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Symmetry
Volume 1
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Symmetry, Volume 1, Issue 1 (September 2009) – 6 articles , Pages 1-105

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563 KiB  
Article
Nuclei, Primes and the Random Matrix Connection
by Frank W. K. Firk and Steven J. Miller
Symmetry 2009, 1(1), 64-105; https://doi.org/10.3390/sym1010064 - 20 Sep 2009
Cited by 36 | Viewed by 10174
Abstract
In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting [...] Read more.
In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics – random matrix theory – provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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<p>Molecules in a box</p> Full article ">Figure 2
<p>An energy-level diagram showing the location of highly-excited resonances in the compound nucleus formed by the interaction of a neutron, n, with a nucleus of mass number A. Nature provides us with a narrow energy region in which the resonances are clearly separated, and are observable.</p> Full article ">Figure 3
<p>High resolution studies of the total neutron cross section of <math display="inline"> <semantics> <mrow> <msup> <mrow/> <mn>238</mn> </msup> <mi mathvariant="normal">U</mi> </mrow> </semantics> </math>, in the energy range 400eV - 1800eV (12). The vertical scale (in units of "barns") is a measure of the effective area of the target nucleus.</p> Full article ">Figure 4
<p>A Wigner distribution fitted to the spacing distribution of 932 s-wave resonances in the interaction <math display="inline"> <semantics> <mrow> <msup> <mrow/> <mn>238</mn> </msup> <mi mathvariant="normal">U</mi> <mo>+</mo> <mi mathvariant="normal">n</mi> </mrow> </semantics> </math> at energies up to 20 keV.</p> Full article ">Figure 5
<p>Odlyzko’s test of the Montgomery conjecture, involving 70 million Riemann zeros near <math display="inline"> <semantics> <msup> <mn>10</mn> <mn>20</mn> </msup> </semantics> </math>.</p> Full article ">Figure 6
<p>A histogram plot of the normalized eigenvalues for 500 matrices, each <math display="inline"> <semantics> <mrow> <mn>400</mn> <mo>×</mo> <mn>400</mn> </mrow> </semantics> </math>. The entries are chosen independently from the standard normal <math display="inline"> <semantics> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mn>2</mn> <mi>π</mi> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> <mo form="prefix">exp</mo> <mrow> <mo>(</mo> <mo>−</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math>.</p> Full article ">Figure 7
<p>A histogram plot of the normalized eigenvalues for 500 matrices, each <math display="inline"> <semantics> <mrow> <mn>400</mn> <mo>×</mo> <mn>400</mn> </mrow> </semantics> </math>. The entries are drawn from the Cauchy distribution <math display="inline"> <semantics> <msup> <mrow> <mo>(</mo> <mi>π</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </semantics> </math>. The bin on the extreme right represents all normalized eigenvalues that larger or large (and similarly for the bin on the extreme left).</p> Full article ">Figure 8
<p>Spacings between normalized eigenvalues of 5000 uniform matrices on <math display="inline"> <semantics> <mrow> <mo>[</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics> </math> (size is 300).</p> Full article ">Figure 9
<p>Spacings between normalized eigenvalues of 5000 Cauchy matrices; the left are <math display="inline"> <semantics> <mrow> <mn>100</mn> <mo>×</mo> <mn>100</mn> </mrow> </semantics> </math> and the right are <math display="inline"> <semantics> <mrow> <mn>300</mn> <mo>×</mo> <mn>300</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 10
<p>70 million spacings between adjacent zeros of <math display="inline"> <semantics> <mrow> <mi>ζ</mi> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics> </math>, starting at the <math display="inline"> <semantics> <msup> <mn>10</mn> <mrow> <mn>20</mn> <mi>th</mi> </mrow> </msup> </semantics> </math> zero, versus the corresponding results for eigenvalues of complex Hermitian matrices (from Odlyzko).</p> Full article ">
149 KiB  
Article
A Stochastic Poisson Structure
by Rémi Léandre
Symmetry 2009, 1(1), 55-63; https://doi.org/10.3390/sym1010055 - 20 Aug 2009
Cited by 5 | Viewed by 5829
Abstract
We define a Poisson structure on the Nualart-Pardoux test algebra associated to the path space of a finite dimensional Lie algebra. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
926 KiB  
Review
Symmetry-Break in Voronoi Tessellations
by Valerio Lucarini
Symmetry 2009, 1(1), 21-54; https://doi.org/10.3390/sym1010021 - 20 Aug 2009
Cited by 18 | Viewed by 8821
Abstract
We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of [...] Read more.
