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Symmetry
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Symmetry: Anniversary Feature Papers 2018
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Symmetry: Anniversary Feature Papers 2018

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (30 December 2019) | Viewed by 39103

Special Issue Editor

Prof. Dr. Sergei D. Odintsov

grade E-Mail Website
Guest Editor
1. Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Luis Companys, 23, 08010 Barcelona, Spain
2. Institute of Space Sciences (ICE-CSIC), C. Can Magrans s/n, 08193 Barcelona, Spain
Interests: cosmology; dark energy and inflation; quantum gravity; modified gravity and beyond general relativity; quantum fields at external fields
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Our journal Symmetry celebrates its 10th anniversary. This is Anniversary Feature Papers special issue.

After the success of the Special Issue “Symmetry: Feature Papers 2017” (https://www.mdpi.com/journal/symmetry/special_issues/feature_papers_2017), I am glad to announce the Special Issue “Symmetry: Anniversary Feature Papers 2018” online. In 2017, we cooperated with some excellent scholars/scientific groups and published several very important high-level works which have already been cited according to the data of Web of Science. We aim to introduce a new insight into science development or cutting edge technology related to the symmetry field, which will make a great contribution to the community. Thus, we will continue the Special Issue “Symmetry: Feature Papers” series in 2018. We will strictly select 5–10 papers from excellent scholars around the world to publish for free in order to benefit both authors and readers.

You are welcome to send short proposals for submissions of Feature Papers to our Editorial Office ([email protected]) before submission. They will be evaluated by Editors firstly. Please note that selected full papers will still be subjected to a thorough and rigorous peer review.

Prof. Dr. Sergei D. Odintsov
Editor-in-Chief and Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (10 papers)

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25 pages, 7508 KiB  
Article
Using Reflection Symmetry to Improve the Protection of Radio-Electronic Equipment from Ultrashort Pulses
by Evgeniya B. Chernikova, Anton O. Belousov, Talgat R. Gazizov and Alexander M. Zabolotsky
Symmetry 2019, 11(7), 883; https://doi.org/10.3390/sym11070883 - 5 Jul 2019
Cited by 27 | Viewed by 3150
Abstract
The paper considers the protection of critical radio-electronic equipment (REE) from ultrashort pulses (USP) by means of modal filters (MFs). A new approach to improve modal filtration by using reflection symmetry is analyzed. The results of a sophisticated research into protective devices based [...] Read more.
The paper considers the protection of critical radio-electronic equipment (REE) from ultrashort pulses (USP) by means of modal filters (MFs). A new approach to improve modal filtration by using reflection symmetry is analyzed. The results of a sophisticated research into protective devices based on reflection symmetric MFs are presented: improving the characteristics of four MFs through optimization both by one and simultaneously by several criteria; calculating the per-unit-length time delays matrix of a reflection symmetric MF using the obtained analytical expressions; calculating the time and frequency responses of an MF with and without losses in conductors and dielectric; developing the laboratory evaluation board; analyzing the effect of moisture protection on the characteristics; analyzing the features of reflection symmetry structures; comparing microstrip and reflection symmetric four-conductor MFs. The obtained results allow us to argue that the reflection symmetric MF protects REE from a USP due to its decomposition into a sequence of pulses with pairwise equalized voltage amplitudes and close time intervals between decomposition pulses with an attenuation coefficient of four times with a controlled bandwidth of a useful signal. This research helps take advantage of the possibilities of using the symmetry to improve modal filtering and opens the ways to create a large number of new MF designs, applying only the principles of the symmetry described in the work. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
►▼ Show Figures

