Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps

In this paper, we introduce a few notions of inverse topological pressure [Formula: see text], defined in terms of backward orbits (prehistories) instead of forward orbits. This inverse topological pressure has some properties similar to the regular (forward) pressure but, in general, if the map is not a homeomorphism, they do not coincide. In fact, there are several ways to define inverse topological pressure; for instance, we show that the Bowen type definition coincides with the one using spanning sets. Then we consider the case of a holomorphic map [Formula: see text] which is Axiom A and such that its critical set does not intersect a particular basic set of saddle type Λ. We will prove that, under a technical condition, the Hausdorff dimension of the intersection between the local stable manifold and the basic set is equal to ts, i.e. [Formula: see text], for all points x belonging to Λ. Here ts represents the unique zero of the function t→P-(tϕs), with P- denoting the inverse topological pressure and [Formula: see text], y∈Λ. In general, [Formula: see text] will be estimated above by ts and below by [Formula: see text], where [Formula: see text] is the unique zero of the map t→P_(tϕs). As a corollary we obtain that, if the stable dimension is non-zero, then Λ must be a non-Jordan curve, and also, if f|Λ happens to be a homeomorphism (like in the examples from [13]), then the stable dimension cannot be zero.

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Rilevare e curare la violenza all’infanzia: l’esperienza pugliese di GIADA

QUADERNI ACP ◽

10.53141/qacp.2021.263-265 ◽

2021 ◽

Vol 28 (5) ◽

pp. 263

Author(s):

Maria Grazia Foschino Barbaro ◽

Michele Pellegrini

Keyword(s):

Diagnosis And Treatment ◽

Abused Women ◽

Pediatric Hospital ◽

Violence Against Children ◽

Stable Dimension ◽

Women And Children ◽

Forms Of Violence

This article illustrates the path built over a decade by the Interdisciplinary Assistance Group for Abused Women and Children (GIADA) which has gradually taken on a stable dimension at the “Giovanni XXIII” Pediatric Hospital in Bari. The article describes the functions performed and the architecture of the Apulian model in which GIADA acts as a Level III Center for the prevention, diagnosis and treatment of the different forms of violence against children.

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Test-Retest Reliability of the Bender-Gestalt for First-Grade Children

Perceptual and Motor Skills ◽

10.2466/pms.1976.42.3.743 ◽

1976 ◽

Vol 42 (3) ◽

pp. 743-746 ◽

Cited By ~ 5

Author(s):

Fred H. Wallbrown ◽

Jane D. Wallbrown ◽

Ann W. Engin

Keyword(s):

School District ◽

Scoring System ◽

First Grade ◽

Working Time ◽

Reliability Estimate ◽

Suburban School District ◽

Reliability Estimates ◽

Stable Dimension ◽

Test Retest Reliability ◽

Reliability Coefficients

The reliability of the Bender was investigated for 144 first-grade children from a suburban school district. The test-retest interval ranged from 9 through 14 days. The Koppitz scoring system was used; reliability coefficients were determined for errors of distortion, rotation, integration, and perseveration as well as for the total score. The total time to reproduce the designs was also included. Magnitudes of the reliability estimates for the separate error categories were too small to justify interpreting scores individually. The reliability estimate for total working time was comparable to that for the total Koppitz score. Working time may be a stable dimension of Bender performance suitable for use in research.

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Relations between stable dimension and the preimage counting function on basic sets with overlaps

Bulletin of the London Mathematical Society ◽

10.1112/blms/bdp092 ◽

2009 ◽

Vol 42 (1) ◽

pp. 15-27 ◽

Cited By ~ 6

Author(s):

Eugen Mihailescu ◽

Mariusz Urbanski

Keyword(s):

Counting Function ◽

Stable Dimension ◽

Basic Sets

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Preparation and characterization of silicone rubber cured via catalyst-free aza-Michael reaction

RSC Advances ◽

10.1039/c6ra23016d ◽

2016 ◽

Vol 6 (113) ◽

pp. 111648-111656 ◽

Cited By ~ 10

Author(s):

Linglong Feng ◽

Lin Zhou ◽

Shengyu Feng

Keyword(s):

Silicone Rubber ◽

Michael Reaction ◽

High Strength ◽

Catalyst Free ◽

Stable Dimension

A novel silicone rubber of high strength and stable dimension was cured via catalyst-free aza-Michael reaction.

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On a class of stable conditional measures

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385710000477 ◽

2010 ◽

Vol 31 (5) ◽

pp. 1499-1515 ◽

Cited By ~ 7

Author(s):

EUGEN MIHAILESCU

Keyword(s):

Invariant Measures ◽

Reversible Systems ◽

Absolutely Continuous ◽

Stable Manifolds ◽

Equilibrium Measures ◽

Basic Set ◽

Stable Dimension ◽

Saddle Type ◽

Local Stable Manifolds ◽

Basic Sets

AbstractThe dynamics of endomorphisms (smooth non-invertible maps) presents many differences from that of diffeomorphisms or that of expanding maps; most methods from those cases do not work if the map has a basic set of saddle type with self-intersections. In this paper we study the conditional measures of a certain class of equilibrium measures, corresponding to a measurable partition subordinated to local stable manifolds. We show that these conditional measures are geometric probabilities on the local stable manifolds, thus answering in particular the questions related to the stable pointwise Hausdorff and box dimensions. These stable conditional measures are shown to be absolutely continuous if and only if the respective basic set is a non-invertible repeller. We find also invariant measures of maximal stable dimension, on folded basic sets. Examples are given, too, for such non-reversible systems.