We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomaly in the scaling indicates that large cells preferentially feature large isoperimetric quotients. As the number of faces is strongly correlated with the sphericity (cells with more faces are bulkier), in 3D it is shown that the anomalous scaling is heavily reduced when we perform power law fits separately on cells with a specific number of faces. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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Figure 1

Figure 1
<p>Ensemble mean of the mean - (a) - and of the standard deviation - (b) - of the number of sides (n) of the Voronoi cells. Note that in (a) the number of sides of all cells is 4 (3) - out of scale - for α=0 in the case of regular square (triangular) tessellation. Half-width of the error bars is twice the standard deviation computed over the ensemble. Poisson-Voronoi limit is indicated.</p> Full article ">Figure 2
<p>Ensemble mean of the mean - (a) - and of the standard deviation - (b) - of the area (A) of the Voronoi cells. Half-width of the error bars is twice the standard deviation computed over the ensemble. Poisson-Voronoi limit is indicated. In (b), linear approximation for small values of α is also shown. Values are multiplied times the appropriate power of the density in order to obtain universal functions.</p> Full article ">Figure 2 Cont.
<p>Ensemble mean of the mean - (a) - and of the standard deviation - (b) - of the area (A) of the Voronoi cells. Half-width of the error bars is twice the standard deviation computed over the ensemble. Poisson-Voronoi limit is indicated. In (b), linear approximation for small values of α is also shown. Values are multiplied times the appropriate power of the density in order to obtain universal functions.</p> Full article ">Figure 3
<p>Ensemble mean of the mean - (a) - and of the standard deviation - (b) - of the perimeter (P) of the Voronoi cells. Half-width of the error bars is twice the standard deviation computed over the ensemble. Poisson-Voronoi limit is indicated. In (b), linear approximation for small values of α is also shown. Values are multiplied times the appropriate power of the density in order to obtain universal functions.</p> Full article ">Figure 4
<p>Ensemble mean of A - (a) - and of P - (b) - of n-sided Voronoi cells. Half-width of the error bars is twice the ensemble standard deviation. Full ensemble mean is indicated. Linear (a) and square root (b) fits of the Poisson-Voronoi limit results as a function of n is shown. Values are multiplied times the appropriate power of the density in order to obtain universal functions.</p> Full article ">Figure 5
<p>Joint distribution of the perimeter and of the area of the Voronoi cells in the Poisson-Voronoi tessellation limit in 2D. The black solid line indicates the best log-log least squares fit, with ensemble mean of the exponent <math display="inline"> <semantics> <mrow> <mrow> <mo>〈</mo> <mi>η</mi> <mo>〉</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>+</mo> <mrow> <mo>〈</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>〉</mo> </mrow> <mo>=</mo> <mn>2.17</mn> </mrow> </semantics> </math>. The dashed black line reports the fit of isoperimetric quotient (see right vertical axis), which scales with the area with exponent <math display="inline"> <semantics> <mrow> <mrow> <mo>〈</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>〉</mo> </mrow> <mo>=</mo> <mn>0.17</mn> </mrow> </semantics> </math>. The effective range of applicability of the scaling law is between 1.5 and 6 in units of normalized area. Correspondingly, q ranges between 0.65 and 0.78, and ε between 0.052 and 0.062. Details in the text.</p> Full article ">Figure 6
<p>Ensemble mean of the scaling exponent fitting the power-law relation <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>∝</mo> <msup> <mi>P</mi> <mi>η</mi> </msup> </mrow> </semantics> </math> for the perturbed square, hexagonal and triangular Voronoi tessellations. The anomalous scaling (<math display="inline"> <semantics> <mrow> <mi>η</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </semantics> </math>) is apparent. The error bars, whose half-width is twice the ensemble standard deviation, are too small to be plotted. The Poisson-Voronoi limit (see <a href="#symmetry-01-00021-f005" class="html-fig">Figure 5</a>) is indicated. Details in the text.</p> Full article ">Figure 7