Figure 1

Figure 1
<p>Photograph of a flexible printed cable (FPC).</p> Full article ">Figure 2
<p>Cross-sections of (<b>a</b>) symmetric and (<b>b</b>) asymmetric structures. Conductors: P—passive, A—active, R—reference.</p> Full article ">Figure 3
<p>Cross-sections of structures with (<b>a</b>) strong (<b>b</b>) and weak couplings and (<b>c</b>) a connection diagram of coupled lines.</p> Full article ">Figure 4
<p>Cross-section of a two-conductor MF structure with reflection symmetry.</p> Full article ">Figure 5
<p>Cross-section of the reflection symmetric structure.</p> Full article ">Figure 6
<p>Cross-sections of reflection symmetric MF structures: (<b>a</b>) 1; (<b>b</b>) 2; (<b>c</b>) 3; (<b>d</b>) 4.</p> Full article ">Figure 7
<p>(<b>a</b>) Schematic diagram for simulation; (<b>b</b>) the forms of the exciting EMF (––) and the voltage on the MF input (- -) when simulating the time response.</p> Full article ">Figure 8
<p>Voltage waveforms at the near and far ends of the active conductor of the structure from <a href="#symmetry-11-00883-f003" class="html-fig">Figure 3</a>b obtained with rare (<b>─</b>) and fine (––) segmentations in CST MWS and in TALGAT (– –) [<a href="#B33-symmetry-11-00883" class="html-bibr">33</a>].</p> Full article ">Figure 9
<p>Waveforms at the output of the structures: 1 (––); 2 (– –); 3 (· ·) and 4 (–·–) after optimization by criteria (<b>a</b>) 1 and (<b>b</b>) 2.</p> Full article ">Figure 10
<p>Voltage waveforms at the output of the structures 1 (––), 2 (– –), 3 (· ·) and 4 (–·–) after multicriteria optimization.</p> Full article ">Figure 11
<p>The voltage waveforms at the reflection symmetric MF output with quasi-static (––) and electrodynamic (– –) approaches (<b>a</b>) without and (<b>b</b>) with considering the losses.</p> Full article ">Figure 12
<p>Frequency dependences of |S<sub>21</sub>| for the reflection symmetric MF calculated by quasi-static (––) and electrodynamic (– –) approaches (<b>a</b>) without and (<b>b</b>) with taking into account the losses.</p> Full article ">Figure 13
<p>CP which provides connection to the reference conductor.</p> Full article ">Figure 14
<p>Type 0603 resistor.</p> Full article ">Figure 15
<p>SMA connector.</p> Full article ">Figure 16
<p>Top layer layout of the reflection symmetric MF.</p> Full article ">Figure 17
<p>The cross-section of the reflection symmetric MF covered with a single layer of lacquer.</p> Full article ">Figure 18
<p>Waveforms at the reflection symmetric MF output for <span class="html-italic">h<sub>o</sub></span> = 0 (<b>––</b>), 20 (––), 40 (<b>- -</b>), 60 (– –) μm.</p> Full article ">Figure 19
<p>Cross-section of the four-conductor microstrip MF.</p> Full article ">
19 pages, 302 KiB  
Article
Approximate Solutions and Symmetry of a Two-Component Nonlocal Reaction-Diffusion Population Model of the Fisher–KPP Type
by Alexander V. Shapovalov and Andrey Yu. Trifonov
Symmetry 2019, 11(3), 366; https://doi.org/10.3390/sym11030366 - 12 Mar 2019
Cited by 6 | Viewed by 2238
Abstract
We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the [...] Read more.
We propose an approximate analytical approach to a ( 1 + 1 ) dimensional two-component system consisting of a nonlocal generalization of the well-known Fisher–Kolmogorov–Petrovskii– Piskunov (KPP) population equation and a diffusion equation for the density of the active substance solution surrounding the population. Both equations of the system have terms that describe the interaction effects between the population and the active substance. The first order perturbation theory is applied to the system assuming that the interaction parameter is small. The Wentzel–Kramers–Brillouin (WKB)–Maslov semiclassical approximation is applied to the generalized nonlocal Fisher–KPP equation with the diffusion parameter assumed to be small, which corresponds to population dynamics under certain conditions. In the framework of the approach proposed, we consider symmetry operators which can be used to construct families of special approximate solutions to the system of model equations, and the procedure for constructing the solutions is illustrated by an example. The approximate solutions are discussed in the context of the released activity effect variously debated in the literature. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
13 pages, 319 KiB  
Article
Relativistic Neutron Stars: Rheological Type Extensions of the Equations of State
by Alexander Balakin, Alexei Ilin, Anna Kotanjyan and Levon Grigoryan
Symmetry 2019, 11(2), 189; https://doi.org/10.3390/sym11020189 - 8 Feb 2019
Viewed by 2448
Abstract
Based on the Rheological Paradigm, we extend the equations of state for relativistic spherically symmetric static neutron stars, taking into consideration the derivative of the matter pressure along the so-called director four-vector. The modified equations of state are applied to the model of [...] Read more.
Based on the Rheological Paradigm, we extend the equations of state for relativistic spherically symmetric static neutron stars, taking into consideration the derivative of the matter pressure along the so-called director four-vector. The modified equations of state are applied to the model of a zero-temperature neutron condensate. This model includes one new parameter with the dimensionality of length, which describes the rheological type screening inside the neutron star. As an illustration of the new approach, we consider the rheological type generalization of the non-relativistic Lane–Emden theory and find numerically the profiles of the pressure for a number of values of the new guiding parameter. We have found that the rheological type self-interaction makes the neutron star more compact, since the radius of the star, related to the first null of the pressure profile, decreases when the modulus of the rheological type guiding parameter grows. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
►▼ Show Figures

