#### Symmetry

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#### Latest articles in this journal

Published: 27 November 2021
by MDPI
Abstract:
Baoding Liu created uncertainty theory to describe the information represented by human language. In turn, Yuhan Liu founded chance theory for modelling phenomena where both uncertainty and randomness are present. The first theory involves an uncertain measure and variable, whereas the second one introduces the notions of a chance measure and an uncertain random variable. Laws of large numbers (LLNs) are important theorems within both theories. In this paper, we prove a law of large numbers (LLN) for uncertain random variables being continuous functions of pairwise independent, identically distributed random variables and regular, independent, identically distributed uncertain variables, which is a generalisation of a previously proved version of LLN, where the independence of random variables was assumed. Moreover, we prove the Marcinkiewicz–Zygmund type LLN in the case of uncertain random variables. The proved version of the Marcinkiewicz–Zygmund type theorem reflects the difference between probability and chance theory. Furthermore, we obtain the Chow type LLN for delayed sums of uncertain random variables and formulate counterparts of the last two theorems for uncertain variables. Finally, we provide illustrative examples of applications of the proved theorems. All the proved theorems can be applied for uncertain random variables being functions of symmetrically or asymmetrically distributed random variables, and symmetrical or asymmetrical uncertain variables. Furthermore, in some special cases, under the assumption of symmetry of the random and uncertain variables, the limits in the first and the third theorem have forms of symmetrical uncertain variables.
Published: 27 November 2021
by MDPI
Abstract:
The paper analyzes a two-stage oligopoly game of semi-collusion in production described by a system with a symmetric structure. We examine the local stability of a Nash equilibrium and the presence of bifurcations. We discover that the model is capable of exhibiting extremely complicated dynamic behaviors.
Published: 27 November 2021
by MDPI
Abstract:
The maximum weighted independent set (MWIS) problem is important since it occurs in various applications, such as facility location, selection of non-overlapping time slots, labeling of digital maps, etc. However, in real-life situations, input parameters within those models are often loosely defined or subject to change. For such reasons, this paper studies robust variants of the MWIS problem. The study is restricted to cases where the involved graph is a tree. Uncertainty of vertex weights is represented by intervals. First, it is observed that the max–min variant of the problem can be solved in linear time. Next, as the most important original contribution, it is proved that the min–max regret variant is NP-hard. Finally, two mutually related approximation algorithms for the min–max regret variant are proposed. The first of them is already known, but adjusted to the considered situation, while the second one is completely new. Both algorithms are analyzed and evaluated experimentally.
Published: 27 November 2021
by MDPI
Abstract:
The target detection of smoke through remote sensing images obtained by means of unmanned aerial vehicles (UAVs) can be effective for monitoring early forest fires. However, smoke targets in UAV images are often small and difficult to detect accurately. In this paper, we use YOLOX-L as a baseline and propose a forest smoke detection network based on the parallel spatial domain attention mechanism and a small-scale transformer feature pyramid network (PDAM–STPNNet). First, to enhance the proportion of small forest fire smoke targets in the dataset, we use component stitching data enhancement to generate small forest fire smoke target images in a scaled collage. Then, to fully extract the texture features of smoke, we propose a parallel spatial domain attention mechanism (PDAM) to consider the local and global textures of smoke with symmetry. Finally, we propose a small-scale transformer feature pyramid network (STPN), which uses the transformer encoder to replace all CSP_2 blocks in turn on top of YOLOX-L’s FPN, effectively improving the model’s ability to extract small-target smoke. We validated the effectiveness of our model with recourse to a home-made dataset, the Wildfire Observers and Smoke Recognition Homepage, and the Bowfire dataset. The experiments show that our method has a better detection capability than previous methods.
Published: 27 November 2021
by MDPI
Abstract:
In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers $\left\{1,2,\cdots ,|V\left(G\right)|\right\}$ in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.
Published: 26 November 2021
by MDPI
Abstract:
The spatial properties of solutions for a class of thermoelastic plate with biharmonic operator were studied. The energy method was used. We constructed an energy expression. A differential inequality which the energy expression was controlled by a second-order differential inequality is deduced. The $Phragm\stackrel{´}{e}n$-$Lindel\stackrel{¨}{o}f$ alternative results of the solutions were obtained by solving the inequality. These results show that the Saint-Venant principle is also valid for the hyperbolic–hyperbolic coupling equations. Our results can been seen as a version of symmetry in inequality for studying the $Phragm\stackrel{´}{e}n$-$Lindel\stackrel{¨}{o}f$ alternative results.
