On the axiomatic definition of real JB*–triples

In the last twenty years, a theory of real Jordan triples has been developed. In 1994 T. Dang and B. Russo introduced the concept of J*B–triple. These J*B–triples include real C*–algebras and complex JB*–triples. However, concerning J*B–triples, an important problem was left open. Indeed, the question was whether the complexification of a J*B–triple is a complex JB*–triple in some norm extending the original norm. T. Dang and B. Russo solved this problem for commutative J*B–triples. In this paper we characterize those J*B–triples with a unitary element whose complexifications are complex JB*–triples in some norm extending the original one. We actually find a necessary and sufficient new axiom to characterize those J*B–triples with a unitary element which are J*B–algebras in the sense of [1] or real JB*–triples in the sense of [4].     

Perfectness of Conelike *–Semigroups in ℚk

Let S be an abelian *–semigroup in ℚ^k. We give a sufficient condition for every positive definite function on S to have a unique representing measure on the dual semigroup of S (i.e. S is perfect). To characterize perfectness for any abelian *–semigroupis a challenging, but not yet generally solved problem. In this paper, we characterize the structure of involutions on an abelian *–semigroup which is a subset of ℚ^k, and show that any conelike *–semigroups in ℚ^k are perfect.     

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.     

Intuitionistic ϵ- and τ-calculi

There are several open problems in the study of the calculi which result from adding either of Hilbert's ?- or τ-operators to the first order intuitionistic predicate calculus. This paper provides answers to several of them. In particular, the first complete and sound semantics for these calculi are presented, in both a “quasi-extensional” version which uses choice functions in a straightforward way to interpret the ?- or τ-terms, and in a form which does not require extensionality assumptions. Unlike the classical case, the addition of either operator to intuitionistic logic is non-conservative. Several interesting consequences of the addition of each operator are proved. Finally, the independence of several other schemes in either calculus are also proved, making use of the semantics supplied earlier in the paper.     

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

This study presents a robust modification of Chebyshev ? ‐weighted Crank–Nicolson method for analyzing the sub‐diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub‐fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of sub‐diffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.     

Fully discrete A‐ϕ finite element method for Maxwell's equations with nonlinear conductivity

This article is devoted to the study of a fully discrete A ‐ finite element method to solve nonlinear Maxwell's equations based on backward Euler discretization in time and nodal finite elements in space. The nonlinearity is owing to a field‐dependent conductivity with the power‐law form . We design a nonlinear time‐discrete scheme for approximation in suitable function spaces. We show the well‐posedness of the problem, prove the convergence of the semidiscrete scheme based on the boundedness of the second derivative in the dual space and derive its error estimate. The Minty–Browder technique is introduced to obtain the convergence of the nonlinear term. Finally, we discuss the error estimate for the fully discretized problem and support the theoretical result by two numerical experiments. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2083–2108, 2014     

On Interpolation of Vector – Valued Banach Lattices and Calderón – Lozanovskii Construction

We establish the connection between the variants of Schechter's complex interpolation methods, Peetre – Gustavsson's interpolation methods, and the Calderón – Lozanovskii construction on vector – valued Banach lattices. As applications, we show that the uniform convexity and the UMD property are stable by interpolation.     

Stability analysis of higher order nonlinear differential equations in β–normed spaces

In this paper, we interrogate different Ulam type stabilities, ie, β–Ulam–Hyers stability, generalized β–Ulam–Hyers stability, β–Ulam–Hyers–Rassias stability, and generalized β–Ulam–Hyers–Rassias stability, for nth order nonlinear differential equations with integrable impulses of fractional type. The existence and uniqueness of solutions are investigated by using the Banach contraction principle. In the end, we give an example to support our main result.     

Hyers–Ulam–Rassias stability of a generalized Apollonius–Jensen type additive mapping and isomorphisms between C *‐algebras

Let X, Y be Banach modules over a C *‐algebra. We prove the Hyers–Ulam–Rassias stability of the following functional equation in Banach modules over a unital C *‐algebra: It is shown that a mapping f: X → Y satisfies the above functional equation and f (0) = 0 if and only if the mapping f: X → Y is Cauchy additive. As an application, we show that every almost linear bijection h: A → B of a unital C *‐algebra A onto a unital C *‐algebra B is a C *‐algebra isomorphism when h (2^d uy) = h (2^d u) h (y) for all unitaries u ∈ A, all y ∈ A, and all d ∈ Z . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Facial and Inner Ideal Structure of a Real JBW*–Triple

