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About 10,000 papers for “Function space”

Function space completions

John W. Brace, J. D. Thomison · Illinois Journal of Mathematics · 1978

Many problems in analysis require the introduction of a linear space G of functions. The next step is to equip G with a norm which is appropriate for the specific problem. This in turn leads to the necessity of constructing a complete space ( (Banach space) by adjoining functions to G. We assume that the domain and range spaces remain fixed. Ordinarily the norm of a function in G will be defined in terms of the values the function assumes at points in the domain. If the same statement cannot be made for functions in (, then it is unlikely that G will be of much use. The purpose of this paper is to overcome this difficulty.

Cited by 2Open access

Space and function

Hazel Conway, Rowan Roenisch · 2010

A comprehensive introduction to architecture and architectural history and exceptional in its approach, this book explores architecture as a current practice in relation to history and in relation to the wider context of cultures, conservation and the environment. This new edition brings in the new emphasis on sustainability, urbanism, urban regeneration and cultural identity, in order to take a holistic approach to the subject of architecture. Highly illustrated, this book enables the reader to make sense of the experience of architecture and the built environment by understanding more about the form, construction, meaning and history of the subject.

Cited by 0

Function Space Optimization

Marianna B. Accomando, Bob Chamberlin · Apple Academic Press eBooks · 2020

This chapter examines a revenue stream that isn’t as clear-cut as rooms: Function Space, and the multiple revenue streams associated with banquet/catering events. Technological advances have allowed us to understand that Function Space revenue is not an ancillary byproduct of filling sleeping rooms but a symbiotic and lucrative partner in the success of a property. Function space optimization is an analytical process which is evolving quickly to meet the challenges of a highly competitive environment providing the foundation to develop a winning strategy. The best available rate pricing of sleeping rooms allows to quickly adjust to changes in demand but with function space, the revenue streams are fixed. Weekly pattern management recognizes that each day contributes uniquely to the Function Space Optimization Strategy. A logical extension of Revenue Management is applying Yield strategies to an equally perishable product, function space.

Cited by 0

Function space topologies

Nader Vakil · Cambridge University Press eBooks · 2011

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Cited by 0

The Universal Teichmüller Space and Function Space

Yutong Liu, Yi Qi · Journal of Function Spaces · 2020

In this paper, a subspace <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>2,1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:msub></mml:mrow></mml:math> of the universal Teichmüller space, which is related to the analytic function space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>2,1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math>, is introduced and the holomorphy of the Bers map is shown. It is also proved that the pre-Bers map is holomorphic and the prelogarithmic derivative model <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>T</mml:mi><mml:mo>˜</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>2,1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:msub></mml:mrow></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>2,1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:msub></mml:mrow></mml:math> is a disconnected subset of the function space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>2,1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:math>. Moreover, several equivalent descriptions of elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mn>2,1</mml:mn><mml:mo>−</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:msub></mml:mrow></mml:math> are obtained and the holomorphy of higher Bers maps is proved.

Cited by 0Open access

Function of empty space

Krištof, Michal · Brno University of Technology Digital Library (Brno University of Technology) · 2013

The city generally consists of built and unbuilt spaces. An unbuilt space could be defined in other words as "void" which is essentially a public or semi-public space. Is it possible to precisely define the meaning of this "external" space of the city? Wh

Cited by 0

FOUR FUNCTION SPACE TOPOLOGIES

Lynn Taylor, Winter Master Arts, Name (degree et al. · 1970

Abstract approved: (pp-ftd&amp;quot;sof B.H. Arnbld) This paper defines four function space topologies, characterizes two of them in terms of more familiar concepts, and compares the four topologies. Then in the cases of the two less familiar topologies we have con-sidered several common properties of topological spaces and attempted to answer the following question: If the range space has a given property, does the function space necessarily have the same property?

Cited by 0

Function-Space Learning Rates

Edward Milsom, Ben Anson, Laurence Aitchison · ArXiv.org · 2025

We consider layerwise function-space learning rates, which measure the magnitude of the change in a neural network's output function in response to an update to a parameter tensor. This contrasts with traditional learning rates, which describe the magnitude of changes in parameter space. We develop efficient methods to measure and set function-space learning rates in arbitrary neural networks, requiring only minimal computational overhead through a few additional backward passes that can be performed at the start of, or periodically during, training. We demonstrate two key applications: (1) analysing the dynamics of standard neural network optimisers in function space, rather than parameter space, and (2) introducing FLeRM (Function-space Learning Rate Matching), a novel approach to hyperparameter transfer across model scales. FLeRM records function-space learning rates while training a small, cheap base model, then automatically adjusts parameter-space layerwise learning rates when training larger models to maintain consistent function-space updates. FLeRM gives hyperparameter transfer across model width, depth, initialisation scale, and LoRA rank in various architectures including MLPs with residual connections and transformers with different layer normalisation schemes.

Cited by 0Open access