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## Banach Space (banach + space)
## Selected Abstracts## ON THE STRONG LAW OF LARGE NUMBERS UNDER REARRANGEMENTS FOR SEQUENCES OF BLOCKWISE ORTHOGONAL RANDOM ELEMENTS IN BANACH SPACES AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 4 2007Nguyen Van QuangSummary The condition of the strong law of large numbers is obtained for sequences of random elements in type p Banach spaces that are blockwise orthogonal. The current work extends a result of Chobanyan & Mandrekar (2000)[On Kolmogorov SLLN under rearrangements for orthogonal random variables in a B -space. J. Theoret. Probab. 13, 135,139.] Special cases of the main results are presented as corollaries, and illustrative examples are provided. [source] ## Stability and linearization: discrete-time systems INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, Issue 5 2002Irwin W. SandbergAbstract A theorem by Hadamard gives a two-part condition under which a map from one Banach space to another is a homeomorphism. The theorem, while often very useful, is incomplete in the sense that it does not explicitly specify the family of maps for which the condition is met. Recently, under a typically weak additional assumption on the map, it was shown that Hadamard's condition is met if and only if the map is a homeomorphism with a Lipschitz continuous inverse. Here an application is given concerning the relation between the stability of a discrete-time non-linear system and the stability of related linear systems. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## Linear functionals on nonlinear spaces and applications to problems from viscoplasticity theory MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2008Waldemar PompeAbstract A classical result in the theory of monotone operators states that if C is a reflexive Banach space, and an operator A: C,C* is monotone, semicontinuous and coercive, then A is surjective. In this paper, we define the ,dual space' C* of a convex, usually not linear, subset C of some Banach space X (in general, we will have C*,X*) and prove an analogous result. Then, we give an application to problems from viscoplasticity theory, where the natural space to look for solutions is not linear. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Starlike and convex rational mappings on infinite dimensional domains MATHEMATISCHE NACHRICHTEN, Issue 2 2009Cho-Ho ChuAbstract We give starlike criteria for a class of rational mappings on the open unit ball of a complex Banach space. We also give a sufficient condition for these mappings to be convex when they are defined in Hilbert spaces. These criteria facilitate the construction of concrete examples of starlike and convex mappings on infinite dimensional domains (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Notes on the geometry of the space of polynomials MATHEMATISCHE NACHRICHTEN, Issue 16 2007Han Ju LeeAbstract We show that the symmetric injective tensor product space is not complex strictly convex if E is a complex Banach space of dim E , 2 and if n , 2 holds. It is also reproved that ,, is finitely represented in if E is infinite-dimensional and if n , 2 holds, which was proved in the other way in [3]. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Evolution operators generated by non-densely defined operators MATHEMATISCHE NACHRICHTEN, Issue 11 2005Hirokazu OkaAbstract In this paper it is shown that an evolution operator is generated by a family of closed linear operators whose common domain is not necessarily dense in the underlying Banach space, under the stability condition proposed by the second author from the viewpoint of finite difference approximations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces MATHEMATISCHE NACHRICHTEN, Issue 10 2005Ljubomir B.Abstract The concept of the operators of generalized monotone type is introduced and iterative approximation methods for a fixed point of such operators by the Ishikawa and Mann iteration schemes {xn} and {yn} with errors is studied. Let X be a real Banach space and T : D , X , 2D be a multi-valued operator of generalized monotone type with fixed points. A new general lemma on the convergence of real sequences is proved and used to show that {xn} converges strongly to a unique fixed point of T in D. This result is applied to the iterative approximation method for solutions of nonlinear equations with generalized strongly accretive operators. Our results generalize many of know results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Measurable selectors for the metric projection MATHEMATISCHE NACHRICHTEN, Issue 1 2003B. CascalesAbstract Let Y and Z be two topological spaces and F : Y × Z , , a function that is upper semi,continuous in the first variable and lower semi,continuous in the second variable. If Z is Polish and for every y , Y there is a point z , Z with