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In this paper, we introduce novel extensions of the reversed Minkowski inequality for various functions defined on time scales. Our approach involves the application of Jensen's and Hölder's inequalities on time scales. Our results encompass the continuous inequalities established by Benaissa as special cases when the time scale T corresponds to the real numbers (when T = R). Additionally, we derive distinct inequalities within the realm of time scale calculus, such as cases T = N and qN for q > 1. These findings represent new and significant contributions for the reader. [ABSTRACT FROM AUTHOR]

Let (V, E) be a finite connected graph. We are concerned about the Chern–Simons Higgs model 0.1 Δ u = λ e u (e u - 1) + f , where Δ is the graph Laplacian, λ is a real number and f is a function on V. When λ > 0 and f = 4 π ∑ i = 1 N δ p i , N ∈ N , p 1 , ⋯ , p N ∈ V , the equation (0.1) was investigated by Huang et al. (Commun Math Phys 377:613–621, 2020) and Hou and Sun (Calc Var 61:139, 2022) via the upper and lower solutions principle. We now consider an arbitrary real number λ and a general function f, whose integral mean is denoted by f ¯ , and prove that when λ f ¯ < 0 , the equation (0.1) has a solution; when λ f ¯ > 0 , there exist two critical numbers Λ ∗ > 0 and Λ ∗ < 0 such that if λ ∈ (Λ ∗ , + ∞) ∪ (- ∞ , Λ ∗) , then (0.1) has at least two solutions, including one local minimum solution; if λ ∈ (0 , Λ ∗) ∪ (Λ ∗ , 0) , then (0.1) has no solution; while if λ = Λ ∗ or Λ ∗ , then (0.1) has at least one solution. Our method is calculating the topological degree and using the relation between the degree and the critical group of a related functional. Similar method is also applied to the Chern–Simons Higgs system, and a partial result for the multiple solutions of the system is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following elliptic system - Δ u = | v | p - 1 v + ϵ (α u + β 1 v) , in Ω , - Δ v = | u | q - 1 u + ϵ (β 2 u + α v) , in Ω , u = v = 0 , on ∂ Ω , where Ω is a smooth bounded domain in R N , N ≥ 3 , ϵ is a small parameter, α , β 1 and β 2 are real numbers, (p, q) is a pair of positive numbers lying on the critical hyperbola 1 p + 1 + 1 q + 1 = N - 2 N. We first revisited the blowing-up solutions constructed in Kim and Pistoia (J Funct Anal 281(2):58, 2021) and then we proved its non-degeneracy. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems involving critical Halmitonian system and the construction of new solutions. [ABSTRACT FROM AUTHOR]

This article explores the concept of luck and the proportions of infinite sets. The author challenges the idea that luck can be measured proportionally, arguing that it is mathematically incorrect to claim that there are more worlds where a proposition is true than false. The article introduces measure theory, a branch of mathematics that compares the sizes of uncountable sets, and highlights its connection to probability theory. It explains how probability measures can be used to determine the likelihood of certain outcomes, such as the probability of Smith winning or losing. The probabilities presented are based on mathematical calculations and should not be interpreted as subjective judgments. [Extracted from the article]

Probabilistic databases (PDBs) model uncertainty in data. The current standard is to view PDBs as finite probability spaces over relational database instances. Since many attributes in typical databases have infinite domains, such as integers, strings, or real numbers, it is often more natural to viewPDBs as infinite probability spaces over database instances. In this article, we lay the mathematical foundations of infinite probabilistic databases. Our focus then is on independence assumptions. Tuple-independent PDBs play a central role in theory and practice of PDBs. Here we study infinite tuple-independent PDBs as well as related models such as infinite block-independent disjoint PDBs. While the standardmodel of PDBs focuses on a set-based semantics, we also study tuple-independent PDBs with a bag semantics and independence in PDBs over uncountable fact spaces. We also propose a new approach to PDBs with an open-world assumption, addressing issues raised by Ceylan et al. (Proc. KR 2016) and generalizing their work, which is still rooted in finite tuple-independent PDBs. Moreover, for countable PDBs we propose an approximate query answering algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers' comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus. [ABSTRACT FROM AUTHOR]

