A local factorization theorem for arcs with finite penetration index

1975 ◽  
pp. 259-275
Author(s):  
J. Mcpherson ◽  
Geoffrey Hemion
Keyword(s):  
Factorization Theorem ◽  
Penetration Index

scholarly journals Kolmogorov’s factorization theorem for von Neumann algebras

2013 ◽  
Vol 401 (1) ◽  
pp. 289-292 ◽  
Author(s):  
Kohei Nakade ◽  
Tomoyoshi Ohwada ◽  
Kichi-Suke Saito
Keyword(s):  
Von Neumann Algebras ◽  
Factorization Theorem ◽  
Von Neumann

CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

2006 ◽  
Vol 05 (02) ◽  
pp. 231-243
Author(s):  
DONGVU TONIEN
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Additive Group ◽  
New Product ◽  
Binary String ◽  
Factorization Theorem ◽  
General Criterion ◽  
Unique Factorization ◽  
Product Operation ◽  
Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

The modification of bitumen with styrene–butadiene–styrene, ethylene vinyl acetate and varying the amount of reactive ethylene terpolymer

2017 ◽  
Vol 50 (3) ◽  
pp. 241-255 ◽  
Author(s):  
Seyfullah Keyf
Keyword(s):  
Vinyl Acetate ◽  
Softening Point ◽  
Elastic Recovery ◽  
Ethylene Vinyl Acetate ◽  
Modified Bitumen ◽  
Penetration Index ◽  
Different Types ◽  
Styrene Butadiene Styrene ◽  
Styrene Butadiene ◽  
Butadiene Styrene

In this article, 50/70 penetration grade TUPRAS bitumen was modified. Reactive elastomeric terpolymer (Elvaloy RET; DuPont Company), ethylene vinyl acetate (EVA) and styrene–butadiene–styrene (SBS) polymers were used in bitumen modification. Set hours of the tests were applied to samples taken from the obtained modified bitumen mixture. Varying amount of reactive ethylene terpolymer with weight ratios of 0.5%, 1.0%, 1.5%, 2.0% and 2.5% were introduced to the mixture of raw bitumen with 1.0% SBS and 1.0% EVA. Penetration, penetration index, softening point, ductility and elastic recovery tests were performed with these modified bitumen and raw bitumen. The samples of raw bitumen and modified bitumens of 2.0% Elvaloy RET, 1.0% SBS and 1.0% EVA were investigated by means of IR spectroscopy. The raw bitumen was modified with SBS, EVA and RET, and it was determined that penetration and ductility values were decreased while penetration index, softening point and elastic recovery were increased. The purpose of this study is to research the improving properties of 50/70 penetration grade–modified bitumen used in highways of Turkey. The most important characteristics (such as softening point, penetration and % elastic recovery) of new polymer-modified bitumen (NPMB) containing 2.5% EVA, 1% RET and 1% SBS were compared with eight different types of polymer-modified bitumens in Turkey (TPMB). NPMB provided all required parameters (softening point, penetration and % elastic recovery) for five different types of TPMBs (TPMB 70-16, TPMB 70-22, TPMB 76-16, TPMB 76-22 and TPMB 82-16).

scholarly journals Factorization at subleading power and endpoint divergences in h → γγ decay. Part II. Renormalization and scale evolution

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ze Long Liu ◽  
Bianka Mecaj ◽  
Matthias Neubert ◽  
Xing Wang
Keyword(s):  
Decay Amplitude ◽  
Factorization Theorem ◽  
Effective Theory ◽  
Operator Matrix ◽  
Quark Loop ◽  
Perfect Agreement ◽  
Soft Collinear Effective Theory ◽  
Convolution Integrals ◽  
Factorization Formula ◽  
Massless Quark

Abstract Building on the recent derivation of a bare factorization theorem for the b-quark induced contribution to the h → γγ decay amplitude based on soft-collinear effective theory, we derive the first renormalized factorization theorem for a process described at subleading power in scale ratios, where λ = mb/Mh « 1 in our case. We prove two refactorization conditions for a matching coefficient and an operator matrix element in the endpoint region, where they exhibit singularities giving rise to divergent convolution integrals. The refactorization conditions ensure that the dependence of the decay amplitude on the rapidity regulator, which regularizes the endpoint singularities, cancels out to all orders of perturbation theory. We establish the renormalized form of the factorization formula, proving that extra contributions arising from the fact that “endpoint regularization” does not commute with renormalization can be absorbed, to all orders, by a redefinition of one of the matching coefficients. We derive the renormalization-group evolution equation satisfied by all quantities in the factorization formula and use them to predict the large logarithms of order $$ {\alpha \alpha}_s^2{L}^k $$ αα s 2 L k in the three-loop decay amplitude, where $$ L=\ln \left(-{M}_h^2/{m}_b^2\right) $$ L = ln − M h 2 / m b 2 and k = 6, 5, 4, 3. We find perfect agreement with existing numerical results for the amplitude and analytical results for the three-loop contributions involving a massless quark loop. On the other hand, we disagree with the results of previous attempts to predict the series of subleading logarithms $$ \sim {\alpha \alpha}_s^n{L}^{2n+1} $$ ∼ αα s n L 2 n + 1 .

The Mardešić Factorization Theorem and the Dimension of Metrizable Spaces

2019 ◽  
pp. 139-146
Author(s):  
Michael G. Charalambous
Keyword(s):  
Factorization Theorem ◽  
Metrizable Spaces

scholarly journals A factorization theorem for compact operators

1972 ◽  
Vol 34 (1) ◽  
pp. 201-201 ◽  
Author(s):  
Daniel J. Randtke
Keyword(s):  
Compact Operators ◽  
Factorization Theorem

The unique factorization theorem

1968 ◽  
pp. 1-10
Author(s):  
K. Chandrasekharan
Keyword(s):  
Factorization Theorem ◽  
Unique Factorization

Quadratic Surplus

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
Convex Analysis ◽  
Scalar Product ◽  
Optimal Transport ◽  
Point Of View ◽  
Factorization Theorem ◽  
Polar Factorization

This chapter considers the case when the attributes are d-dimensional vectors and the surplus is the scalar product; it assumes that the distribution of the workers' attributes is continuous, but it relaxes the assumption that the distribution of the firms' attributes is discrete. This setting allows us to entirely rediscover convex analysis, which is introduced from the point of view of optimal transport. As a consequence, Brenier's polar factorization theorem is given, which provides a vector extension for the scalar notions of quantile and rearrangement.

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