F(y, z) = infw,ZF(y, w) we prove that there is a nice measurable function h : Y , Z satisfying F(y, h(y)) = infz,ZF(y, z) for every y , Y . As an application we obtain the existence of universally measurable selectors for the metric projection onto weakly K,analytic convex proximinal subsets of a Banach space, which then allows us to prove that Lp(,, Y ) is proximinal in Lp(,, X) for every proximinal weakly K,analytic subspace Y of a Banach space X. [source] ## On some nonself mappings MATHEMATISCHE NACHRICHTEN, Issue 1 2003LJ. B.Abstract Let X be a Banach space, let K be a non,empty closed subset of X and let T : K , X be a non,self mapping. The main result of this paper is that if T satisfies the contractive,type condition (1.1) below and maps ,K (,K the boundary of K) into K then T has a unique fixed point in K. [source] ## Operator,valued Fourier multiplier theorems on Besov spaces MATHEMATISCHE NACHRICHTEN, Issue 1 2003Maria GirardiPresented is a general Fourier multiplier theorem for operator,valued multiplier functions on vector,valued Besov spaces where the required smoothness of the multiplier functions depends on the geometry of the underlying Banach space (specifically, its Fourier type). The main result covers many classical multiplier conditions, such as Mihlin and Hörmander conditions. [source] ## Computability of compact operators on computable Banach spaces with bases MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4-5 2007Vasco BrattkaAbstract We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non-constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable Schauder bases that are well-behaved. The conditions imposed on the bases are such that they generalize the Hilbert space case. In particular, we prove that the space of compact operators on Banach spaces with monotone, computably shrinking, and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable. Moreover, we provide a computable version of the Theorem of Schauder on adjoints in this framework and we discuss a non-uniform result on composition with bounded linear operators from right. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A review on the use of the adjoint method in four-dimensional atmospheric-chemistry data assimilation THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 576 2001K.-Y. WangAbstract In this paper we review a theoretical formulation of the adjoint method to be used in four-dimensional (4D) chemistry data assimilation. The goal of the chemistry data assimilation is to combine an atmospheric-chemistry model and actual observations to produce the best estimate of the chemistry of the atmosphere. The observational dataset collected during the past decades is an unprecedented expansion of our knowledge of the atmosphere. The exploitation of these data is the best way to advance our understanding of atmospheric chemistry, and to develop chemistry models for chemistry-climate prediction. The assimilation focuses on estimating the state of the chemistry in a chemically and dynamically consistent manner (if the model allows online interactions between chemistry and dynamics). In so doing, we can: produce simultaneous and chemically consistent estimates of all species (including model parameters), observed and unobserved; fill in data voids; test the photochemical theories used in the chemistry models. In this paper, the Hilbert space is first formulated from the geometric structure of the Banach space, followed by the development of the adjoint operator in Hilbert space. The principle of the adjoint method is described, followed by two examples which show the relationship of the gradient of the cost function with respect to the output vector and the gradient of the cost function with respect to the input vector. Applications to chemistry data assimilation are presented for both continuous and discrete cases. The 4D data variational adjoint method is then tested in the assimilation of stratospheric chemistry using a simple catalytic ozone-destruction mechanism, and the test results indicate that the performance of the assimilation method is good. [source] ## On some subsets of Schechter's essential spectrum of a matrix operator and application to transport operator MATHEMATISCHE NACHRICHTEN, Issue 9 2010Naouel Ben AliAbstract This paper is devoted to the investigation of the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a product of Banach spaces. The obtained results are applied to a two-group transport operators with general boundary conditions in the Banach space Lp ([,a, a ] × [,1, 1]) × Lp ([,a, a ] × [,1, 1]), a > 0, p , 1 (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A limit theorem for randomly stopped independent increment processes on separable metrizable groups MATHEMATISCHE NACHRICHTEN, Issue 15 2007Kern, Peter BeckerAbstract In the spirit of the classical random central limit theorem a general limit theorem for random stopping in the scheme of infinitesimal triangular arrays on a separable metrizable group is presented. The approach incorporates and generalizes earlier results for normalized sequences of independent random variables on both separable Banach spaces and simply connected nilpotent Lie groups originated by Siegel and Hazod, respectively. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On factorization of Schatten class type mappings MATHEMATISCHE NACHRICHTEN, Issue 9-10 2006Cristiane de Andrade MendesAbstract We present some results on factorization of multilinear mappings and polynomials of Schatten class type ,,2 through infinite dimensional Banach spaces, ,1 and ,, spaces. We conclude this work with a factorization result for holomorphic mappings of Schatten class type ,,2. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Vector valued Fourier analysis on unimodular groups MATHEMATISCHE NACHRICHTEN, Issue 8 2006Hun Hee LeeAbstract The notion of Fourier type and cotype of linear maps between operator spaces with respect to certain unimodular (possibly nonabelian and noncompact) group is defined here. We develop analogous theory compared to Fourier types with respect to locally compact abelian groups of operators between Banach spaces. We consider the Heisenberg group as an example of nonabelian and noncompact groups and prove that Fourier type and cotype with respect to the Heisenberg group implies Fourier type with respect to classical abelian groups. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Spectral problems for operator matrices MATHEMATISCHE NACHRICHTEN, Issue 12-13 2005A. BátkaiAbstract We study spectral properties of 2 × 2 block operator matrices whose entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components. We investigate closability in the product space, essential spectra and generation of holomorphic semigroups. Application is given to several models governed by ordinary and partial differential equations, for example containing delays, floating singularities or eigenvalue dependent boundary conditions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces MATHEMATISCHE NACHRICHTEN, Issue 10 2005Ljubomir B.Abstract The concept of the operators of generalized monotone type is introduced and iterative approximation methods for a fixed point of such operators by the Ishikawa and Mann iteration schemes {xn} and {yn} with errors is studied. Let X be a real Banach space and T : D , X , 2D be a multi-valued operator of generalized monotone type with fixed points. A new general lemma on the convergence of real sequences is proved and used to show that {xn} converges strongly to a unique fixed point of T in D. This result is applied to the iterative approximation method for solutions of nonlinear equations with generalized strongly accretive operators. Our results generalize many of know results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Computability of compact operators on computable Banach spaces with bases MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4-5 2007Vasco BrattkaAbstract We develop some parts of the theory of compact operators from the point of view of computable analysis. While computable compact operators on Hilbert spaces are easy to understand, it turns out that these operators on Banach spaces are harder to handle. Classically, the theory of compact operators on Banach spaces is developed with the help of the non-constructive tool of sequential compactness. We demonstrate that a substantial amount of this theory can be developed computably on Banach spaces with computable Schauder bases that are well-behaved. The conditions imposed on the bases are such that they generalize the Hilbert space case. In particular, we prove that the space of compact operators on Banach spaces with monotone, computably shrinking, and computable bases is a computable Banach space itself and operations such as composition with bounded linear operators from left are computable. Moreover, we provide a computable version of the Theorem of Schauder on adjoints in this framework and we discuss a non-uniform result on composition with bounded linear operators from right. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## ON THE STRONG LAW OF LARGE NUMBERS UNDER REARRANGEMENTS FOR SEQUENCES OF BLOCKWISE ORTHOGONAL RANDOM ELEMENTS IN BANACH SPACES AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 4 2007Nguyen Van QuangSummary The condition of the strong law of large numbers is obtained for sequences of random elements in type p Banach spaces that are blockwise orthogonal. The current work extends a result of Chobanyan & Mandrekar (2000)[On Kolmogorov SLLN under rearrangements for orthogonal random variables in a B -space. J. Theoret. Probab. 13, 135,139.] Special cases of the main results are presented as corollaries, and illustrative examples are provided. [source] |