Figure 1

Figure 1
<p>Plot of the function <math display="inline"><semantics> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi mathvariant="sans-serif">Γ</mi> <mo>*</mo> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>P</mi> <mo>*</mo> </msub> </mrow> </semantics></math> in the domain of the first nulls; this plot depicts the profiles of the reduced pressure as the function of the dimensionless rheological parameter <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mo>*</mo> </msub> </semantics></math>. For negative <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mo>*</mo> </msub> </semantics></math>, the radius of the star, predetermined by the condition <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, becomes smaller than the radius predicted by the Lane–Emden theory; for positive <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mo>*</mo> </msub> </semantics></math> there are no roots of the function <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi mathvariant="sans-serif">Γ</mi> <mo>*</mo> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">
33 pages, 1626 KiB  
Article
Information Security Methods—Modern Research Directions
by Alexander Shelupanov, Oleg Evsyutin, Anton Konev, Evgeniy Kostyuchenko, Dmitry Kruchinin and Dmitry Nikiforov
Symmetry 2019, 11(2), 150; https://doi.org/10.3390/sym11020150 - 29 Jan 2019
Cited by 22 | Viewed by 7639
Abstract
In Tomsk University of Control Systems and Radioelectronics (TUSUR) one of the main areas of research is information security. The work is carried out by a scientific group under the guidance of Professor Shelupanov. One of the directions is the development of a [...] Read more.
In Tomsk University of Control Systems and Radioelectronics (TUSUR) one of the main areas of research is information security. The work is carried out by a scientific group under the guidance of Professor Shelupanov. One of the directions is the development of a comprehensive approach to assessing the security of the information systems. This direction includes the construction of an information security threats model and a protection system model, which allow to compile a complete list of threats and methods of protection against them. The main directions of information security tools development are dynamic methods of biometrics, methods for generating prime numbers for data encryption, steganography, methods and means of data protection in Internet of Things (IoT) systems. The article presents the main results of research in the listed areas of information security. The resultant properties in symmetric cryptography are based on the properties of the power of the generating functions. The authors have obtained symmetric principles for the development of primality testing algorithms, as discussed in the Appendix. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
►▼ Show Figures

Figure 1

Figure 1
<p>Technique for the formation of a recommended list of information security tools.</p> Full article ">Figure 2
<p>Decomposition of the technique for the formation of a recommended list of information security tools.</p> Full article ">Figure 3
<p>Document flow model.</p> Full article ">Figure 4
<p>Elementary document flow.</p> Full article ">Figure 5
<p>Automatic Electric Meter Reading (AMR) system structure.</p> Full article ">
45 pages, 1336 KiB  
Article
The Ambiguity in the Definition and Behavior of the Gravitational and Cosmological ‘Coupling Constants’ in the Theory of Induced Gravity
by Farkhat Zaripov
Symmetry 2019, 11(1), 81; https://doi.org/10.3390/sym11010081 - 12 Jan 2019
Cited by 3 | Viewed by 3464
Abstract
This work is the extension of author’s research, where the modified theory of induced gravity (MTIG) is proposed. The theory describes two systems (stages): Einstein (ES) and “restructuring” (RS). We consider equations with quadratic potential that are symmetric with respect to scale transformations. [...] Read more.
This work is the extension of author’s research, where the modified theory of induced gravity (MTIG) is proposed. The theory describes two systems (stages): Einstein (ES) and “restructuring” (RS). We consider equations with quadratic potential that are symmetric with respect to scale transformations. The solutions of the equations obtained for the case of spaces defined by the Friedman-Robertson-Walker metric, as well as for a centrally symmetric space are investigated. In our model arise effective gravitational and cosmological “constants”, which are defined by the “mean square” of the scalar fields. In obtained solutions the values of such parameters as “Hubble parameter”, gravitational and cosmological “constants” in the RS stage fluctuate near monotonically evolving mean values. These parameters are matched with observational data, described as phenomena of dark energy and dark matter. The MTIG equations for the case of a centrally symmetric gravitational field, in addition to the Schwarzschild-de Sitter solutions, contain solutions that lead to the new physical effects at large distances from the center. The Schwarzschild-Sitter solution becomes unstable and enters the oscillatory regime. For distances greater than a certain critical value, the following effects can appear: deviation from General relativity and Newton’s law of gravitational interaction, antigravity. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
►▼ Show Figures