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Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-029-2 ◽

2008 ◽

Vol 60 (3) ◽

pp. 658-684 ◽

Cited By ~ 10

Author(s):

Eugen Mihailescu ◽

Mariusz Urbański

Keyword(s):

Hausdorff Dimension ◽

Arbitrary Point ◽

Base Point ◽

Topological Pressure ◽

Axiom A ◽

Hyperbolic Diffeomorphisms ◽

Base Points ◽

Basic Set ◽

Fixed Positive Number ◽

Stable Dimension

AbstractWe study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) mapf: ℙ2ℂ → ℙ2ℂ, where ℙ2ℂ stands for the complex projective space of dimension 2. Letδs(x)denote a basic set for f of unstable index 1, and x an arbitrary point of Λ; we denote byδs(x)the Hausdorff dimension of∩ Λ, whereris some fixed positive number andis the local stable manifold atxof sizer;δs(x)is calledthe stable dimension at x. Mihailescu and Urba ńnski introduced a notion of inverse topological pressure, denoted by P−, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates off, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on Λ. When each pointxfrom Λ has the same numberd′of preimages in Λ, then we show thatδs(x)is independent of x; in factδs(x)is shown to be equal in this case with the unique zero of the mapt → P(tϕs−log d′). We also prove the Lipschitz continuity of the stable vector spaces over Λ; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

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Socjologia krytyczna na peryferiach

Kultura i Spoleczenstwo ◽

10.35757/kis.2009.53.1.6 ◽

2009 ◽

Vol 53 (1) ◽

pp. 105-121

Author(s):

Tomasz Zarycki

Keyword(s):

Theoretical Model ◽

Cultural Capital ◽

Eastern Europe ◽

Central And Eastern Europe ◽

Pierre Bourdieu ◽

Social Resources ◽

Political Capital ◽

Cultural Dimension ◽

Critical Sociology ◽

Stable Dimension

The paper proposes a redefinition of the rules of critical sociology in the context of peripheral countries, among them Poland and also Russia and other countries of Central and Eastern Europe. The proposed theoretical model refers to the notions of cultural and political capital as understood and defined by Pierre Bourdieu. The cultural capital in particular is believed to be the key and most stable dimension of inequality in Poland, as well as an important source of inequalities in other dimensions. It has been suggested that critical sociology of the Polish periphery should focus its interests precisely on this issue. At the same time the position that overlooks the cultural dimension of inequalities and treats interests defined in terms of culture as “irrational” is considered to be a manifestation of “Orientalism” and lack of respect for the important social resources of the population.

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TINJAUAN AL-QUR’AN DAN PRINSIP PEMBEBANAN DALAM PENENTUAN DIMENSI KESTABILAN PONDASI BANGUNAN

ULUL ALBAB Jurnal Studi Islam ◽

10.18860/ua.v9i1.6223 ◽

2018 ◽

Vol 9 (1) ◽

pp. 61-71

Author(s):

Agung Sedayu

Keyword(s):

Ground Water ◽

Structure Formation ◽

Power Factor ◽

Type Structure ◽

Building Construction ◽

Building Foundation ◽

Stable Dimension ◽

Foundation Type ◽

Stability Of Structure ◽

Foundation Structure

Foundation is a part of building construction having function continuously to support entire or all the building burden to ground. One of the factor of buildingcon truction should be supported by the sturdy foundation, and sturdy foundation depend on his compiler composition and dimension of foundation. Therefore it is needed tire calculations of foundation so that we get well guaranted of his stability. Besides that, ground factor also have stability of structure, because the ground sustain the foundation and place where burden given by foundation. Power factor support the ground against the foundation action is highly varied, it is _depend on composition and nature of ground, information concerning energy number support of this ground can be shown passing investigation _of ground. To guarantee security in a building one of the among others is to looking for comparison between foundations and well-balanced ground. The dimension of foundation must be enough and fit with the power of ground. Detennination of building foundation is very influenced by energy support the ground. And energy support the ground depict strength of ground to accepted burden. Energy support the ground is very influenced by character, nature, type, structure, formation, and component compiler of ground, besides also usage of foundation type is also influenced by contour, ground water face and topography. Energy support the ground explained by some concept or theories, among others is Terzaghi theory and accepted encumbering principle or apply at the foundation structure. From both this concept determinable of enough and stable dimension of strength sustain a building. The foundation dimension is reckoned by the minimum or smallest measure or dimension, but have ever been optimal to sustain the burden befall it.

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