scholarly journals Coincidence Results for Summing Multilinear Mappings

2016 ◽  
Vol 59 (4) ◽  
pp. 877-897 ◽  
Author(s):  
Oscar Blasco ◽  
Geraldo Botelho ◽  
Daniel Pellegrino ◽  
Pilar Rueda
Keyword(s):  
Banach Spaces ◽  
A Priori ◽  
New Techniques ◽  
Multilinear Mappings

AbstractIn this paper we prove coincidence results concerning spaces of absolutely summing multilinear mappings between Banach spaces. The nature of these results arises from two distinct approaches: the coincidence of two a priori different classes of summing multilinear mappings, and the summability of all multilinear mappings defined on products of Banach spaces. Optimal generalizations of known results are obtained. We also introduce and explore new techniques in the field: for example, a technique to extend coincidence results for linear, bilinear and even trilinear mappings to general multilinear ones.

scholarly journals Precision vs. Accuracy in Star Catalogs

1995 ◽  
Vol 166 ◽  
pp. 372-372
Author(s):  
L. G. Taff ◽  
J. E. Morrison ◽  
R. L. Smart
Keyword(s):  
Random Error ◽  
A Priori ◽  
Systematic Errors ◽  
A Posteriori ◽  
New Techniques ◽  
Star Catalog ◽  
True Value ◽  
Reduction Methods ◽  
Star Catalogs ◽  
Comparable Amplitude

As better precision is achieved and more sophisticated reduction methods are created previously invisible biases surface. This has been especially true in astrometric Schmidt plate work. The problem of their amelioration is not fully solved and precision per se is meaningless in the presence of poor accuracy of comparable amplitude. Continuing to benignly neglect this issue puts us in the position of standing on only one statistical leg. New techniques have been designed to further minimize systematic errors. Of especial interest to star catalog analysis is the method of infinitely overlapping circles (Taff, Bucciarelli & Lattanzi, ApJ 361, 667, 1990; Taff, Bucciarelli & Lattanzi, ApJ 392, 746 1992; Bucciarelli, Taff & Lattanzi, J. Stat. Comp. and Sim. 48, 29 1993). With it almost complete success has occurred with regard to the removal of systematic errors which creep into compilation catalogs as a result of inadequate treatment of catalog-to-catalog systematic errors; they can essentially be eliminated a priori or a posteriori (Bucciarelli, Lattanzi & Taff, in press in ApJ 1994; Taff & Bucciarelli, in press in ApJ 1994). What infinitely overlapping circles does can be briefly described as follows: Let X (x) be the measured (true) value of a standard coordinate, S(x,y) (ε) be the systematic (random) error in x at this point, let w∞ be the infinitely overlapping circle weight, a be the standard deviation of the random error in x, N be the total number of stars in this circle which has radius R, and x0,y0 be the coordinates of the center of this circle.

scholarly journals $B$-CONVEX OPERATOR SPACES

2003 ◽  
Vol 46 (3) ◽  
pp. 649-668 ◽  
Author(s):  
Javier Parcet
Keyword(s):  
Banach Spaces ◽  
A Priori ◽  
Operator Space ◽  
Subject Classification ◽  
Operator Spaces ◽  
Secondary 42C15 ◽  
Convex Operator ◽  
Primary 46L07

AbstractThe notion of $B$-convexity for operator spaces, which a priori depends on a set of parameters indexed by $\sSi$, is defined. Some of the classical characterizations of this geometric notion for Banach spaces are studied in this new context. For instance, an operator space is $B_{\sSi}$-convex if and only if it has $\sSi$-subtype. The class of uniformly non-$\mathcal{L}^1(\sSi)$ operator spaces, which is also the class of $B_{\sSi}$-convex operator spaces, is introduced. Moreover, an operator space having non-trivial $\sSi$-type is $B_{\sSi}$-convex. However, the converse is false. The row and column operator spaces are nice counterexamples of this fact, since both are Hilbertian. In particular, this result shows that a version of the Maurey–Pisier Theorem does not hold in our context. Some other examples of Hilbertian operator spaces will be considered. In the last part of this paper, the independence of $B_{\sSi}$-convexity with respect to $\sSi$ is studied. This provides some interesting problems, which will be posed.AMS 2000 Mathematics subject classification: Primary 46L07. Secondary 42C15

scholarly journals On symmetric ideals of multilinear mappings between Banach spaces