<p>Ensemble mean of the mean and of the standard deviation of the number of faces of the Voronoi cells for perturbed SC, BCC and FCC cubic crystals. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated.</p> Full article ">Figure 8
<p>Ensemble mean of the standard deviation of the volume (V) of the Voronoi cells for perturbed SC, BCC and FCC cubic crystal. The ensemble mean of the mean is set to the inverse of the density. Values are multiplied times the appropriate power of the density in order to obtain universal functions. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated.</p> Full article ">Figure 9
<p>Ensemble mean of the mean and of the standard deviation of the area (A) of the Voronoi cells for perturbed SC, BCC and FCC cubic crystals. Values are multiplied times the appropriate power of the density in order to obtain universal functions. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated.</p> Full article ">Figure 10
<p>Ensemble mean of the mean and of the standard deviation (see the different scales) of the isoperimetric quotient <math display="inline"> <semantics> <mrow> <mi>Q</mi> <mo>=</mo> <mn>36</mn> <mi>π</mi> <mrow> <mrow> <msup> <mi>V</mi> <mn>2</mn> </msup> </mrow> <mo>/</mo> <mrow> <msup> <mi>A</mi> <mn>3</mn> </msup> </mrow> </mrow> </mrow> </semantics> </math> of the Voronoi cells for perturbed SC, BCC and FCC cubic crystals The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit is indicated. Details in the text.</p> Full article ">Figure 11
<p>Ensemble mean of the isoperimetric quotient of the Voronoi cells for perturbed SC (a), BCC (b) and FCC(c) cubic crystals, where averages are taken over cells having f faces. A white shading indicates that the corresponding ensemble is empty. More faceted cells are typically bulkier.</p> Full article ">Figure 12
<p>Joint distribution of the area and of the volume of the Voronoi cells in the Poisson-Voronoi tessellation limit. The black solid line indicates the best log-log least squares fit, with ensemble mean of the exponent <math display="inline"> <semantics> <mrow> <mrow> <mo>〈</mo> <mi>η</mi> <mo>〉</mo> </mrow> <mo>=</mo> <mrow> <mn>3</mn> <mo>/</mo> <mrow> <mn>2</mn> <mo>+</mo> </mrow> </mrow> <mrow> <mo>〈</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>〉</mo> </mrow> <mo>=</mo> <mn>1.67</mn> </mrow> </semantics> </math>. The dashed black line reports the corresponding fit of isoperimetric quotient (see right vertical axis), which scales with the area with exponent <math display="inline"> <semantics> <mrow> <mn>2</mn> <mrow> <mo>〈</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>〉</mo> </mrow> <mo>=</mo> <mn>0.34</mn> </mrow> </semantics> </math>. The effective range of applicability of the scaling law is between 2 and 10 in units of normalized area. Correspondingly, Q ranges between 0.35 and 0.65, and ε between 0.056 and 0.076. Details in the text.</p> Full article ">Figure 13
<p>Ensemble mean of the scaling exponent <math display="inline"> <semantics> <mi>η</mi> </semantics> </math> fitting the power-law relation <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>∝</mo> <msup> <mi>A</mi> <mi>η</mi> </msup> </mrow> </semantics> </math> for the Voronoi cells of perturbed SC, BCC and FCC cubic crystals. The presence of an anomalous scaling (<math display="inline"> <semantics> <mrow> <mrow> <mo>〈</mo> <mi>η</mi> <mo>〉</mo> </mrow> <mo>&gt;</mo> <mrow> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </mrow> </semantics> </math>) due to the fluctuations in the shape of the cells is apparent. The error bars, whose half-width is twice the standard deviation computed over the ensemble, are too small to be plotted. The Poisson-Voronoi limit (see <a href="#symmetry-01-00021-f012" class="html-fig">Figure 12</a>) is indicated. Details in the text.</p> Full article ">
190 KiB  
Article
Using Group Theory to Obtain Eigenvalues of Nonsymmetric Systems by Symmetry Averaging
by Marion L. Ellzey, Jr.
