scholarly journals On the extensibility of mappings, their local properties and some of their connections with the dimension theory

1960 ◽  
Vol 48 (3) ◽  
pp. 287-305 ◽  
Author(s):  
J. Jaroń
2018 ◽  
Vol 17 (06) ◽  
pp. 1850106
Author(s):  
Samir Bouchiba

Our main goal in this paper is to set the general frame for studying the dimension theory of tensor products of algebras over an arbitrary ring [Formula: see text]. Actually, we translate the theory initiated by Grothendieck and Sharp and subsequently developed by Wadsworth on Krull dimension of tensor products of algebras over a field [Formula: see text] into the general setting of algebras over an arbitrary ring [Formula: see text]. For this sake, we introduce and study the notion of a fibered AF-ring over a ring [Formula: see text]. This concept extends naturally the notion of AF-ring over a field introduced by Wadsworth in [The Krull dimension of tensor products of commutative algebras over a field, J. London Math. Soc. 19 (1979) 391–401.] to algebras over arbitrary rings. We prove that Wadsworth theorems express local properties related to the fiber rings of tensor products of algebras over a ring. Also, given a triplet of rings [Formula: see text] consisting of two [Formula: see text]-algebras [Formula: see text] and [Formula: see text] such that [Formula: see text], we introduce the inherent notion to [Formula: see text] of a [Formula: see text]-fibered AF-ring which allows to compute the Krull dimension of all fiber rings of the considered tensor product [Formula: see text]. As an application, we provide a formula for the Krull dimension of [Formula: see text] when either [Formula: see text] or [Formula: see text] is zero-dimensional as well as for the Krull dimension of [Formula: see text] when [Formula: see text] is a fibered AF-ring over the ring of integers [Formula: see text] with nonzero characteristic and [Formula: see text] is an arbitrary ring. This enables us to answer a question of Jorge Martinez on evaluating the Krull dimension of [Formula: see text] when [Formula: see text] is a Boolean ring. Actually, we prove that if [Formula: see text] and [Formula: see text] are rings such that [Formula: see text] is not trivial and [Formula: see text] is a Boolean ring, then dim[Formula: see text].


2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.


1982 ◽  
Vol 43 (6) ◽  
pp. 961-971 ◽  
Author(s):  
J.A. Hodges ◽  
G. Jéhanno ◽  
D. Debray ◽  
F. Holtzberg ◽  
M. Loewenhaupt
Keyword(s):  
X Ray ◽  

2020 ◽  
Vol 63 (1) ◽  
pp. 116-135
Author(s):  
Anton V. Kuznetsov

The articles examines the teleofunctional solution to the problem of mental causation, presented by Dmitry Volkov in his recently published book Free Will. An Illusion or an Opportunity. D.B. Volkov proposes solutions to three big metaphysical problems – mental causation, personal identity, and free will. Solving the first problem, Volkov creatively combines the advantages of Dennett’s teleofunctional model and Vasilyev’s local interactionism. Volkov’s teleofunctional model of mental causation seeks to prove the causal relevance of mental properties as non-local higher order properties. In my view, its substantiation is based on three points: (a) critics of the exclusion problem and Kim’s model of mental causation, (b) “Library of first editions” argument, (c) reduction of the causal trajectories argument (CTA 1) by Vasilyev to the counterpart argument (CTA 2) by Volkov. Each of these points faces objections. Kim’s criticism is based on an implicit confusion of two types of reduction – reduction from supervenience and from multiple realizability. The latter type does not threaten Kim’s ideas, but Volkov uses this very type in his criticism. The “Library of first editions” argument does not achieve its goal due to compositional features and because non-local relational properties are a type of external properties that cannot be causally relevant. The reduction of CTA 1 to CTA 2 is unsuccessful since, in the case of this reduction, important features of CTA 1 are lost – these are local mental properties, due to which the influence of non-local physical factors occurs. My main objection is that the concept of causally relevant non-local properties is incompatible with the very concept of cause. The set of causally relevant properties of cause can only be local.


Fractals ◽  
2007 ◽  
Vol 15 (01) ◽  
pp. 63-72 ◽  
Author(s):  
JÖRG NEUNHÄUSERER

We develop the dimension theory for a class of linear solenoids, which have a "fractal" attractor. We will find the dimension of the attractor, proof formulas for the dimension of ergodic measures on this attractor and discuss the question of whether there exists a measure of full dimension.


2013 ◽  
Vol 577-578 ◽  
pp. 593-596 ◽  
Author(s):  
Václav Mentl

The steam turbine rotors represent large components both in radial and axial directions. Their local properties generally differ from one forging to another, or if we compare head and bottom parts of the original ingot, or central and circumferential localities of one rotor body respectively, or if we compare the properties of separate discs e.g. in the case of welded rotors. These differences stem from both even slight changes in the chemical composition (of separate heats or even within one ingot) and thermo-mechanical treatment and in the differences in technology with respect to the real shape and size of the forgings in question. In the paper, the consequences of the differences in fracture toughness characteristics in various rotor localities are discussed with respect to the rotors operational safety taking into account the existence of cracks and material degradation.


2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.


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