Symmetry

251 KiB

Open AccessArticle

by Michael G. Titelbaum

Symmetry 2011, 3(3), 680-698; https://doi.org/10.3390/sym3030680 - 16 Sep 2011

Cited by 3 | Viewed by 5069

This article proves that formal theories of evidential favoring must fail because they are inevitably language dependent. I begin by describing Carnap’s early confirmation theories to show how language dependence problems (like Goodman’s grue problem) arise. I then generalize to showthat any formal [...] Read more.

This article proves that formal theories of evidential favoring must fail because they are inevitably language dependent. I begin by describing Carnap’s early confirmation theories to show how language dependence problems (like Goodman’s grue problem) arise. I then generalize to showthat any formal favoring theory satisfying minimal plausible conditions will yield different judgments about the same evidence and hypothesis when they are expressed in alternate languages. This does not just indict formal theories of favoring; it also shows that something beyond our evidence must be invoked to substantively favor one hypothesis over another. Full article

(This article belongs to the Special Issue Symmetry in Probability and Inference)

449 KiB

Open AccessArticle

Lattices of Graphical Gaussian Models with Symmetries

by Helene Gehrmann

Symmetry 2011, 3(3), 653-679; https://doi.org/10.3390/sym3030653 - 7 Sep 2011

Cited by 9 | Viewed by 5028

Abstract

In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge [...] Read more.

In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards–Havránek model selection procedure [2]. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other. Full article

(This article belongs to the Special Issue Symmetry in Probability and Inference)

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278 KiB

Open AccessArticle

Symmetry and the Brown-Freiling Refutation of the Continuum Hypothesis

by Paul Bartha

Symmetry 2011, 3(3), 636-652; https://doi.org/10.3390/sym3030636 - 6 Sep 2011

Viewed by 5805

Abstract

Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for [...] Read more.

Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for two reasons. First, the failure is unusual in that the symmetry intuitions are demonstrably coherent, even though other constraints make it impossible to find a probability model for the scenario. Second, the best probability models have properties analogous to non-conglomerability, motivating a proposed extension of that concept (and corresponding limits on Bayesian conditionalization). Full article

(This article belongs to the Special Issue Symmetry in Probability and Inference)

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303 KiB

Open AccessArticle

Symmetry, Invariance and Ontology in Physics and Statistics

by Julio Michael Stern

Symmetry 2011, 3(3), 611-635; https://doi.org/10.3390/sym3030611 - 1 Sep 2011

Cited by 23 | Viewed by 7245

Abstract

This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in [...] Read more.

This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion. Full article

(This article belongs to the Special Issue Symmetry in Probability and Inference)

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227 KiB

Open AccessArticle

High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument

by Donald St. P. Richards

Symmetry 2011, 3(3), 600-610; https://doi.org/10.3390/sym3030600 - 26 Aug 2011

Cited by 8 | Viewed by 4640

Abstract

Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. [...] Read more.

Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconis, and Newman: Let the random matrix H_n be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let A_n be a non-random n × n real matrix such that tr (A'_nA_n) = 1. Then, as n→∞, √n tr A_nH_n converges in distribution to the standard normal distribution. Full article

(This article belongs to the Special Issue Symmetry in Probability and Inference)

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Open AccessArticle

Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas

by Michael D. Perlman and Jon A. Wellner

Symmetry 2011, 3(3), 574-599; https://doi.org/10.3390/sym3030574 - 23 Aug 2011

Cited by 7 | Viewed by 5266

Abstract

Do there exist circular and spherical copulas in ℝ^d? That is, do there exist circularly symmetric distributions on the unit disk in ℝ² and spherically symmetric distributions on the unit ball in ℝ^d, d ≥ 3, whose one-dimensional [...] Read more.

Do there exist circular and spherical copulas in ℝ^d? That is, do there exist circularly symmetric distributions on the unit disk in ℝ² and spherically symmetric distributions on the unit ball in ℝ^d, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in ℝ² by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in ℝ^d are also described, and determined explicitly for d = 2. Full article

(This article belongs to the Special Issue Symmetry in Probability and Inference)

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331 KiB

Open AccessArticle

Green’s Symmetries in Finite Digraphs

by Allen D. Parks

Symmetry 2011, 3(3), 564-573; https://doi.org/10.3390/sym3030564 - 15 Aug 2011

Cited by 2 | Viewed by 5328

Abstract

The semigroup D_V of digraphs on a set V of n labeled vertices is defined. It is shown that D_V is faithfully represented by the semigroup B_n of n ´ n Boolean matrices and that the Green’s L, R, H, [...] Read more.