2006 ◽  
Vol 81 (1) ◽  
pp. 141-148 ◽  
Author(s):  
Geraldo Botelho ◽  
Daniel M. Pellegrino
Keyword(s):  
Banach Spaces ◽  
Operator Ideals ◽  
Multilinear Mappings

AbstractIn this paper we provide examples and counterexamples of symmetric ideals of multilinear mappings between Banach spaces and prove that if I1, …, In are operator ideals, then the ideals of multilinear mappings L(I1, …, In) and /I1, …, In/ are symmetric if and only if I1 = … = In.

scholarly journals The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

2019 ◽  
Vol 27 (4) ◽  
pp. 539-557
Author(s):  
Barbara Kaltenbacher ◽  
Andrej Klassen ◽  
Mario Luiz Previatti de Souza
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Newton Method ◽  
Noise Level ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

Error estimates for the simplified iteratively regularized Gauss–Newton method in Banach spaces under a Morozov-type stopping rule

2018 ◽  
Vol 26 (3) ◽  
pp. 311-333 ◽  
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Banach Spaces ◽  
Error Estimates ◽  
Newton Method ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Parameter Identification Problem ◽  
Ill Posed

AbstractJin Qinian and Min Zhong [10] considered an iteratively regularized Gauss–Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss–Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper.

scholarly journals A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

2018 ◽  
Vol 26 (5) ◽  
pp. 639-646 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Banach Spaces ◽  
Convergence Rates ◽  
A Priori ◽  
Linear Equations ◽  
Rate Function ◽  
Bregman Distances ◽  
Error Measures ◽  
Ill Posed ◽  
General Convex ◽  
Linear Operator Equations

Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

scholarly journals Global existence for abstract nonlinear Volterra–Fredholm functional integrodifferential equation

2012 ◽  
Vol 45 (1) ◽  
Author(s):  
M. B. Dhakne ◽  
Kishor D. Kucche
Keyword(s):  
Global Existence ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
A Priori ◽  
A Priori Bounds ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
Fredholm Functional

AbstractIn the present paper, we investigate the global existence of solutions to initial value problem for nonlinear mixed Volterra–Fredholm functional integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray–Schauder alternative and rely on a priori bounds of solution.

scholarly journals Avances hacia una nueva metodología de trabajo en proyectos arqueológicos: El caso de la Villa Romana de La Ontavia (Terrinches, Ciudad Real)

2011 ◽  
Vol 2 (3) ◽  
pp. 157 ◽  
Author(s):  
Luis Benítez de Lugo Enrich ◽  
Víctor Manuel López-Menchero Bendicho
Keyword(s):  
Cultural Heritage ◽  
A Priori ◽  
Archaeological Remains ◽  
New Techniques ◽  
Archaeological Heritage ◽  
Ciudad Real ◽  
Comprehensive Management ◽  
Roman Villa

<p>The growing international need for new tools to facilitate the comprehensive management of archaeological heritage -ie the coordinated processes of research, conservation and presentation of archaeological remains- opens the door for the use of new techniques in some cases based on the use of ICT and, in other cases, based in the use of elements "a priori" more traditional but equally effective, such as volumetric reconstructions. The project being developed now in the Roman villa of La Ontavia (Terrinches, Ciudad Real) intends to advance the development of new techniques and working methods more in tune with the needs of the comprehensive management of cultural heritage in the XXI century.</p>

scholarly journals Norm Attaining Arens Extensions onℓ1

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Javier Falcó ◽  
Domingo García ◽  
Manuel Maestre ◽  
Pilar Rueda
Keyword(s):  
Banach Spaces ◽  
A Priori ◽  
Multilinear Form ◽  
Multilinear Forms ◽  
Norm Attaining

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form onℓ1with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the property that only some of their Arens extensions attain their norms.

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