Construction and Simulation of Composite Measures and Condensation Model for Designing Probabilistic Computational Applications

The probabilistic algorithms are widely applied in designing computational applications such as distributed systems and probabilistic databases, to determine distributed consensus in the presence of random failures of nodes or networks. In distributed computing, symmetry breaking is performed by employing probabilistic algorithms. In general, probabilistic symmetry breaking without any bias is preferred. Thus, the designing of randomized and probabilistic algorithms requires modeling of associated probability spaces to generate control-inputs. It is required that discrete measures in such spaces are computable and tractable in nature. This paper proposes the construction of composite discrete measures in real as well as complex metric spaces. The measures are constructed on different varieties of continuous smooth curves having distinctive non-linear profiles. The compositions of discrete measures consider arbitrary functions within metric spaces. The measures are constructed on 1-D interval and 2-D surfaces and, the corresponding probability metric product is defined. The associated sigma algebraic properties are formulated. The condensation measure of the uniform contraction map is constructed as axioms. The computational evaluations of the proposed composite set of measures are presented.

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Some New Fixed-Point Theorems for a ()-Pair Meir-Keeler-Type Set-Valued Contraction Map in Complete Metric Spaces

Journal of Applied Mathematics ◽

10.1155/2011/327941 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11

Author(s):

Chi-Ming Chen

Keyword(s):

Fixed Point ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Complete Metric Spaces ◽

Contraction Map

We obtain some new fixed point theorems for a ()-pair Meir-Keeler-type set-valued contraction map in metric spaces. Our main results generalize and improve the results of Klim and Wardowski, (2007).

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Algebraic properties of SU(4) ⊗ SU(4) symmetry breaking

Nuclear Physics B ◽

10.1016/0550-3213(74)90387-3 ◽

1974 ◽

Vol 76 (2) ◽

pp. 301-309 ◽

Cited By ~ 3

Author(s):

R.E. Mott

Keyword(s):

Symmetry Breaking ◽

Algebraic Properties

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On disjointness of dynamical systems

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100055936 ◽

1979 ◽

Vol 85 (3) ◽

pp. 477-491 ◽

Cited By ~ 2

Author(s):

J. Auslander ◽

Y. N. Dowker

Keyword(s):

Dynamical System ◽

Metric Spaces ◽

Probability Measures ◽

Compact Metric Spaces ◽

Probability Spaces ◽

Dynamical Properties ◽

Borel Probability ◽

The Given ◽

Special Case ◽

Measure Preserving Transformations

By a dynamical system we mean one of several related objects: measure preserving transformations on probability spaces (processes), self homeomorphisms of compact metric spaces (compact systems), or a combination of these, namely compact systems provided with invariant Borel probability measures. It is the latter, which we call compact processes, which will be of most interest in this paper. In particular, we will study the dynamical properties of the product of two processes with respect to compatible measures – those measures which project to the given measures on the component spaces. This leads to the notion of disjointness of two processes – the only compatible measure is the product measure. As an application we obtain a theorem, a special case of which gives rise to a class of transformations which preserve normal sequences. Finally, we study a topological analog (topological disjointness) and briefly consider the relation between the two notions of disjointness.

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ON THE PROBABILITY DISTRIBUTION OF THE OVERLAP IN SPIN GLASSES

International Journal of Modern Physics B ◽

10.1142/s0217979204024355 ◽

2004 ◽

Vol 18 (04n05) ◽

pp. 733-743 ◽

Cited By ~ 10

Author(s):

GIORGIO PARISI

Keyword(s):

Probability Distribution ◽

Symmetry Breaking ◽

Spin Glasses ◽

Replica Symmetry ◽

Replica Symmetry Breaking ◽

Rigorous Approach ◽

Algebraic Properties ◽

Replica Approach

In this paper we review the predictions of the replica approach on the probability distribution of the overlaps among replicas and on the sample to sample fluctuations of this probability. We stress the role of replica equivalence in obtaining relations that do not depend on the form of replica symmetry breaking. A comparison is done with the results obtained with a different rigorous approach. The role of the ultrametricity and of other algebraic properties in discussed.

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Metric Versus Topological Receptive Entropy of Semigroup Actions

Qualitative Theory of Dynamical Systems ◽

10.1007/s12346-021-00485-7 ◽

2021 ◽

Vol 20 (2) ◽

Author(s):

Andrzej Biś ◽

Dikran Dikranjan ◽

Anna Giordano Bruno ◽

Luchezar Stoyanov

Keyword(s):

Variational Principle ◽

Topological Entropy ◽

Metric Spaces ◽

Amenable Group ◽

Metric Entropy ◽

Semigroup Actions ◽

Compact Metric Spaces ◽

Classical Setting ◽

Probability Spaces ◽

Similar Notion

AbstractWe study the receptive metric entropy for semigroup actions on probability spaces, inspired by a similar notion of topological entropy introduced by Hofmann and Stoyanov (Adv Math 115:54–98, 1995). We analyze its basic properties and its relation with the classical metric entropy. In the case of semigroup actions on compact metric spaces we compare the receptive metric entropy with the receptive topological entropy looking for a Variational Principle. With this aim we propose several characterizations of the receptive topological entropy. Finally we introduce a receptive local metric entropy inspired by a notion by Bowen generalized in the classical setting of amenable group actions by Zheng and Chen, and we prove partial versions of the Brin–Katok Formula and the local Variational Principle.