Figure 1

Figure 1
<p>Oscillatory solutions for flat cosmological model. <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>t</mi> <mo>/</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>144.517</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.2</mn> </mrow> </semantics></math>. Border conditions: (<a href="#FD76-symmetry-11-00081" class="html-disp-formula">76</a>), <math display="inline"><semantics> <mrow> <msub> <mi>Z</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1.0040965</mn> <mo>.</mo> </mrow> </semantics></math></p> Full article ">Figure 2
<p>The dependence comparison <math display="inline"><semantics> <msub> <mi>g</mi> <mn>00</mn> </msub> </semantics></math> (the metric component) for the centrally symmetric space with the Schwarzschild de Sitter’s solution (dashed line); r is given in au. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.0059986</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <msub> <mi>k</mi> <mn>1</mn> </msub> </semantics></math> = −9.263854653 · 10<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>31</mn> </mrow> </msup> </semantics></math> au<math display="inline"><semantics> <msup> <mrow/> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>M</mi> <mo>=</mo> <mi>G</mi> <msub> <mi>M</mi> <mo>⨀</mo> </msub> </mrow> </semantics></math>. Border conditions: (<a href="#FD97-symmetry-11-00081" class="html-disp-formula">97</a>).</p> Full article ">Figure 3
<p>Plot of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mi>Y</mi> <mo>/</mo> <msub> <mi>Y</mi> <mn>0</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>—current age of the universe. After reaching the point A, the function Y (t) branches. It may evolve on the straight line <span class="html-italic">d</span> or the curve <math display="inline"><semantics> <msub> <mi>d</mi> <mn>1</mn> </msub> </semantics></math>.</p> Full article ">Figure 4
<p>Plot of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mi>Y</mi> <mo>/</mo> <msub> <mi>Y</mi> <mn>0</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>—current age of the universe. Double transition: from RS into ES and back from ES into RS.</p> Full article ">Figure 5
<p>Relation of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>Z</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1.40908675530186</mn> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>t</mi> <mo>/</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math>—current age of the Universe, <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>1.44517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Relation of <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>—the current value of the scale factor. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>1.44517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Relation of <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mover accent="true"> <mi>b</mi> <mo>˙</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>—the current value of the Hubble constant. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>1.44517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Relation of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mi>Y</mi> <mo>/</mo> <msub> <mi>Y</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>Z</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1.040908675530</mn> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>t</mi> <mo>/</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math>—current age of the Universe. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>14.4517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Relation of <math display="inline"><semantics> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>—the current value of the scale factor. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>14.4517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>Relation of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mi>Y</mi> <mo>/</mo> <msub> <mi>Y</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>Z</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1.04072460</mn> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>t</mi> <mo>/</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math>—current age of the Universe. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>14.4517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Relation of <math display="inline"><semantics> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>b</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>—the current value of the scale factor. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>14.4517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Relation of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mi>Y</mi> <mo>/</mo> <msub> <mi>Y</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>Z</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1.0000407246</mn> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>t</mi> <mo>/</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math>—current age of the Universe. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>14517.022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>Relation of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mi>Y</mi> <mo>/</mo> <msub> <mi>Y</mi> <mn>0</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>Z</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1.0041022218</mn> <mo>…</mo> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>t</mi> <mo>/</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math>—current age of the Universe. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>144.517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>Relation of <math display="inline"><semantics> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>Z</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>1.0041022218</mn> <mo>…</mo> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mi>t</mi> <mo>/</mo> <msub> <mi>t</mi> <mi>m</mi> </msub> </mrow> </semantics></math>—current age of the Universe. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>144.517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 15
<p>Relation of <math display="inline"><semantics> <mrow> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mover accent="true"> <mi>b</mi> <mo>˙</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>: <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>—the current value of the Hubble constant. <math display="inline"><semantics> <mrow> <msub> <mover accent="true"> <mi>B</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> </msub> <mo>=</mo> <mn>144.517022939</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 16
<p>Graph of <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>. r is measured in astronomical units. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.0059986</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>9.263854653</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>31</mn> </mrow> </msup> <mi>a</mi> <msup> <mi>u</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt"/> <mi>G</mi> <mi>M</mi> <mo>=</mo> <mi>G</mi> <msub> <mi>M</mi> <mo>⨀</mo> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 17
<p>The “additional acceleration” <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> graph. r is measured in astronomical units. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.0059986</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>9.263854653</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>31</mn> </mrow> </msup> <mi>a</mi> <msup> <mi>u</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt"/> <mi>G</mi> <mi>M</mi> <mo>=</mo> <mi>G</mi> <msub> <mi>M</mi> <mo>⨀</mo> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 18
<p>The graph of the function <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>. r is measured in astronomical units. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.0059986</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>9.263854653</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>31</mn> </mrow> </msup> <mi>a</mi> <msup> <mi>u</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt"/> <mi>G</mi> <mi>M</mi> <mo>=</mo> <mi>G</mi> <msub> <mi>M</mi> <mo>⨀</mo> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 19
