A definable continuous rank for nonmultidimensional superstable theories

1996 ◽  
Vol 61 (3) ◽  
pp. 967-984
Author(s):  
Ambar Chowdhury ◽  
James Loveys ◽  
Predrag Tanović
Keyword(s):  
Finite Rank ◽  
Atomic Models ◽  
Original Idea ◽  
Superstable Theories

Pillay studied nonmultidimensional superstable theories in [8], among other things defining a certain hierarchy of regular types in terms of which all other types may be analysed. Using this hierarchy, he showed that after naming a suitable ‘base’ of parameters, there are j-constructible (hence locally atomic) models over arbitrary sets (see Section 2 for definitions). It is asked at the end of [8] whether the parameter set can be removed. On a different note, it has been known for some time that in nonmultidimensional superstable theories, R∞-rank is definable for formulas having finite rank (see for example [9]). Definability of R∞-rank has had various applications in the literature, and so it is natural to ask whether the restriction to finite rank is necessary. In this paper we do not quite answer this question, but instead use Pillay's analysis to establish the existence of a ‘new’ continuous rank (the original idea for which is due to Tanović) which is defined on all complete types, reflects forking as does R∞-rank and satisfies certain definability properties.

Some remarks on nonmultidimensional superstable theories

1994 ◽  
Vol 59 (1) ◽  
pp. 151-165 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
General Theory ◽  
Finite Rank ◽  
Atomic Model ◽  
Elementary Substructure ◽  
Finite Dimensional ◽  
Definable Groups ◽  
Superstable Theories ◽  
Finite Dimensional Case ◽  
Language Context ◽  
Minimal Formula

In this paper we study nonmultidimensional superstable theories T, possibly in an uncountable language, and develop some techniques permitting the generalisation of certain results from the finite rank (and/or countable language) context to the general case.We prove, among other things, the following: there is a set A0 of parameters, which has cardinality at most ∣T∣, and in the finite-dimensional case is finite, such that over any B ⊇ A0 there is a locally atomic model. One of the consequences of this is that if C is the monster model of T, φ(x) is a formula over A0, φC ⊇ X and (X, φC) satisfies the Tarski-Vaught condition after adding names for A0, then there is an elementary substructure M of C containing A0 such that φM = X. Applications to the spectrum problem will appear in [Ch-P].In fact, all the components of the machinery we develop are already present in the general theory. One such component involves a stratification of the regular types of T using a generalized notion of weakly minimal formula. This appears in [Sh, Chapter V and the proof of IX.2.4] and also in [P2]. A second component involves definable groups which arise as ‘binding” groups. The existence of such groups, under certain hypotheses on the behavior of nonorthogonality, is due to Hrushovski [Hr1], and our use of them to help obtain “j-constructible” models is similar to their use in [Bu-Sh].

Combine Nanoprobing Technique with Difference Analysis to Identify Nonvisual Failures

2006 ◽  
Author(s):  
Cha-Ming Shen ◽  
Tsan-Cheng Chuang ◽  
Jie-Fei Chang ◽  
Jin-Hong Chou
Keyword(s):  
Failure Mode ◽  
Failure Modes ◽  
Electrical Characteristic ◽  
The Novel ◽  
Difference Analysis ◽  
Original Idea ◽  
Deductive Method ◽  
Complete Measurement

Abstract This paper presents a novel deductive methodology, which is accomplished by applying difference analysis to nano-probing technique. In order to prove the novel methodology, the specimens with 90nm process and soft failures were chosen for the experiment. The objective is to overcome the difficulty in detecting non-visual, erratic, and complex failure modes. And the original idea of this deductive method is based on the complete measurement of electrical characteristic by nano-probing and difference analysis. The capability to distinguish erratic and invisible defect was proven, even when the compound and complicated failure mode resulted in a puzzling characteristic.

Jozef Duysans gevecht met den engel. Schermutselingen rond het graf van Paul Van Ostaijen

2012 ◽  
Vol 71 (4) ◽  
pp. 299-328
Author(s):  
Armand Van Nimmen
Keyword(s):  
Crucial Role ◽  
Second World War ◽  
World War ◽  
Nineteen Thirties ◽  
Public Monuments ◽  
Deep Waters ◽  
Original Idea ◽  
The City ◽  
Second World

Deze bijdrage handelt over de perikelen in de jaren dertig rond het plan om het lichamelijk overschot van de Vlaamse dichter Paul Van Ostaijen over te brengen uit het klein Waals dorp waar hij in vergetelheid begraven lag onder een houten kruis naar zijn geboortestad Antwerpen. Daar zou hij herbegraven worden op de stedelijke begraafplaats Schoonselhof onder een gepaste denksteen. Zoals meermaals het geval is bij het oprichten van publieke monumenten, verliepen – wegens onderling gekibbel en gebrek aan financiële middelen – meer dan zes jaren vooraleer de oorspronkelijke idee kon verwezenlijkt worden.Aandacht in dit artikel gaat naar Jozef Duysan, bewonderaar van de dichter en uitgesproken flamingant, die een cruciale rol speelde in de conceptie en uitvoering van het initiatief. Ten slotte beschrijft het artikel hoe deze nu bijna totaal vergeten man tijdens de Tweede Wereldoorlog in het vaarwater geraakte van de collaboratie, fungeerde als directeur van het Arbeidsamt in Antwerpen, na de oorlog veroordeeld werd en jaren lang ondergedoken leefde in die stad.________Jozef Duysan’s battle with the angel: Skirmishes around the tomb of Paul Van OstaijenThis contribution reports the vicissitudes concerning the plan dating from the nineteen thirties to transfer the mortal remains of the Flemish poet Paul Van Ostaijen from the small Walloon village where he was buried in oblivion under a wooden cross to Antwerp, the city of his birth. He was to be reburied there on the municipal cemetery Schoonselhof under a fitting memorial headstone. As frequently happens on the occasion of creating public monuments, more than six years passed before the original idea could be carried out – because of internal bickering and lack of financial means. This article focuses on Jozef Duysan, an admirer of the poet and an explicit Flemish militant, who played a crucial role in the concept and realisation of the initiative. In conclusion the article recounts how this man who has been practically completely forgotten now,  ventured into the deep waters of the collaboration during the Second World War, how he acted as director of the Arbeidsamt in Antwerp and how he was convicted after the war and lived for many years in hiding in that city.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang
Keyword(s):  
Finite Rank ◽  
Intrinsic Geometry ◽  
Bounded Geometry ◽  
Higher Dimensional ◽  
Median Algebra ◽  
Asymptotic Geometry ◽  
Median Operator

AbstractThis paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

scholarly journals Mutual information for low-rank even-order symmetric tensor estimation

2020 ◽  
Author(s):  
Clément Luneau ◽  
Jean Barbier ◽  
Nicolas Macris
Keyword(s):  
Mutual Information ◽  
Finite Rank ◽  
Interpolation Method ◽  
Symmetric Tensor ◽  
Low Rank ◽  
Tensor Factorization ◽  
Adaptive Interpolation ◽  
Odd Order ◽  
Rank One ◽  
Even Order

Abstract We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its asymptotic mutual information when the tensor is of even order. The proof applies the adaptive interpolation method originally invented for rank-one factorization. Here we show how to extend the adaptive interpolation to finite-rank and even-order tensors. This requires new non-trivial ideas with respect to the current analysis in the literature. We also underline where the proof falls short when dealing with odd-order tensors.

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