Published: 26 November 2021
by MDPI
Abstract:
One fundamental step towards grasping the global dynamic structure of a population system involves characterizing the convergence behavior (specifically, how to characterize the convergence behavior). This paper focuses on the neutral functional differential equations arising from population dynamics. With the help of monotonicity techniques and functional methods, we analyze the subtle relations of both the $\omega$-limited set and special point. Meanwhile, we prove that every bounded solution converges to a constant vector, as t tends to positive infinity. Our results correlate with the findings from earlier publications, and our proof yields an improved Haddock conjecture.
Published: 26 November 2021
by MDPI
Abstract:
The general dynamic characteristics of the acoustic cavity with multiple partial partitions are presented in this thesis. A theoretical model has been developed for predictions, and several configurations are analyzed. To describe the apertures on the interface of subcavities, the virtual air panel assumption is introduced into the improved Fourier series system. The governing equations of the coupling system are derived by using the energy principle. The results obtained with the proposed model are firstly compared with the numerical calculations based on the finite element method (FEM). Subsequently, a configuration made up from a rigid cavity partitioned by a partial steel panel has been specifically built, and the forced responses of the coupling system have been measured for comparison and model validation. The present results are excellent over most of the studied frequency range. Furthermore, the visualizations of the interior sound intensity field of the acoustic cavity with three partial partitions under different frequencies are researched to illustrate the energy transmission paths and vibro-acoustic coupling mechanism of the complicated system. The obtained results are believed to be helpful in the optimal design of the vibro-acoustic coupling system with optimal sound insulation capacity.
Published: 26 November 2021
by MDPI
Abstract:
This review aims to cover the history and recent developments on cryogenic bolometers for neutrinoless double beta decay (0$\nu$2$\beta$) searches. A 0$\nu$2$\beta$ decay observation would confirm the total lepton charge non-conservation, which is related to a global U(1)${}_{LC}$ symmetry. This discovery would also provide essential information on neutrino masses and nature, opening the door to new physics beyond the Standard Model. The bolometric technology shows good prospects for future ton-scale experiments that aim to fully investigate the inverted ordering region of neutrino masses. The big advantage of bolometers is the high energy resolution and the possibility of particle identification, as well as various methods of additional background rejection. The CUORE experiment has proved the feasibility of ton-scale cryogenic experiments, setting the most stringent limit on ${}^{130}$Te 0$\nu$2$\beta$ decay. Two CUPID demonstrators (CUPID-0 and CUPID-Mo) have set the most stringent limits on ${}^{82}$Se and ${}^{100}$Mo isotopes, respectively, with compatibly low exposures. Several experiments are developing new methods to improve the background in the region of interest with bolometric detectors. CUPID and AMoRE experiments aim to cover the inverted hierarchy region, using scintillating bolometers with hundreds of kg of ${}^{100}$Mo. We review all of these efforts here, with a focus on the different types of radioactive background and the measures put in place to mitigate them.
Published: 26 November 2021
by MDPI
Abstract:
It is shown that the inflationary model is the result of the symmetry of the generalized $F\left(R,T,X,\phi \right)$-cosmological model using the Noether symmetry. It leads to a solution, a particular case of which is Starobinsky’s cosmological model. It is shown that even in the more particular case of cosmological models $F\left(R,X,\phi \right)$ and $F\left(T,X,\phi \right)$ the Monge–Ampère equation is still obtained, one of the solutions including the Starobinsky model. For these models, it is shown that one can obtain both power-law and exponential solutions for the scale factor from the Euler–Lagrange equations. In this case, the scalar field $\phi$ has similar time dependences, exponential and exponential. The resulting form of the Lagrangian of the model allows us to consider it as a model with ${R}^{2}$ or ${X}^{2}$. However, it is also shown that previously less studied models with a non-minimal relationship between R and X are important, as one of the possible models. It is shown that in this case the power-law model can have a limited evolutionary period with a negative value of the kinetic term.