<p>The deviation graph of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>ν</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>. r in au. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.0059986</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>9.263854653</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>31</mn> </mrow> </msup> <mi>a</mi> <msup> <mi>u</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt"/> <mi>G</mi> <mi>M</mi> <mo>=</mo> <mi>G</mi> <msub> <mi>M</mi> <mo>⨀</mo> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 20
<p>A comparison of the numerical solution <math display="inline"><semantics> <msup> <mi>g</mi> <mn>11</mn> </msup> </semantics></math> (solid line) with the Schwarzschild-de Sitter solution (dashed line); r in au.</p> Full article ">Figure 21
<p>This figure shows the acceleration vs. distance; r—in astronomical units.</p> Full article ">Figure 22
<p>This figure shows the “circular” orbit velocity vs. distance; r—in astronomical units.</p> Full article ">Figure 23
<p>The “gravity” (“in”) and “antigravity” (“out”) bands formation scheme with different directions of acceleration in relation to the center.</p> Full article ">Figure 24
<p>“Observed circular rotational velocities” vs. distance circled in pencil, signs-(bottom) mean “antigravity”; <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>100</mn> <mo>,</mo> <mn>700</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>a</mi> <mi>u</mi> </mrow> </semantics></math>.</p> Full article ">Figure 25
<p>The field Z deviation vs. distance. r is measured in kiloparsec (kpc). <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.2665</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>4.035487823</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt"/> <mi>G</mi> <mi>M</mi> <mo>=</mo> <mi>G</mi> <msub> <mi>M</mi> <mo>⨀</mo> </msub> <mo>·</mo> <msup> <mn>10</mn> <mn>10</mn> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 26
<p>“Observed circular rotational velocities” vs. distance; <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0.5</mn> <mo>,</mo> <mn>25</mn> <mo stretchy="false">)</mo> <mi>k</mi> <mi>p</mi> <mi>c</mi> </mrow> </semantics></math>. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>0.2665</mn> <mo>…</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>4.035487823</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo>,</mo> <mspace width="4pt"/> <mi>G</mi> <mi>M</mi> <mo>=</mo> <mi>G</mi> <msub> <mi>M</mi> <mo>⨀</mo> </msub> <mo>·</mo> <msup> <mn>10</mn> <mn>10</mn> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 27
<p>Image of the Galaxy (Milky Way), seen from above. The distribution of 52 stars.</p> Full article ">Figure 28
<p>A typical spiral galaxies rotation curve, with parameters close to the Milky Way.</p> Full article ">Figure 29
<p><math display="inline"><semantics> <mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>—solutions of Equations (<a href="#FD70-symmetry-11-00081" class="html-disp-formula">70</a>) and (<a href="#FD71-symmetry-11-00081" class="html-disp-formula">71</a>) for time interval that equals to two periods of oscillations <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>t</mi> <mo>=</mo> <mn>0.0000024750</mn> </mrow> </semantics></math> hu.</p> Full article ">Figure 30
<p>A comparison of the numerical solution <math display="inline"><semantics> <msub> <mi>g</mi> <mn>00</mn> </msub> </semantics></math> (solid line) with the Schwarzschild-de Sitter solution (dashed line); r—in the parsecs. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>1.599</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <msup> <mi>c</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>. The solid graph is obtained for the boundary condition: <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 1.00009833247, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 0.</p> Full article ">Figure 31
<p>Test body acceleration <math display="inline"><semantics> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. distance; r—in the parsecs. For the case of solar mass. <math display="inline"><semantics> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mn>1.599</mn> <mo>·</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <msup> <mi>c</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 1.00009833247, <math display="inline"><semantics> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 0, <math display="inline"><semantics> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> = 0.</p> Full article ">
9 pages, 203 KiB  
Article
On Non-Linear Behavior of Viscosity in Low-Concentration Solutions and Aggregate Structures
by Vasilis K. Oikonomou
Symmetry 2018, 10(9), 368; https://doi.org/10.3390/sym10090368 - 31 Aug 2018
Cited by 5 | Viewed by 2278
Abstract
In this paper, an experimental method that may reveal possible aggregate symmetrical structures in highly diluted solutions is proposed, generated by the method of the release activity, which is not yet completely proven. The release activity phenomenon (regardless of whether or not it [...] Read more.
In this paper, an experimental method that may reveal possible aggregate symmetrical structures in highly diluted solutions is proposed, generated by the method of the release activity, which is not yet completely proven. The release activity phenomenon (regardless of whether or not it is real) could be viewed as being quite controversial. However, the focus of this paper is to reveal any possible higher-order, pragmatic, underlying symmetry or structure supporting this theory, by proposing an experiment based on viscosity. Our proposal is based on the sequential measurement of the viscosity of a highly diluted solution and the perturbative expansion of the viscosity as a function of the concentration. The coefficients of this perturbative expansion directly quantify the modification of the hydrodynamic flow around particles and around higher-order structures. Any deviation from a linear dependence of the viscosity, as a function of the concentration, could potentially reveal a collective structure of some sort, or some symmetrical pattern in the solvent. We describe our experimental proposal for non-electrolyte solutes, and future directions for revealing collective structures in solutions are discussed as related to the release activity method. Regardless of whether or not the release activity is pragmatic, it needs to be scrutinized in order to reveal its inner workings. Finally, some theoretical arguments are presented to support the proposal. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
18 pages, 345 KiB  
Article
Inflation in Mimetic f(G) Gravity
by Yi Zhong and Diego Sáez-Chillón Gómez
Symmetry 2018, 10(5), 170; https://doi.org/10.3390/sym10050170 - 17 May 2018
Cited by 37 | Viewed by 3264
Abstract
Mimetic gravity is analysed in the framework of some extensions of general relativity (GR), whereby a function of the Gauss–Bonnet invariant in four dimensions is considered. By assuming the mimetic condition, the conformal degree of freedom is isolated, and a pressureless fluid naturally [...] Read more.
Mimetic gravity is analysed in the framework of some extensions of general relativity (GR), whereby a function of the Gauss–Bonnet invariant in four dimensions is considered. By assuming the mimetic condition, the conformal degree of freedom is isolated, and a pressureless fluid naturally arises. Then, the complete set of field equations for mimetic Gauss–Bonnet gravity is established, and some inflationary models are analysed, for which the corresponding gravitational action is reconstructed. The spectral index and tensor-to-scalar ratio are obtained and compared with observational bounds from Planck and BICEP2/Keck array data. Full agreement with the above data is achieved for several versions of the mimetic Gauss–Bonnet gravity. Finally, some extensions of Gauss–Bonnet mimetic gravity are considered, and the possibility of reproducing inflation is also explored. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
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Figure 1

