scholarly journals Finite products of limits of direct systems induced by maps

2015 ◽  
Vol 16 (2) ◽  
pp. 209 ◽  
Author(s):  
Ivan Ivansic ◽  
Leonard R. Rubin
Keyword(s):  
Direct Limit ◽  
Finite Product ◽  
Relative Topology ◽  
Finite Products

Let Z, H be spaces. In previous work, we introduced the<br />direct (inclusion) system induced by the set of maps between the spaces Z and H. Its direct limit is a subset of Z × H, but its topology is different from the relative topology. We found that many of the spaces constructed from this method are pseudo-compact and Tychonoff. We are going to show herein that these spaces are typically not sequentially compact and we will explore conditions under which a finite product of them will be pseudo-compact.

The finite-product method in the theory of waves and stability

2009 ◽  
Vol 466 (2114) ◽  
pp. 471-491 ◽  
Author(s):  
C. J. Chapman ◽  
S. V. Sorokin
Keyword(s):  
Dispersion Relation ◽  
Elastic Waves ◽  
Low Frequency ◽  
Wave Theory ◽  
Finite Product ◽  
High Frequencies ◽  
Polynomial Approximations ◽  
Wide Range ◽  
Product Method ◽  
Finite Products

This paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. The finite products chosen must be consistent with the low-frequency, low-wavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge’s phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability theory.

scholarly journals Relative Full Completeness for Bicategorical Cartesian Closed Structure

2020 ◽  
pp. 277-298 ◽  
Author(s):  
Marcelo Fiore ◽  
Philip Saville
Keyword(s):  
Programming Languages ◽  
Lambda Calculus ◽  
Equational Theory ◽  
Higher Order ◽  
Finite Product ◽  
Semantic Framework ◽  
Finite Products ◽  
Closed Structure ◽  
Cartesian Closed ◽  
Comma Category

AbstractThe glueing construction, defined as a certain comma category, is an important tool for reasoning about type theories, logics, and programming languages. Here we extend the construction to accommodate ‘2-dimensional theories’ of types, terms between types, and rewrites between terms. Taking bicategories as the semantic framework for such systems, we define the glueing bicategory and establish a bicategorical version of the well-known construction of cartesian closed structure on a glueing category. As an application, we show that free finite-product bicategories are fully complete relative to free cartesian closed bicategories, thereby establishing that the higher-order equational theory of rewriting in the simply-typed lambda calculus is a conservative extension of the algebraic equational theory of rewriting in the fragment with finite products only.

Uniform Finite Generation of Threedimensional Linear Lie Groups

1975 ◽  
Vol 27 (2) ◽  
pp. 396-417 ◽  
Author(s):  
R. M. Koch ◽  
Franklin Lowenthal
Keyword(s):  
Positive Integer ◽  
Lie Group ◽  
Finite Product ◽  
Connected Subgroup ◽  
Arcwise Connected ◽  
Finite Products ◽  
The One ◽  
Lie Subgroup ◽  
Infinitesimal Transformations ◽  
Connected Lie Group

A connected Lie group G is generated by one-parameter subgroups exp(tX1), … , exp(tXk) if every element of G can be written as a finite product of elements chosen from these subgroups. This happens just in case the Lie algebra of G is generated by the corresponding infinitesimal transformations X1, … , Xk ; indeed the set of all such finite products is an arcwise connected subgroup of G, and hence a Lie subgroup by Yamabe's theorem [9]. If there is a positive integer n such that every element of G possesses such a representation of length at most n, G is said to be uniformly finitely generated by the one-parameter subgroups.

Uniform Finite Generation of SU(2) and SL(2, R)

1972 ◽  
Vol 24 (4) ◽  
pp. 713-727 ◽  
Author(s):  
Franklin Lowenthal
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Lie Group ◽  
Finitely Generated ◽  
Finite Product ◽  
Finite Generation ◽  
Arcwise Connected ◽  
Finite Products ◽  
Infinitesimal Transformations ◽  
Connected Lie Group

A connected Lie group H is generated by a pair of one-parameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups, i.e., if and only if the subalgebra generated by the corresponding pair of infinitesimal transformations is equal to the whole Lie algebra h of H (observe that the subgroup of all finite products is arcwise connected and hence, by Yamabe's theorem [5], is a sub-Lie group). If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n ; otherwise define it as infinity.