The semigroup D_V of digraphs on a set V of n labeled vertices is defined. It is shown that D_V is faithfully represented by the semigroup B_n of n ´ n Boolean matrices and that the Green’s L, R, H, and D equivalence classifications of digraphs in D_V follow directly from the Green’s classifications already established for B_n. The new results found from this are: (i) L, R, and H equivalent digraphs contain sets of vertices with identical neighborhoods which remain invariant under certain one-sided semigroup multiplications that transform one digraph into another within the same equivalence class, i.e., these digraphs exhibit Green’s isoneighborhood symmetries; and (ii) D equivalent digraphs are characterized by isomorphic inclusion lattices that are generated by their out-neighborhoods and which are preserved under certain two-sided semigroup multiplications that transform digraphs within the same D equivalence class, i.e., these digraphs are characterized by Green’s isolattice symmetries. As a simple illustrative example, the Green’s classification of all digraphs on two vertices is presented and the associated Green’s symmetries are identified. Full article

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414 KiB

Open AccessArticle

Symmetry Aspects of the Band Structure and Motion Equations Applied in Calculating the Cyclotron Frequency of Electrons in Metals

by Stanislaw Olszewski and Tomasz Roliński

Symmetry 2011, 3(3), 541-563; https://doi.org/10.3390/sym3030541 - 10 Aug 2011

Cited by 1 | Viewed by 4686

Abstract

Cyclotron frequency of a crystal electron is, in general, not an easily accessible parameter. Nevertheless, its calculation can be simplified when the symmetry properties of the band structure and those of the motion equations in the magnetic field are simultaneously taken into account. [...] Read more.

Cyclotron frequency of a crystal electron is, in general, not an easily accessible parameter. Nevertheless, its calculation can be simplified when the symmetry properties of the band structure and those of the motion equations in the magnetic field are simultaneously taken into account. In effect, a combined symmetry of the electron Hamiltonian and that of the Lorentz equation provide us with a non-linear oscillator problem of high symmetry. In the next step, the kinetic energy of the oscillator can be separated from the whole of electron energy and applied in a new kind of calculation of the cyclotron frequency which is much more simple than before. In consequence, a detailed approach to the electron circulation, also in more complex band structures, becomes a relatively easy task. For different crystal lattices of cubic symmetry taken as examples the cyclotron frequency of the present and a former method are compared numerically giving the same results. Full article

(This article belongs to the Special Issue Symmetries of Electronic Order)

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322 KiB

Open AccessArticle

Action Duality: A Constructive Principle for Quantum Foundations

by Ken B. Wharton, David J. Miller and Huw Price

Symmetry 2011, 3(3), 524-540; https://doi.org/10.3390/sym3030524 - 27 Jul 2011

Cited by 21 | Viewed by 6870

Abstract

An analysis of the path integral approach to quantum theory motivates the hypothesis that two experiments with the same classical action should have dual ontological descriptions. If correct, this hypothesis would not only constrain realistic interpretations of quantum theory, but would also act [...] Read more.

An analysis of the path integral approach to quantum theory motivates the hypothesis that two experiments with the same classical action should have dual ontological descriptions. If correct, this hypothesis would not only constrain realistic interpretations of quantum theory, but would also act as a constructive principle, allowing any realistic model of one experiment to generate a corresponding model for its action-dual. Two pairs of action-dual experiments are presented, including one experiment that violates the Bell inequality and yet is action-dual to a single particle. The implications generally support retrodictive and retrocausal interpretations. Full article

(This article belongs to the Special Issue Quantum Symmetry)

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2910 KiB

Open AccessArticle

Folded Sheet Versus Transparent Sheet Models for Human Symmetry Judgments

by Jacques Ninio

Symmetry 2011, 3(3), 503-523; https://doi.org/10.3390/sym3030503 - 22 Jul 2011

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Abstract

As a contribution to the mysteries of human symmetry perception, reaction time data were collected on the detection of symmetry or repetition violations, in the context of short term visual memory studies. The histograms for reaction time distributions are rather narrow in the [...] Read more.