Figure 1
<p>Confidence region for the parameter values of <math display="inline"><semantics> <mrow> <mi>h</mi> <mo>=</mo> <mfrac> <msub> <mi>G</mi> <mn>1</mn> </msub> <msub> <mi>G</mi> <mn>0</mn> </msub> </mfrac> <mo>+</mo> <mi>N</mi> </mrow> </semantics></math> and <span class="html-italic">b</span> to be consistent with the Planck data and Background Imaging of Cosmic Extragalactic Polarisation (BICEP2)/Keck Array; <span class="html-italic">h</span> and <span class="html-italic">b</span> are constrained inside the blue region.</p> Full article ">Figure 2
<p>Constraints for the parameters <math display="inline"><semantics> <mfrac> <msub> <mi>G</mi> <mn>2</mn> </msub> <msub> <mi>G</mi> <mn>3</mn> </msub> </mfrac> </semantics></math> and <math display="inline"><semantics> <mi>β</mi> </semantics></math>. We have assumed <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> (red). To be consistent with the Planck data and Background Imaging of Cosmic Extragalactic Polarisation (BICEP2)/Keck Array, <math display="inline"><semantics> <mfrac> <msub> <mi>G</mi> <mn>2</mn> </msub> <msub> <mi>G</mi> <mn>3</mn> </msub> </mfrac> </semantics></math> and <math display="inline"><semantics> <mi>β</mi> </semantics></math> are constrained inside the coloured areas.</p> Full article ">
22 pages, 882 KiB  
Article
The Landau-Lifshitz Equation, the NLS, and the Magnetic Rogue Wave as a By-Product of Two Colliding Regular “Positons”
by Artyom V. Yurov and Valerian A. Yurov
Symmetry 2018, 10(4), 82; https://doi.org/10.3390/sym10040082 - 27 Mar 2018
Cited by 6 | Viewed by 3782
Abstract
In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear Schrödinger equation (NLS), and is aimed at resolving [...] Read more.
In this article we present a new method for construction of exact solutions of the Landau-Lifshitz-Gilbert equation (LLG) for ferromagnetic nanowires. The method is based on the established relationship between the LLG and the nonlinear Schrödinger equation (NLS), and is aimed at resolving an old problem: how to produce multiple-rogue wave solutions of NLS using just the Darboux-type transformations. The solutions of this type—known as P-breathers—have been proven to exist by Dubard and Matveev, but their technique heavily relied on using the solutions of yet another nonlinear equation, the Kadomtsev-Petviashvili I equation (KP-I), and its relationship with NLS. We have shown that in fact one doesn’t have to use KP-I but can instead reach the same results just with NLS solutions, but only if they are dressed via the binary Darboux transformation. In particular, our approach allows us to construct all the Dubard-Matveev P-breathers. Furthermore, the new method can lead to some completely new, previously unknown solutions. One particular solution that we have constructed describes two “positon”-like waves, colliding with each other and in the process producing a new, short-lived rogue wave. We called this unusual solution (in which a rogue wave is begotten after the impact of two solitons) the “impacton”. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
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Figure 1