Symmetry 2009, 1(1), 10-20; https://doi.org/10.3390/sym1010010 - 6 Aug 2009
Cited by 1 | Viewed by 6502
Abstract
If the Hamiltonian in the time independent Schrödinger equation, HΨ = EΨ, is invariant under a group of symmetry transformations, the theory of group representations can help obtain the eigenvalues and eigenvectors of H. A finite group that is not [...] Read more.
If the Hamiltonian in the time independent Schrödinger equation, HΨ = EΨ, is invariant under a group of symmetry transformations, the theory of group representations can help obtain the eigenvalues and eigenvectors of H. A finite group that is not a symmetry group of H is nevertheless a symmetry group of an operator Hsym projected from H by the process of symmetry averaging. In this case H = Hsym + HR where HR is the nonsymmetric remainder. Depending on the nature of the remainder, the solutions for the full operator may be obtained by perturbation theory. It is shown here that when H is represented as a matrix [H] over a basis symmetry adapted to the group, the reduced matrix elements of [Hsym] are simple averages of certain elements of [H], providing a substantial enhancement in computational efficiency. A series of examples are given for the smallest molecular graphs. The first is a two vertex graph corresponding to a heteronuclear diatomic molecule. The symmetrized component then corresponds to a homonuclear system. A three vertex system is symmetry averaged in the first case to Cs and in the second case to the nonabelian C3v. These examples illustrate key aspects of the symmetry-averaging process. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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<p></p> Full article ">Figure 2
<p></p> Full article ">
180 KiB  
Commentary
Symmetry at the Foundation of Science and Nature
by Joe Rosen
Symmetry 2009, 1(1), 3-9; https://doi.org/10.3390/sym1010003 - 5 Jun 2009
Cited by 6 | Viewed by 8449
Abstract
This article demonstrates that science is founded on symmetry and that Nature must have symmetry at its foundation. Full details are given in the book: Rosen, J. Symmetry Rules: How Science and Nature Are Founded on Symmetry; Springer-Verlag: Berlin, Germany, 2008. Full article
164 KiB  
Editorial
Symmetry – An International and Interdisciplinary Scientific Open Access Journal
by Shu-Kun Lin
Symmetry 2009, 1(1), 1-2; https://doi.org/10.3390/sym1010001 - 5 Jun 2009
Cited by 2 | Viewed by 5934
Abstract
As the publisher of MDPI journals, I am pleased to launch Symmetry (ISSN 2073-8994), an international and interdisciplinary open access scientific journal. Twenty years ago, a journal entitled Symmetry – An Interdisciplinary and International Journal was launched by VCH Publishers, Inc. in New [...] Read more.
As the publisher of MDPI journals, I am pleased to launch Symmetry (ISSN 2073-8994), an international and interdisciplinary open access scientific journal. Twenty years ago, a journal entitled Symmetry – An Interdisciplinary and International Journal was launched by VCH Publishers, Inc. in New York, with Professor Istvan Hargittai as Editor-in-Chief. I submitted a paper which was processed by Professor Sven J. Cyvin from The University of Trondheim – The Norwegian Institute of Technology. The paper was accepted and scheduled for publication in the printed issue 4 of volume 1, 1990. I still keep a copy of the galley proofs. However, the publication of this journal was terminated after just the release of the first issue of volume 1, and this paper was finally published elsewhere [1]. [...] Full article
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