As a contribution to the mysteries of human symmetry perception, reaction time data were collected on the detection of symmetry or repetition violations, in the context of short term visual memory studies. The histograms for reaction time distributions are rather narrow in the case of symmetry judgments. Their analysis was performed in terms of a simple kinetic model of a mental process in two steps, a slow one for the construction of the representation of the images to be compared, and a fast one, in the 50 ms range, for the decision. There was no need for an additional ‘mental rotation’ step. Symmetry seems to facilitate the construction step. I also present here original stimuli showing a color equalization effect across a symmetry axis, and its counterpart in periodic patterns. According to a “folded sheet model”, when a shape is perceived, the brain automatically constructs a mirror-image representation of the shape. Based in part on the reaction time analysis, I present here an alternative “transparent sheet” model in which the brain constructs a single representation, which can be accessed from two sides, thus generating simultaneously a pattern and its mirror-symmetric partner. Filtering processes, implied by current models of symmetry perception could intervene at an early stage, by nucleating the propagation of similar perceptual groupings in the two symmetric images. Full article

(This article belongs to the Special Issue Symmetry Processing in Perception and Art)

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357 KiB

Open AccessArticle

Classifying Entropy Measures

by Angel Garrido

Symmetry 2011, 3(3), 487-502; https://doi.org/10.3390/sym3030487 - 20 Jul 2011

Cited by 12 | Viewed by 5984

Abstract

Our paper analyzes some aspects of Uncertainty Measures. We need to obtain new ways to model adequate conditions or restrictions, constructed from vague pieces of information. The classical entropy measure originates from scientific fields; more specifically, from Statistical Physics and Thermodynamics. With time [...] Read more.

Our paper analyzes some aspects of Uncertainty Measures. We need to obtain new ways to model adequate conditions or restrictions, constructed from vague pieces of information. The classical entropy measure originates from scientific fields; more specifically, from Statistical Physics and Thermodynamics. With time it was adapted by Claude Shannon, creating the current expanding Information Theory. However, the Hungarian mathematician, Alfred Rényi, proves that different and valid entropy measures exist in accordance with the purpose and/or need of application. Accordingly, it is essential to clarify the different types of measures and their mutual relationships. For these reasons, we attempt here to obtain an adequate revision of such fuzzy entropy measures from a mathematical point of view. Full article

(This article belongs to the Special Issue Symmetry Measures on Complex Networks)

275 KiB

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On Symmetry of Independence Polynomials

by Vadim E. Levit and Eugen Mandrescu

Symmetry 2011, 3(3), 472-486; https://doi.org/10.3390/sym3030472 - 15 Jul 2011

Cited by 5 | Viewed by 5452

Abstract

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum [...] Read more.

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum matching. If sk is the number of independent sets of size k in G, then I(G; x) = s₀ + s₁x + s₂x₂ + ... + s_αx^α, α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If sj = s_αj for all 0 ≤ j ≤ [α/2], then I(G; x) is called symmetric (or palindromic). It is known that the graph G _° 2K₁, obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer k ≤ μ (G), one can build a graph H, such that: G is a subgraph of H, I (H; x) is symmetric, and I (G _° 2K1; x) = (1 + x)^k · I (H; x). Full article

(This article belongs to the Special Issue Symmetry Measures on Complex Networks)

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625 KiB

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Mirror Symmetry Is Subject to Crowding

by Gabrielle Roddy and Rick Gurnsey

Symmetry 2011, 3(3), 457-471; https://doi.org/10.3390/sym3030457 - 13 Jul 2011

Cited by 5 | Viewed by 4915

Abstract

Mirror symmetry is often thought to be particularly salient to human observers because it engages specialized mechanisms that evolved to sense symmetrical objects in nature. Although symmetry is indeed present in many of our artifacts and markings on wildlife, studies have shown that [...] Read more.