Figure 1
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mrow> </semantics> </math>. The parameter <math display="inline"> <semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 2
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 4
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 5
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>0.25</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 6
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>0.2</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 7
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 8
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 9
<p>A graph of function <math display="inline"> <semantics> <msup> <mfenced separators="" open="|" close="|"> <msup> <mi>u</mi> <mrow> <mo>(</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mn>2</mn> </msup> </semantics> </math> at <math display="inline"> <semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>+</mo> <mn>10</mn> </mrow> </semantics> </math>.</p> Full article ">
26 pages, 367 KiB  
Article
Some Aspects of Nonlinearity and Self-Organization In Biosystems on Examples of Localized Excitations in the DNA Molecule and Generalized Fisher–KPP Model
by A. V. Shapovalov and V. V. Obukhov
Symmetry 2018, 10(3), 53; https://doi.org/10.3390/sym10030053 - 27 Feb 2018
Cited by 11 | Viewed by 3615
Abstract
This review deals with ideas and approaches to nonlinear phenomena, based on different branches of physics and related to biological systems, that focus on how small impacts can significantly change the state of the system at large spatial scales. This problem is very [...] Read more.
This review deals with ideas and approaches to nonlinear phenomena, based on different branches of physics and related to biological systems, that focus on how small impacts can significantly change the state of the system at large spatial scales. This problem is very extensive, and it cannot be fully resolved in this paper. Instead, some selected physical effects are briefly reviewed. We consider sine-Gordon solitons and nonlinear Schrodinger solitons in some models of DNA as examples of self-organization at the molecular level, as well as examine features of their formation and dynamics under the influence of external influences. In addition, the formation of patterns in the generalized Fisher–KPP model is viewed as a simple example of self-organization in a system with nonlocal interaction at the cellular level. Symmetries of model equations are employed to analyze the considered nonlinear phenomena. In this context the possible relations between phenomena considered and released activity effect, which is assessed differently in the literature, are discussed. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
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Figure 1