Comparative studies regarding the use of recycled PVC in finite products

2017 ◽  
Vol 62 (2) ◽  
pp. 45-52
Author(s):  
Zosin Sergiu Petri ◽  
◽  
Dumitru Ristoiu ◽  
Mihail Simion Beldean-Galea ◽  
Radu Mihăiescu ◽  
...  
Keyword(s):  
Comparative Studies ◽  
Finite Products

Closed form evaluation of sums containing squares of Fibonomial coefficients

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Closed Form ◽  
Systematic Approach ◽  
Sums Of Squares ◽  
Lucas Numbers ◽  
Fibonomial Coefficients ◽  
Finite Products ◽  
Fibonacci And Lucas Numbers ◽  
Form Evaluation

AbstractWe give a systematic approach to compute certain sums of squares of Fibonomial coefficients with finite products of generalized Fibonacci and Lucas numbers as coefficients. The technique is to rewrite everything in terms of a variable

scholarly journals Some categorical aspects of the inverse limits in ditopological context

2018 ◽  
Vol 19 (1) ◽  
pp. 101
Author(s):  
Filiz Yildiz
Keyword(s):  
Compatibility Condition ◽  
Natural Transformation ◽  
Inverse Limits ◽  
Infinite Products ◽  
Inverse Systems ◽  
Finite Products

<p>This paper considers some various categorical aspects of the inverse systems (projective spectrums) and inverse limits described in the category if<strong>PDitop</strong>, whose objects are ditopological plain texture spaces and morphisms are bicontinuous point functions satisfying a compatibility condition between those spaces. In this context, the category <strong>Inv<sub>ifPDitop</sub></strong> consisting of the inverse systems constructed by the objects and morphisms of if<strong>PDitop</strong>, besides the inverse systems of mappings, described between inverse systems, is introduced, and the related ideas are studied in a categorical - functorial setting. In conclusion, an identity natural transformation is obtained in the context of inverse systems - limits constructed in if<strong>PDitop</strong> and the ditopological infinite products are characterized by the finite products via inverse limits.</p>

Undecidability of the identity problem for finite semigroups

1992 ◽  
Vol 57 (1) ◽  
pp. 179-192 ◽  
Author(s):  
Douglas Albert ◽  
Robert Baldinger ◽  
John Rhodes
Keyword(s):  
Word Problem ◽  
Finite Semigroup ◽  
Identity Problem ◽  
Finite Semigroups ◽  
Or Groups ◽  
Finite Set ◽  
Variety Of Semigroups ◽  
Finite Products

In 1947 E. Post [28] and A. A. Markov [18] independently proved the undecidability of the word problem (or the problem of deducibility of relations) for semigroups. In 1968 V. L. Murskil [23] proved the undecidability of the identity problem (or the problem of deducibility of identities) in semigroups.If we slightly generalize the statement of these results we can state many related results in the literature and state our new results proved here. Let V denote either a (Birkhoff) variety of semigroups or groups or a pseudovariety of finite semigroups. By a very well-known theorem a (Birkhoff) variety is defined by equations or equivalently closed under substructure, surmorphisms and all products; see [7]. It is also well known that V is a pseudovariety of finite semigroups iff V is closed under substructure, surmorphism and finite products, or, equivalently, determined eventually by equations w1 = w1′, w2 = w2′, w3 = w3′,… (where the finite semigroup S eventually satisfies these equations iff there exists an n, depending on S, such that S satisfies Wj = Wj′ for j ≥ n). See [8] and [29]. All semigroups form a variety while all finite semigroups form a pseudovariety.We now consider a table (see the next page). In it, for example, the box denoting the “word” (identity) problem for the psuedovariety V” means, given a finite set of relations (identities) E and a relation (identity) u = ν, the problem of whether it is decidable that E implies u = ν inside V.

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