Effective Uniform Bounds from Proofs in Abstract Functional Analysis

In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. `Uniform' here means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, normed linear spaces, uniformly convex spaces as well as inner product spaces.

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Strongly Uniform Bounds from Semi-Constructive Proofs

BRICS Report Series ◽

10.7146/brics.v11i31.21856 ◽

2004 ◽

Vol 11 (31) ◽

Author(s):

Philipp Gerhardy ◽

Ulrich Kohlenbach

Keyword(s):

Functional Analysis ◽

Fixed Point Theory ◽

Classical Case ◽

Point Theory ◽

Prima Facie ◽

Uniform Bounds ◽

Linear Spaces ◽

Compact Spaces ◽

Constructive Proofs ◽

First Time

In 2003, the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie non-constructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to the classical case, there are practically no restrictions on the logical complexity of theorems for which bounds can be extracted. Again, our metatheorems guarantee very general uniformities, even in cases where the existence of uniform bounds is not obtainable by (ineffective) straightforward functional analytic means. Already in the purely intuitionistic case, where the existence of effective bounds is implicit, the metatheorems allow one to derive uniformities that may not be obvious at all from a given constructive proofs. Finally, we illustrate our main metatheorem by an example from metric fixed point theory.

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Clinical Issues: What’s in a Word? A Functional Analysis of Word Learning

Perspectives on Language Learning and Education ◽

10.1044/lle12.3.4 ◽

2005 ◽

Vol 12 (3) ◽

pp. 4-8 ◽

Cited By ~ 7

Author(s):

Prahlad Gupta

Keyword(s):

Functional Analysis ◽

Word Learning ◽

Clinical Issues

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Review for "High‐throughput single‐cell sequencing of paired TCRα and TCRβ genes for the direct expression‐cloning and functional analysis of murine T cell receptors"

10.1002/eji.201848030/v1/review1 ◽

2019 ◽

Keyword(s):

Functional Analysis ◽

T Cell ◽

Single Cell ◽

High Throughput ◽

T Cell Receptors ◽

Expression Cloning ◽

Cell Receptors ◽

Single Cell Sequencing

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Case Formulation and Design of Behavioral Treatment Programs

European Journal of Psychological Assessment ◽

10.1027//1015-5759.19.3.164 ◽

2003 ◽

Vol 19 (3) ◽

pp. 164-174 ◽

Cited By ~ 27

Author(s):

Stephen N. Haynes ◽

Andrew E. Williams

Keyword(s):

Functional Analysis ◽

Behavior Problems ◽

Behavioral Treatment ◽

Clinical Case ◽

Relative Magnitude ◽

Mechanisms Of Action ◽

Treatment Programs ◽

Case Formulation ◽

Functional Relations ◽

Causal Variables

Summary: We review the rationale for behavioral clinical case formulations and emphasize the role of the functional analysis in the design of individualized treatments. Standardized treatments may not be optimally effective for clients who have multiple behavior problems. These problems can affect each other in complex ways and each behavior problem can be influenced by multiple, interacting causal variables. The mechanisms of action of standardized treatments may not always address the most important causal variables for a client's behavior problems. The functional analysis integrates judgments about the client's behavior problems, important causal variables, and functional relations among variables. The functional analysis aids treatment decisions by helping the clinician estimate the relative magnitude of effect of each causal variable on the client's behavior problems, so that the most effective treatments can be selected. The parameters of, and issues associated with, a functional analysis and Functional Analytic Clinical Case Models (FACCM) are illustrated with a clinical case. The task of selecting the best treatment for a client is complicated because treatments differ in their level of specificity and have unequally weighted mechanisms of action. Further, a treatment's mechanism of action is often unknown.

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