Figure 1
<p>Relative kink velocity <math display="inline"> <semantics> <mrow> <mi>v</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </semantics> </math> (the solid curve) and the trend velocity <math display="inline"> <semantics> <mrow> <msub> <mi>v</mi> <mi>trend</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> (the open diamonds curve) for <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>: (<b>a</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 2
<p>Trend velocity <math display="inline"> <semantics> <mrow> <msub> <mi>v</mi> <mi>trend</mi> </msub> <mrow> <mo>(</mo> <mi>τ</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> (the dotted curve) and the average relative velocity <math display="inline"> <semantics> <mrow> <mo>〈</mo> <mi>v</mi> <mo>(</mo> <mi>τ</mi> <mo>)</mo> <mo>〉</mo> </mrow> </semantics> </math> (the open diamonds curve) for <math display="inline"> <semantics> <mrow> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.6</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <mn>10</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>Root-mean-square value of the kink momentum for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>D</mi> <mo stretchy="false">˜</mo> </mover> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 4
<p>Average value of the kink momentum for <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mi>γ</mi> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics> </math>, and <math display="inline"> <semantics> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>. (<b>a</b>) The solid curve is for <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; the dashed curve is for <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>. (<b>b</b>) The solid curve is for <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics> </math>; the dashed curve is for <math display="inline"> <semantics> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 5
<p>Graph of the function <math display="inline"> <semantics> <mrow> <msub> <mi>ρ</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> for <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math>, <math display="inline"> <semantics> <mrow> <mi>a</mi> <mo>&gt;</mo> <mi>κ</mi> <msub> <mi>λ</mi> <mn>0</mn> </msub> <msub> <mi>v</mi> <mn>0</mn> </msub> <msub> <mi>β</mi> <mn>00</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 6
<p>Graph of the function <math display="inline"> <semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>t</mi> <mo>,</mo> <mi>s</mi> <mo>)</mo> </mrow> </semantics> </math> on the manifold <math display="inline"> <semantics> <msubsup> <mo>Λ</mo> <mrow> <mi>t</mi> </mrow> <mn>1</mn> </msubsup> </semantics> </math>.</p> Full article ">

Review

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14 pages, 251 KiB  
Review
The Spatial Homeostasis Hypothesis
by Oleg Epstein
Symmetry 2018, 10(4), 103; https://doi.org/10.3390/sym10040103 - 10 Apr 2018
Cited by 59 | Viewed by 5969
Abstract
From studies on the effects of “high dilutions” on organisms, it was found that their administration induces a delicate physiological (molecular and cellular) response. Occasionally, physiological reactions can become atypical (pathological) individual reactions. To resolve this paradox, the spatial homeostasis hypothesis has been [...] Read more.
From studies on the effects of “high dilutions” on organisms, it was found that their administration induces a delicate physiological (molecular and cellular) response. Occasionally, physiological reactions can become atypical (pathological) individual reactions. To resolve this paradox, the spatial homeostasis hypothesis has been proposed. It considers pathological processes as tools used by living systems, in order to retain their spatial integrity (symmetry), allowing them to properly reflect the geometry of the surrounding world and thus, to be a part of the evolutionary process. This article addresses an interdisciplinary subject and is aimed at natural scientists (physicists, chemists, and biologists) as well as philosophers. Full article
(This article belongs to the Special Issue Symmetry: Anniversary Feature Papers 2018)
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