Let B be a real JBW*–triple with predual B_* and canonical hermitification the JBW*–triple A It is shown that the set 𝒰(B)^∼ consisting of the partially ordered set 𝒰(B) of tripotents in B with a greatest element adjoined forms a sub–complete lattice of the complete lattice 𝒰(A)^∼ of tripotents in A with the same greatest element adjoined. The complete lattice 𝒰(B)^∼ is shown to be order isomorphic to the complete lattice ℱ_n(B_*1 of norm–closed faces of the unit ball B_*1 in B_* and anti–order isomorphic to the complete lattice ℱ_w*(B₁) of weak*–closed faces of the unit ball B₁ in B. Consequently, every proper norm–closed face of B_*1 is norm–exposed (by a tripotent) and has the property that it is also a norm–closed face of the closed unit ball in the predual of the hermitification of B. Furthermore, every weak*–closed face of B₁ is weak*–semi–exposed, and, if non–empty, of the form u + B₀(u)₁ where u is a tripotent in B and B₀(u)₁ is the closed unit ball in the zero Peirce space B₀(u) corresponding to u. A structural projection on B is a real linear projection R on B such that, for all elements a and b in B, {Ra b Ra}_B is equal to R{a Rb a}_B. A subspace J of B is said to be an inner ideal if {J B J}_B is contained in J and J is said to be complemented if B is the direct sum of J and the subspace Ker(J) defined to be the set of elements b in B such that, for all elements a in J, {a b a}_B is equal to zero. It is shown that every weak*–closed inner ideal in B is complemented or, equivalently, the range of a structural projection. The results are applied to JBW–algebras, real W*–algebras and certain real Cartan factors.     

Calderón – Zygmund Operators on Weighted Weak Hardy Spaces in Locally Compact Vilenkin Groups

Let G be a bounded locally compact Vilenkin group. We study the atomic decom‐position of weighted weak Hardy space. We also define several Calderón – Zygmund type operators and study their boundedness on, spaces like weighted Hardy spaces, weighted weak Hardy spaces and weighted weak Lebesgue spaces. Sharpness of some of our results is also discussed.     

Norm behaviour of solutions to a parabolic–elliptic system modelling chemotaxis in a domain of ℝ3

In this paper, we study a parabolic–elliptic system defined on a bounded domain of ?³, which comes from a chemotactic model. We first prove the existence and uniqueness of local in time solution to this problem in the Sobolev spaces framework, then we study the norm behaviour of solution, which may help us to determine the blow‐up norm of the maximal solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Spectral Problems for Systems of Differential Equations y′ + A0y = λA1y with λ–Polynomial Boundary Conditions

This paper deals with the spectral properties of boundary eigenvalue problems for systems of first order differential equations with boundary conditions which depend on the spectral parameter polynomially. It is not assumed that is injective or surjective. The main results concern the completeness minimality and Riesz basis properties of the corresponding eigenfunctions and associated functions.     

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

This paper is concerned with computing $\mathrm{??}$‐eigenpairs of symmetric tensors. We first show that computing $\mathrm{??}$‐eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex‐valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method.     

Convexity characteristic of Calderón–Lozanovskiĭ sequence spaces

Hudzik, Kamińska and Masty?o obtained some geometric properties of Calderón–Lozanovski? function spaces which are defined on a nonatomic σ‐measure space in Houston. J. Math. 22 (1996), but left the case of atomic measure unsolved. We studied the relevant problems for the sequence spaces and obtained the following main results:

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Multiplicative Decompositions of Holomorphic Fredholm Functions and ψ*-Algebras

In this article we construct multiplicative decompositions of holomorphic Fredholm operator valued functions on Stein manifolds with values in various algebras of differential and pseudo differential operators which are submultiplicative ψ* - algebras, a concept introduced by the first author. For Fredholm functions T(z) satisfying an obvious topological condition we. Prove (0.1) T(z) = A(z)(I + S(z)), where A(z) is holomorphic and invertible and S(z) is holomorphic with values in an “arbitrarily small” operator ideal. This is a stronger condition on S(z) than in the authors' additive decomposition theorem for meromorphic inverses of holomorphic Fredholm functions [12], where the smallness of S(z) depends on the number of complex variables. The Multiplicative Decomposition theorem (0.1) sharpens the authors' Regularization theorem [11]; in case of the Band algebra L(X) of all bounded linear operators on a Band space, (0.1) has been proved by J. Letterer [20] for one complex variable and by M. 0. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov [26] for the Banach ideal of compact operators.     

Mixing Properties and Central Limit Theorem for a Class of Non-Identical Piecewise Monotonic C2 — Transformations

For a sequence T⁽¹⁾, T⁽²⁾,…of piecewise monotonic C² - transformations of the unit interval I onto itself, we prove exponential ψ- mixing, an almost Markov property and other higher-order mixing properties. Furthermore, we obtain optimal rates of convergence in the central limit Theorem and large deviation relations for the sequence f_k oT(k?1)o…oT⁽¹⁾, k=1, 2, …, provided that the real-valued functions f₁, f₂,…on I are of bounded variation and the corresponding probability measure on I possesses a positive, Lipschitz-continuous Lebesgue density.