Spectrally accurate energy‐preserving methods for the numerical solution of the “good” Boussinesq equation

In this paper we study the geometric numerical solution of the so called “good” Boussinesq equation. This goal is achieved by using a convenient space semi‐discretization, able to preserve the corresponding Hamiltonian structure, then using energy‐conserving Runge–Kutta methods in the Hamiltonian boundary value method class for the time integration. Numerical tests are reported, confirming the effectiveness of the proposed method.     

“STRAGEDY” OF THE COMMONS

This paper computes open loop and subgame perfect Nash equilibria for an infinite horizon, common property resource model with congestion and stock externalities. The model permits the comparison of the game-theoretic approach and the traditional commons literature, which preceded the widespread recognition of the games, because the underlying assumptions are equivalent. With access to the commons restricted, the subgame perfect equilibrium captures the inefficiency associated with the strategic scramble to capture the resource reserves the open loop does not. Under sole ownership, the two equilibrium concepts coincide with the surplus maximizing extraction policy. In free access, the extraction strategies under both equilibrium concepts coincide with complete rent dissipation.     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Admissible perturbations of α‐ψ‐pseudocontractive operators: convergence theorems

We prove some convergence theorems for α‐ψ‐pseudocontractive operators in real Hilbert spaces, by using the concept of admissible perturbation. Our results extend and complement some theorems in the existing literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Characterization of g∞,σ‐integral operators

The application of the general tensor norms theory of Defant and Floret to the ideal of (p, σ)‐absolutely continuous operators of Matter, 0 < σ < 1, 1 ≤ p < ∞ leads to the study of g_p′,σ‐nuclear and g_p′,σ‐integral operators. Characterizations of such operators has been obtained previously in the case p > 1. In this paper we characterize the g_∞,σ‐nuclear and g_∞,σ‐integral operators by the existence of factorizations of some special kinds. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A? ?_b,cl (χ,^bΩ^1/2) (cf. [22], [23]) of classical, totally characteristic pseudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spectrally invariant in any C* – algebra. On the other hand, the symbolic structure of A can be extended continuously to the C* – algebra B generated by A as a subalgebra of ζ(σ^bL²(χ, ^bΩ^1/2)) by a generalization of a method of Gohberg and Krupnik. Furthermore, A is densely embedded in a Fréchet algebra A ? B which is a ?* – algebra in the sense of Gramsch [9, Definition 5.1], reflecting also smooth properties of the original algebra A.     

Estimates for n-widths of the Hardy-type operators (Addendum to “Improved estimates for the approximation numbers of the Hardy-type operators”)

Consider the Hardy-type operator T : L^p(a,b)→L^p(a,b),-∞a<b∞, which is defined by

It is shown that

where ρ_n(T) stands for any of the following: the Kolmogorov n-width, the Gel’fand n-width, the Bernstein n-width or the nth approximation number of T.     

A note on ℝ‐linear GMRES for solving a class of ℝ‐linear systems

We prove that ?‐linear GMRES for solving a class of ?‐linear systems is faster than GMRES applied to the related ?‐linear systems in terms of matrix–vector products. Numerical examples are given to demonstrate the theoretical result. Copyright © 2011 John Wiley & Sons, Ltd.     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

Holomorphic liftings and Bergman kernel estimates for 𝒟ℱ𝒩‐domains

Let E be a 𝒟ℱ𝒩‐space and let U ⊂ E be open. By applying the nuclearity of the Fréchet space ℋ︁(U) of holomorphic functions on U we show that there are finite measures μ on U leading to Bergman spaces of μ ‐square integrable holomorphic functions. We give an explicit construction for μ by using infinite dimensional Gaussian measures. Moreover, we prove boundary estimates for the corresponding Bergman kernels K_μ on the diagonal and we give an application of our results to liftings of μ ‐square integrable Banach space valued holomorphic functions over U. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Embedding Subspaces of Lp into 𝓁Np, 0 < p < 1

It is shown that if X is an n – dimensional subspace of L_p, 0 < p < 1, then there exists a subspace Y of 𝓁^N_p such that d(X, Y) ≤ 1 + ε and N ≤ C(ε, p)n(log n)³.     

Uniqueness of the “nonsingular core” for Hermitian and other matrix pencils

Every pencil of hermitian matrices is conjunctive with a pencil of the form L ⊕ M, where L (the "minimal-indices" part) has no elementary divisors and M (the "nonsingular core") is a nonsingular pencil. Here it is shown that the conjunctivity type ofM is determined by that of L ⊕ M. The same method of proof applies to many other types of pencils, e.g. to congruence of pencils based on (1) a pair of symmetric matrices, (2) a pair of alternating matrices, or (3) a symmetric and an alternating matrix.     

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

In the present article, Kantorovich variant of λ‐Bernstein operators with shifted knots are introduced. The advantage of using shifted knot is that one can do approximation on [0,1] as well as on its subinterval. In addition, it adds flexibility to operators for approximation. Some basic results for approximation as well as rate of convergence of the introduced operators are established. The r^th order generalization of the operator is also discussed. Further for comparisons, some graphics and error estimation tables are presented using MATLAB.     

The wi‐club filter on 𝒫κ λ

We develop the theory of C_{κ, λ}ⁱ, a strongly normal filter over ??_κ λ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ ??_κ λ: |x | = |x ∩ κ | and |x | is inaccessible} also contains C_{κ, λ}ⁱ. We also show that functions can be used to obtain a basis for C_{κ, λ}ⁱ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bursting and synchronization transition in the coupled modified ML neurons

Bursting is an important electrical behavior in neuron’s firing. In this paper, based on the fast/slow dynamical bifurcation analysis and the phase plane analysis, two types of bursting are presented in the modified Morris–Lecar neuronal model, and the associated bifurcation mechanisms of switching between the active phase and the silent phase are analyzed. For two coupled bursters, it is found that the same type of coupled bursters may have different synchronization transition path from that of two different types of coupled bursters. The analysis of bursting types and the transition to synchronization may provide us with better insight into neuronal encoding and information transmission.     

On the self‐dual 𝔽5‐codes constructed from Hadamard matrices of order 24

There are exactly 60 inequivalent Hadamard matrices of order 24. In this note, we give a classification of the self‐dual ??₅‐codes of length 48 constructed from the Hadamard matrices of order 24. © 2004 Wiley Periodicals, Inc.     

Discrete operators of sampling type and approximation in ‐variation

The paper deals with approximation results with respect to the φ‐variation by means of a family of discrete operators for φ‐absolutely continuous functions. In particular, for the considered family of operators and for the error of approximation, we first obtain some estimates which are important in order to prove the main result of convergence in φ‐variation. The problem of the rate of approximation is also studied. The discrete operators that we consider are deeply connected to some problems of linear prediction from samples in the past, and therefore have important applications in several fields, such as, for example, in speech processing. Moreover such family of operators coincides, in a particular case, with the generalized sampling‐type series on a subset of the space of the φ‐absolutely continuous functions: therefore we are able to obtain a result of convergence in variation also for the generalized sampling‐type series. Some examples are also discussed.