assumptions, on the relation between λi,μi and their conjugates, we prove that for any real numbers ξ₁,ξ₂, with at least one ξi≠0, the set is dense modulo 1.     

Constructive factorization of some almost periodic triangular matrix functions with a quadrinomial off diagonal entry

Using an abbreviation eμ to denote the function eiμx on the real line R, let , where f is a linear combination of the functions eα, eβ, eα−λ, eβ−λ with some . The criterion for G to admit a canonical factorization was established recently by Avdonin, Bulanova and Moran (2007) [1]. We give an alternative approach to the matter, proving the existence (when it does take place) via deriving explicit factorization formulas. The non-existence of the canonical factorization in the remaining cases then follows from the continuity property of the geometric mean.     

Bounded stationary reflection II

Bounded stationary reflection at a cardinal λ is the assertion that every stationary subset of λ reflects but there is a stationary subset of λ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $\mu >{\mathrm{?}}_{\omega }$ and models in which bounded stationary reflection holds at ${\mu }^{+}$ but the approachability property fails at μ.     

Transient solutions of a software model with imperfect debugging and generation of errors by two servers

In this paper, we obtain the transient solution of probabilities of error in the software, mean number of faults and the expected number of failures remaining at time t, under the assumption that the number of faults is finite, the failure rate is proportional to the number of faults present in the software at any time, debugging is imperfect and error generation will never lead the software to have infinite errors. Moreover, the software is tested by two servers with the first M errors being debugged by first server and the remaining errors (M +1 ≤n ≤N) by the second server. Also, when a failure occurs, instantaneously repair starts with the following probabilities.

• 1.(a) The fault content is reduced by one by the first (second) server with probability μ₁(μ₂),μ₂ ≥μ₁
• 2.(b) The fault content remains unchanged with probability Ψ.
• 3.(c) The fault content is increased by one by the first (second) server with probability λ₁(λ₂), λ₁ ≥λ₂ where μ₁ + Ψ + λ₁ = 1, μ₂ + Ψ + λ₂ = 1, μ₁ ⪢ Ψ ⪢ λ₁, μ₂ ⪢ Ψ ⪢ λ₂. Finally, a numerical example is presented for the transient probabilities for the number of errors in the software, mean number of faults and the expected number of failures remaining in the software.

     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

Ramanujan graphs

A large family of explicitk-regular Cayley graphsX is presented. These graphs satisfy a number of extremal combinatorial properties.

1. For eigenvaluesλ ofX eitherλ=±k or ¦λ¦≦2 √k?1. This property is optimal and leads to the best known explicit expander graphs.
2. The girth ofX is asymptotically ≧4/3 log _k?1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions.