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This paper presents a survey of advanced concepts and research perspectives, of a philosophical-mathematical approach to describe the dynamics of systems of many interacting living entities. The first part introduces the general conceptual framework. Then, a critical analysis of the existing literature is developed and referred to a multiscale view of a mathematics of living organisms. This paper attempts to understand how far the present state-of-the-art is far from the achievement of such challenging objective. The overall study leads to identify research perspectives and possible hints to deal with them. [ABSTRACT FROM AUTHOR]

The study of symmetry is a fascinating and unifying subject that connects various areas of mathematics in the twenty-first century. Algebraic structures offer a framework for comprehending the symmetries of geometric objects in pure mathematics. This paper introduces new concepts in algebraic structures, concentrating on semimodules over semirings and analysing the neutrosophic structure in this context. We explore the properties of neutrosophic subsemimodules and neutrosophic ideals after defining them. We discuss, utilizing neutrosophic products, the representations of neutrosophic ideals and subsemimodules, as well as the relationship between neutrosophic products and intersections. Finally, we derive equivalent criteria in terms of neutrosophic structures for a semiring to be fully idempotent. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of symmetric difference operators in terms of overlap and grouping functions, for which the associativity property is not strongly required. These symmetric difference operators are weaker than symmetric difference operators in terms of positive and continuous t-norms and t-conorms. Therefore, in the sense of the characters of mathematics, these operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. We analyze several related important properties based on two models of symmetric differences. [ABSTRACT FROM AUTHOR]

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR - γ -convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen's inequalities using CR - γ -convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard's, Hermite-Hadamard-Fejer, and Pachpatte's type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations. [ABSTRACT FROM AUTHOR]

In order to find solutions to math word problems, some modules have been designed to check the generated expressions, but they neither take into account the symmetry between math word problems and their corresponding mathematical expressions, nor do they utilize the efficiency of pretrained language models in natural language understanding tasks. Anyway, designing fine-tuning tasks for pretrained language models that encourage cooperation with other modules to improve the performance of math word problem solvers is an unaddressed problem. To solve these problems, in this paper we propose a BERT-based model for solving math word problems with a supervision module. Based on pretrained language models, we present a fine-tuning task to predict the number of different operators in the expressions to learn the potential relationships between the problems and the expressions. Meanwhile, a supervision module is designed to check the incorrect expressions generated and improve the model's performance by optimizing the encoder. A series of experiments are conducted on three datasets, and the experimental results demonstrate the effectiveness of our model and its component's designs. [ABSTRACT FROM AUTHOR]

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths P s 1 , P s 2 , P s 3 into a vertex, and three end vertices into a vertex on the cycle C a 1 , C a 2 , C a 3 , respectively, is denoted as γ (a 1 , a 2 , a 3 , s 1 , s 2 , s 3) . Note that δ (a 1 , a 2 , a 3 , s 1 , s 2) = γ (a 1 , a 2 , a 3 , s 1 , 1 , s 2) , ζ (a 1 , a 2 , a 3 , s) = γ (a 1 , a 2 , a 3 , 1 , 1 , s) , φ (a 1 , a 2 , a 3) = γ (a 1 , a 2 , a 3 , 1 , 1 , 1) . In this paper, we give the necessity and sufficiency that the γ − graph, δ − graph, ζ − graph and φ − graph are singular and prove that the probability that a randomly given γ − graph, δ − graph, ζ − graph or φ − graph being singular is equal to 325 512 , 165 256 , 43 64 , 21 32 , respectively. From our main results, we can conclude that such a γ − graph( δ − graph, ζ − graph, φ − graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems. [ABSTRACT FROM AUTHOR]

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object's symmetries. In this paper, we introduce new concepts in an algebraic structure called BCI-algebra, where we present the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. The relationship between bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy ideals is investigated, and various conditions are provided for a bipolar fuzzy ideal to be a bipolar fuzzy BCI-positive implicative ideal. Furthermore, conditions are presented for a bipolar fuzzy (closed) ideal to be a bipolar fuzzy BCI-commutative ideal. [ABSTRACT FROM AUTHOR]

In mathematics, philosophy, cosmology, and theology, the notion of infinity has generated ample debate. Much less discussion has been generated by the notion of finiteness. However, when we consider finitely supported sets, the notion of finiteness becomes more interesting and richer. We present several independent definitions of finite sets within the framework of finitely supported structures, emphasizing the differences between these definitions. [ABSTRACT FROM AUTHOR]

The complexity of a mathematical expression is a measure that can be used to compare the expression with other mathematical expressions and judge which one is simpler. In the paper, we analyze three effect factors for the complexity of a mathematical expression: representational length, computational time, and intelligibility. Mainly, the paper introduces a binary-lambda-calculus based calculation method for representational complexity and a rule based calculation method for algebraic computation complexity. In the process of calculating the representation complexity of mathematical expressions, we transform the de bruijn notation into the binary lambda calculus of mathematical expressions that is inspired by compressing symmetry strings in Kolmogorov complexity theorem. Furthermore, the application of complexity of mathematical expressions in MACP, a mathematics answer checking protocol, is also addressed. MACP can be used in a computer aided assessment system in order to compute correct answers, verify equivalence of expressions, check user answers whether in a simplification form, and give automatic partial grades. [ABSTRACT FROM AUTHOR]

Recently in the paper [Mediterr. J. Math. 2016, 13, 1535–1553], the authors introduced and studied a new operator which was defined as a convolution of the three popular linear operators, namely the Sǎlǎgean operator, the Ruscheweyh operator and a fractional derivative operator. In the present paper, we consider an operator which is a convolution operator of only two linear operators (with lesser restricted parameters) that yield various well-known operators, defined by a symmetric way, including the one studied in the above-mentioned paper. Several results on the subordination of analytic functions to this operator (defined below) are investigated. Some of the results presented are shown to involve the familiar Appell function and Hurwitz–Lerch Zeta function. Special cases and interesting consequences being in symmetry of our main results are also mentioned. [ABSTRACT FROM AUTHOR]

This paper proposes a novel approach to cope with the multi-criteria group decision-making problems. We give the pairwise comparisons based on the best-worst-method (BWM), which can decrease comparison times. Additionally, our comparison results are determined with the positive and negative aspects. In order to deal with the decision matrices effectively, we consider the elimination and choice translation reality (ELECTRE III) method under the intuitionistic multiplicative preference relations environment. The ELECTRE III method is designed for a double-automatic system. Under a certain limitation, without bothering the decision-makers to reevaluate the alternatives, this system can adjust some special elements that have the most influence on the group's satisfaction degree. Moreover, the proposed method is suitable for both the intuitionistic multiplicative preference relation and the interval valued fuzzy preference relations through the transformation formula. An illustrative example is followed to demonstrate the rationality and availability of the novel method. [ABSTRACT FROM AUTHOR]

This study presents a subclass S (β) of bi-univalent functions within the open unit disk region D . The objective of this class is to determine the bounds of the Hankel determinant of order 3, ( Ⱨ 3 (1) ). In this study, new constraints for the estimates of the third Hankel determinant for the class S (β) are presented, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. Here, we define these bi-univalent functions as S (β) and impose constraints on the coefficients │ a n │ . Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n = 2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These findings contribute to the broader understanding of bi-univalent functions and their potential applications in diverse mathematical contexts. Notably, the results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses. [ABSTRACT FROM AUTHOR]

Reinterpretation of mathematics behind the exactly solvable Calogero's A-particle quantum model is used to propose its generalization. Firstly, it is argued that the strongly singular nature of Calogero's particle–particle interactions makes the original permutation-invariant Hamiltonian tractable as a direct sum H = ⨁ H a of isospectral components, which are mutually independent. Secondly, after the elimination of the center-of-mass motion, the system is reconsidered as existing in the reduced Euclidean space R A − 1 of relative coordinates and decaying into a union of subsets W a called Weyl chambers. The mutual independence of the related reduced forms of operators H a enables us to makes them nonisospectral. This breaks the symmetry and unfolds the spectral degeneracy of H. A new multiparametric generalization of the conventional A-body Calogero model is obtained. Its detailed description is provided up to A = 4 . [ABSTRACT FROM AUTHOR]

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the λ -Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s ( s ∈ N ) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities. [ABSTRACT FROM AUTHOR]

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown. [ABSTRACT FROM AUTHOR]

It is the intention of the current study to suggest a trajectory for the advancement of prospective mathematics teachers' use of meta-cognitive skills in solving mathematics-based programming problems with Scratch. Scratch is a code-based program that can be utilized in teaching various disciplines, especially geometry and its rich range of subjects such as the topic of symmetry. The present study suggests that advancing prospective teachers' meta-cognitive skills in the Scratch environment could be done through problem solving and negotiations. The present paper analyzed the implementation of the trajectory by two pedagogic supervisors who attempted, in the frame of one-year preparation (2018–2019), to educate 18 prospective teachers to use meta-cognitive skills in mathematics-based programming activities, where this attempt was based on problem solving and negotiation processes. Data were collected through videoing and recording the learning sessions of the prospective teachers and was analyzed using deductive and inductive constant comparison methods. The deductive analysis utilized theoretical models of meta-cognitive processes and negotiation processes. The research results indicated that the negotiation processes supported the development of the prospective teachers' meta-cognitive processes in solving mathematics-based programming problems with Scratch. [ABSTRACT FROM AUTHOR]

Let ST L (s) and CV L (s) denote the family of analytic and normalized functions f in the unit disk D : = z : | z | < 1 , such that the quantity z f ′ (z) / f (z) or 1 + z f ″ (z) / f ′ (z) respectively are lying in the region bounded by the limaçon (u − 1) 2 + v 2 − s 4 2 = 4 s 2 u − 1 + s 2 2 + v 2 , where 0 < s ≤ 1 / 2 . The limaçon of Pascal is a curve that possesses properties which qualify it for the several applications in mathematics, statistics (hypothesis testing problem) but also in mechanics (fluid processing applications, known limaçon technology is employed to extract electrical power from low-grade heat, etc.). In this paper we present some results concerning the behavior of f on the classes ST L (s) or CV L (s) . Some appropriate examples are given. [ABSTRACT FROM AUTHOR]

The study of symmetry is one of the most important and beautiful themes uniting various areas of contemporary arithmetic. Algebraic structures are useful structures in pure mathematics for learning a geometrical object's symmetries. In order to provide a mathematical tool for dealing with negative information, a negative-valued function came into existence along with N -structures. In the present analysis, the notion of N -structures is applied to the ideals, especially the commutative ideals of BCI-algebras. Firstly, several properties of N -subalgebras and N -ideals in BCI-algebras are investigated. Furthermore, the notion of a commutative N -ideal is defined, and related properties are investigated. In addition, useful characterizations of commutative N -ideals are established. A condition for a closed N -ideal to be a commutative N -ideal is provided. Finally, it is proved that in a commutative BCI-algebra, every closed N -ideal is a commutative N -ideal. [ABSTRACT FROM AUTHOR]

An obituary for visual mathematics expert Slabik Jablan is presented.

In 1984, Wang and Zheng (J. Comput. Math. 1984, 1, 70–76) introduced a new fourth order iterative method for the simultaneous computation of all zeros of a polynomial. In this paper, we present new local and semilocal convergence theorems with error estimates for Wang–Zheng's method. Our results improve the earlier ones due to Wang and Wu (Computing 1987, 38, 75–87) and Petković, Petković, and Rančić (J. Comput. Appl. Math. 2007, 205, 32–52). [ABSTRACT FROM AUTHOR]

To improve the performance of the arithmetic optimization algorithm (AOA) and solve problems in the AOA, a novel improved AOA using a multi-strategy approach is proposed. Firstly, circle chaotic mapping is used to increase the diversity of the population. Secondly, a math optimizer accelerated (MOA) function optimized by means of a composite cycloid is proposed to improve the convergence speed of the algorithm. Meanwhile, the symmetry of the composite cycloid is used to balance the global search ability in the early and late iterations. Thirdly, an optimal mutation strategy combining the sparrow elite mutation approach and Cauchy disturbances is used to increase the ability of individuals to jump out of the local optimal. The Rastrigin function is selected as the reference test function to analyze the effectiveness of the improved strategy. Twenty benchmark test functions, algorithm time complexity, the Wilcoxon rank-sum test, and the CEC2019 test set are selected to test the overall performance of the improved algorithm, and the results are then compared with those of other algorithms. The test results show that the improved algorithm has obvious advantages in terms of both its global search ability and convergence speed. Finally, the improved algorithm is applied to an engineering example to further verify its practicability. [ABSTRACT FROM AUTHOR]

We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide. [ABSTRACT FROM AUTHOR]

Symmetry is a frequently recurring theme in mathematics, nature, science, etc. In mathematics, its most familiar manifestation appears in geometry, most notably line geometry, and in other closely related areas. In this study, we take advantage of the symmetry properties of both dual space and original space in order to transfer problems in original space to dual space. We use E. Study Mappingas a direct method for analyzing the kinematic geometry of timelike ruled and developable surfaces. Then, the invariants for a spacelike line trajectory are studied and the well-known formulae of Hamilton and Mannheim on the theory of surfaces are provenfor the line space. Meanwhile, a timelike Plücker conoid generated by the Disteli-axis is derived and its kinematic geometry is discussed. Finally, some equations for particular timelike ruled surfaces, such as the general timelike helicoid, the Lorentzian sphere, and the timelike cone, are derived and plotted. [ABSTRACT FROM AUTHOR]

In this paper, we bring together two concepts related to uncertainty and vagueness: fuzzy numbers and intervals. With them, we build a new structure whose elements we call interval fuzzy segments. We have undertaken this based on the conviction that the fuzzy numbers are a correct representation of the real numbers under situations of indeterminacy. We also believe that if it makes sense to consider the set of real numbers between two real bounds, then it also makes sense to consider the set of all the fuzzy numbers between two fuzzy number bounds. In this way, we extend the concept of real interval to the concept of interval fuzzy segment defined by two fuzzy bounds and a transition mapping that leads from the lower fuzzy bound to the upper fuzzy bound and this transition mapping generates the set of all the fuzzy numbers comprised between those fuzzy bounds. At the same time, this transition mapping brings the concept of interval fuzzy segment closer to the concept of line segment. [ABSTRACT FROM AUTHOR]

By using some solvability methods and the contraction mapping principle are investigated bounded, as well as periodic solutions to some classes of nonhomogeneous linear second-order difference equations on domains N0, Z n N2 and Z. The case when the coefficients of the equation are constant and the zeros of the characteristic polynomial associated to the corresponding homogeneous equation do not belong to the unit circle is described in detail. [ABSTRACT FROM AUTHOR]

As the Internet of Things (IoT) has developed, the emerging sensor network (ESN) that integrates emerging technologies, such as autonomous driving, cyber-physical systems, mobile nodes, and existing sensor networks has been in the limelight. Smart homes have been researched and developed by various companies and organizations. Emerging sensor networks have some issues of providing secure service according to a new environment, such as a smart home, and the problems of low power and low-computing capacity for the sensor that previous sensor networks were equipped with. This study classifies various sensors used in smart homes into three classes and contains the hierarchical topology for efficient communication. In addition, a scheme for establishing secure communication among sensors based on physical unclonable functions (PUFs) that cannot be physically cloned is suggested in regard to the sensor's low performance. In addition, we analyzed this scheme by conducting security and performance evaluations proving to constitute secure channels while consuming fewer resources. We believe that our scheme can provide secure communication by using fewer resources in a smart home environment in the future. [ABSTRACT FROM AUTHOR]

The role of the environment initial conditions in the breaking of the time reversal symmetry of effective theories and in generating the soft irreversibility is studied by the help of Closed Time Path formalism. The initial conditions break the time reversal symmetry of the solution of the equation of motion in a trivial manner. When open systems are considered then the initial conditions of the environment must be included in the effective dynamics. This is achieved by means of a generalized ε-prescription where the non-uniform convergence of the limit ε →0 leaves behind a spontaneous breakdown of the time reversal symmetry. [ABSTRACT FROM AUTHOR]

We suggest a diagrammatic model of computation based on an axiom of distributivity. A diagram of a decorated colored tangle, similar to those that appear in low dimensional topology, plays the role of a circuit diagram. Equivalent diagrams represent bisimilar computations. We prove that our model of computation is Turing complete and with bounded resources that it can decide any language in complexity class IP, sometimes with better performance parameters than corresponding classical protocols. [ABSTRACT FROM AUTHOR]

Recently, fractional calculus has been the center of attraction for researchers in mathematical sciences because of its basic definitions, properties and applications in tackling real-life problems. The main purpose of this article is to present some fractional integral inequalities of Ostrowski type for a new class of convex mapping. Specifically, n–polynomial exponentially s–convex via fractional operator are established. Additionally, we present a new Hermite–Hadamard fractional integral inequality. Some special cases of the results are discussed as well. Due to the nature of convexity theory, there exists a strong relationship between convexity and symmetry. When working on either of the concepts, it can be applied to the other one as well. Integral inequalities concerned with convexity have a lot of applications in various fields of mathematics in which symmetry has a great part to play. Finally, in applications, some new limits for special means of positive real numbers and midpoint formula are given. These new outcomes yield a few generalizations of the earlier outcomes already published in the literature. [ABSTRACT FROM AUTHOR]

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids R b n 1 , n 2 , m 1 , m 2 , usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of R b n 1 , n 2 , m 1 , m 2 : the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes O k , m , n and oblate rectangles O r m , n . [ABSTRACT FROM AUTHOR]

In standard quantum theory, symmetry is defined in the spirit of Klein's Erlangen Program—the background space has a symmetry group, and the basic operators should commute according to the Lie algebra of that group. We argue that the definition should be the opposite—background space has a direct physical meaning only on classical level while on quantum level symmetry should be defined by a Lie algebra of basic operators. Then the fact that de Sitter symmetry is more general than Poincare symmetry can be proved mathematically. The problem of explaining cosmological acceleration is very difficult but, as follows from our results, there exists a scenario in which the phenomenon of cosmological acceleration can be explained by proceeding from basic principles of quantum theory. The explanation has nothing to do with existence or nonexistence of dark energy and therefore the cosmological constant problem and the dark energy problem do not arise. We consider finite quantum theory (FQT) where states are elements of a space over a finite ring or field with characteristic p and operators of physical quantities act in this space. We prove that, with the same approach to symmetry, FQT and finite mathematics are more general than standard quantum theory and classical mathematics, respectively: the latter theories are special degenerated cases of the former ones in the formal limit p → ∞ . [ABSTRACT FROM AUTHOR]

We perform Lie symmetry analysis to a zero-coupon bond pricing equation whose price evolution is described in terms of a partial differential equation (PDE). As a result, using the computer software package SYM, run in conjunction with Mathematica, a new family of Lie symmetry group and generators of the aforementioned pricing equation are derived. We furthermore compute the exact invariant solutions which constitute the pricing models for the bond by making use of the derived infinitesimal generators and the associated similarity reduction equations. Using known solutions, we again compute more solutions via group point transformations. [ABSTRACT FROM AUTHOR]

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. 41 (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions. [ABSTRACT FROM AUTHOR]