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Further Ultrastructural Observations on Metastatic Nodules of Human Breast Cancer

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010010086x ◽

1968 ◽

Vol 26 ◽

pp. 224-225

Author(s):

John L. Swedo ◽

R. W. Talley ◽

John H. L. Watson

Keyword(s):

Breast Cancer ◽

Human Breast Cancer ◽

Estrogen Therapy ◽

Specimen Preparation ◽

Human Breast ◽

Autophagic Vacuoles ◽

Previous Communication ◽

Linear Profiles ◽

Metastatic Nodule

Since the report, which described the ultrastructure of a metastatic nodule of human breast cancer after estrogen therapy, additional ultrastructural observations, including some which are correlative with pertinent findings in the literature concerning mycoplasmas, have been recorded concerning the same subject. Specimen preparation was identical to that in.The mitochondria possessed few cristae, and were deteriorated and vacuolated. They often contained particulates and fibrous structures, sometimes arranged in spindle-shaped bundles, Fig. 1. Another apparent aberration was the occurrence, Fig. 2 (arrows) of linear profiles of what seems to be SER, which lie between layers of RER, and are often recognizably continuous with them.It was noted that the structure of the round bodies, interpreted as within autophagic vacuoles in the previous communication, and of vesicular bodies, described morphologically closely resembled those of some mycoplasmas. Specifically, they simulated or reflected the various stages of replication reported for mycoplasmas grown on solid nutrient. Based on this observation, they are referred to here as “mycoplasma-like” structures, in anticipation of confirmatory evidence from investigations now in progress.

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Symmetry breaking in convergent-beam diffraction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100154378 ◽

1989 ◽

Vol 47 ◽

pp. 480-481

Author(s):

D.J. Eaglesham

Keyword(s):

Symmetry Breaking ◽

Point Group ◽

X Ray Diffraction ◽

Multiple Diffraction ◽

Space Groups ◽

Atomic Displacements ◽

Small Probe ◽

Phase Sensitive ◽

Convergent Beam

Convergent Beam Electron Diffraction is now almost routinely used in the determination of the point- and space-groups of crystalline samples. In addition to its small-probe capability, CBED is also postulated to be more sensitive than X-ray diffraction in determining crystal symmetries. Multiple diffraction is phase-sensitive, so that the distinction between centro- and non-centro-symmetric space groups should be trivial in CBED: in addition, the stronger scattering of electrons may give a general increase in sensitivity to small atomic displacements. However, the sensitivity of CBED symmetry to the crystal point group has rarely been quantified, and CBED is also subject to symmetry-breaking due to local strains and inhomogeneities. The purpose of this paper is to classify the various types of symmetry-breaking, present calculations of the sensitivity, and illustrate symmetry-breaking by surface strains.CBED symmetry determinations usually proceed by determining the diffraction group along various zone axes, and hence finding the point group. The diffraction group can be found using either the intensity distribution in the discs

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Centrosome Aurora A gradient ensures single polarity axis in C. elegans embryos

Biochemical Society Transactions ◽

10.1042/bst20200298 ◽

2020 ◽

Vol 48 (3) ◽

pp. 1243-1253 ◽

Cited By ~ 1

Author(s):

Sukriti Kapoor ◽

Sachin Kotak

Keyword(s):

Symmetry Breaking ◽

Cell Fate ◽

Cell Cortex ◽

Axis Formation ◽

Aurora A Kinase ◽

Aurora A ◽

C Elegans ◽

Cell Embryo ◽

Polarity Establishment

Cellular asymmetries are vital for generating cell fate diversity during development and in stem cells. In the newly fertilized Caenorhabditis elegans embryo, centrosomes are responsible for polarity establishment, i.e. anterior–posterior body axis formation. The signal for polarity originates from the centrosomes and is transmitted to the cell cortex, where it disassembles the actomyosin network. This event leads to symmetry breaking and the establishment of distinct domains of evolutionarily conserved PAR proteins. However, the identity of an essential component that localizes to the centrosomes and promotes symmetry breaking was unknown. Recent work has uncovered that the loss of Aurora A kinase (AIR-1 in C. elegans and hereafter referred to as Aurora A) in the one-cell embryo disrupts stereotypical actomyosin-based cortical flows that occur at the time of polarity establishment. This misregulation of actomyosin flow dynamics results in the occurrence of two polarity axes. Notably, the role of Aurora A in ensuring a single polarity axis is independent of its well-established function in centrosome maturation. The mechanism by which Aurora A directs symmetry breaking is likely through direct regulation of Rho-dependent contractility. In this mini-review, we will discuss the unconventional role of Aurora A kinase in polarity establishment in C. elegans embryos and propose a refined model of centrosome-dependent symmetry breaking.

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