Mirror symmetry is often thought to be particularly salient to human observers because it engages specialized mechanisms that evolved to sense symmetrical objects in nature. Although symmetry is indeed present in many of our artifacts and markings on wildlife, studies have shown that sensitivity to mirror symmetry does not serve an alerting function and sensitivity to symmetry decreases in a rather unremarkable way when it is presented away from the centre of the visual field. Here we show that symmetrical targets are vulnerable to the same interference as other stimuli when surrounded by non-target elements. These results provide further evidence that symmetry is not special to the early visual system, and reinforce the notion that our fascination with symmetry is more likely attributable to cognitive and aesthetic factors than to specialized, low level mechanisms in the visual system. Full article

(This article belongs to the Special Issue Symmetry Processing in Perception and Art)

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228 KiB

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Reduction of Image Complexity Explains Aesthetic Preference for Symmetry

by Chien-Chung Chen, Jo-Hsuan Wu and Chia-Ching Wu

Symmetry 2011, 3(3), 443-456; https://doi.org/10.3390/sym3030443 - 11 Jul 2011

Cited by 18 | Viewed by 7342

Abstract

Symmetric patterns are more appealing to human observers than asymmetric ones. Here, we investigate the visual information processing mechanisms underlying this aesthetic preference. All stimuli were derived from phase scrambled versions of forty face or nature images. In addition to the scrambled images, [...] Read more.

Symmetric patterns are more appealing to human observers than asymmetric ones. Here, we investigate the visual information processing mechanisms underlying this aesthetic preference. All stimuli were derived from phase scrambled versions of forty face or nature images. In addition to the scrambled images, there were four other types of test image: symmetric, in which one part of the image was a reflection of another around an axis; repetitive, in which one part of the image was a copy of the other; anti-symmetric, similar to symmetric but with the contrast of one side reversed; and interleaved patterns, in which half of the symmetric pattern was replaced by a scrambled image. The number of axes ranged from 1 to 16 for all image types. The task of our 20 observers was to give a preference rating to each image on a 6-point Lickert scale. The preference rating increased with the number of axes for all stimulus types. The observers showed a similar preference for symmetric and repetitive patterns and slightly less preference for anti-symmetric patterns. The preference for interleaved patterns was much less than for other types of stimuli. Preference for an image cannot be explained by either the ecological significance of its content or the slope of its amplitude spectrum. Instead, preference can be accounted for by the complexity of the image. Full article

(This article belongs to the Special Issue Symmetry Processing in Perception and Art)

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1531 KiB

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Linear Recurrent Double Sequences with Constant Border in M₂(F₂) are Classified According to Their Geometric Content

by Mihai Prunescu

Symmetry 2011, 3(3), 402-442; https://doi.org/10.3390/sym3030402 - 7 Jul 2011

Cited by 4 | Viewed by 5554

Abstract

The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M [...] Read more.

The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M₂(F₂) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M₂(F₂) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M₂(F₂), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version. Full article

(This article belongs to the Special Issue Symmetry in Theoretical Computer Science)

266 KiB

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Is the Notion of Time Really Fundamental?

by Florian Girelli, Stefano Liberati and Lorenzo Sindoni

Symmetry 2011, 3(3), 389-401; https://doi.org/10.3390/sym3030389 - 29 Jun 2011

Cited by 7 | Viewed by 6129

Abstract

From the physics point of view, time is now best described through General Relativity as part of space-time, which is a dynamical object encoding gravity. Time possesses also some intrinsic irreversibility due to thermodynamics and quantum mechanical effects. This irreversibility can look puzzling [...] Read more.

From the physics point of view, time is now best described through General Relativity as part of space-time, which is a dynamical object encoding gravity. Time possesses also some intrinsic irreversibility due to thermodynamics and quantum mechanical effects. This irreversibility can look puzzling since time-like loops (and hence time machines) can appear in General Relativity (for example in the Gödel universe, a solution of Einstein’s equations). We take this apparent discrepancy as a warning bell, pointing out that time as we understand it might not be fundamental and that whatever theory lying beyond General Relativity may not include time as we know it as a fundamental structure. We propose therefore, following the philosophy of analog models of gravity, that time and gravity might not be fundamental per se, but only emergent features. We illustrate our proposal using a toy-model where we show how the Lorentzian signature and Nordström gravity (a diffeomorphisms invariant scalar gravity theory) can emerge from a timeless non-dynamical space. This article received the fourth prize at the essay competition of the Foundational Questions Institute on the nature of time. Full article

(This article belongs to the Special Issue Quantum Symmetry)

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Symmetry, Volume 3, Issue 3 (September 2011) – 16 articles , Pages 389-698

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