A topological index is a real number calculated from the structure of a chemical compound to describe its topology. The use of molecular descriptors has been increasing in recent years, helping to determine the physicochemical and biological properties of drugs. The main purpose of this article is to investigate the properties of the octane isomers using the theoretical method. To study the structures of octane isomers, we have introduced a new approach called "neighborhood product degree" to calculate all the classical degree-based topological indices. The np-degree approach is applied to approximate eight properties of octane isomers, such as the acentric factor, density, refractive index, critical volume, molar volume, radius of curvature, critical pressure, and LogP.The np-degree-based topological indices are the estimated values of the properties of octane structures, so the linear and quadratic regression models and correlation coefficients are applied to check the validity of the estimated results. The quantitative structure property relation are obtained by using the linear, quadratic, exponential, logarithmic and sinusoidal regression methods with the help of SPSS. Two models are applied to all the compuations and three regression models are applied to the np-degree Randic index. The computation showed that quadratic regression model is suitable for study octane isomers and np-degree based graph invariants. If the values of the correlation coefficient r P 0.7, p-values # 0.05, and F-values P 2.5, then the results are significant. The results of np-degree-based topological indices satisfy all the criteria for being significant, so these newly introduced indices are valid to study octane isomers. The information determined in this article is beneficial for chemists and pharmacists. [ABSTRACT FROM AUTHOR]

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of R N { A α u = u + 2 α ⁎ − 1 + λ u − s ¯ φ 1 , u > 0 , in Ω , u = 0 , on ∂ Ω , where A α is the spectral fractional operator, λ and s ¯ are real numbers, Ω ⊂ R N is bounded, 2 α ⁎ = 2 N N − 2 α is a critical exponent, 0 < α < 1 , φ 1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain. [ABSTRACT FROM AUTHOR]

Fuzzy multi-attribute decision-making is a hot research topic in which weight information is one of the conditions for forming a complete decision-making model, and it is also an important factor affecting the decision result. In most fuzzy multi-attribute decision-making problems, the weight information is often given in the form of real numbers. However, in real life, the weight information may not be suitable for specific numerical representation, or we cannot accurately determine the weight information. Therefore, it is very important to use fuzzy numbers to represent weight information. In this paper, we study the problem of disturbing fuzzy multi-attribute decision-making in which the attribute weight, decision-maker weight, and attribute information are given in the form of disturbing fuzzy numbers. Firstly, a new disturbing fuzzy integration operator, namely the disturbing fuzzy ring and multiplication aggregation (DFRMA) operator, is proposed, and its characteristics of closure, monotonicity, and boundary are studied. Then, the general steps of the disturbing fuzzy multi-attribute decision method based on the disturbing fuzzy ring and multiplication aggregation (DFRMA) operator are given, which include the single decision step and group decision step. Finally, an example is given to illustrate the practicability and effectiveness of the method. [ABSTRACT FROM AUTHOR]

Based on the previous studies, we make further research on how fractal dimensions of graphs of fractal continuous functions under operations change and obtain a series of new results in this paper. Initially, it has been proven that a positive continuous function under unary operations of any nonzero real power and the logarithm taking any positive real number that is not equal to one as the base number can keep the fractal dimension invariable. Then, a general method to calculate the Box dimension of two continuous functions under binary operations has been proposed. Using this method, the lower and upper Box dimensions of the product and the quotient of continuous functions without zero points have been investigated. On this basis, these conclusions will be generalized to the ring of rational functions. Furthermore, we discuss the Hölder continuity of continuous functions under operations and then prove that a Lipschitz function can be absorbed by any other continuous functions under certain binary operations in the sense of fractal dimensions. Some elementary results for vector-valued continuous functions have also been given. [ABSTRACT FROM AUTHOR]

We introduce a novel multiparameterized fractional multiplicative integral identity and utilize it to derive a range of inequalities for multiplicatively s-convex mappings in connection with different quadrature rules involving one, two, and three points. Our results cover both new findings and established ones, offering a holistic framework for comprehending these inequalities. To validate our outcomes, we provide an illustrative example with visual aids. Furthermore, we highlight the practical significance of our discoveries by applying them to special means of real numbers within the realm of multiplicative calculus. [ABSTRACT FROM AUTHOR]

We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow. [ABSTRACT FROM AUTHOR]

Kleene's computability theory based on the S1–S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier ∃ n and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from ∃ 2 , while the former are computable in ∃ 3 but not in weaker oracles. Of course, there is a great divide or abyss separating ∃ 2 and ∃ 3 and we identify slight variations of our new non-normal functionals that are again computable in ∃ 2 , i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis. [ABSTRACT FROM AUTHOR]

Although different social crises may eventually favor undemocratic and authoritarian forms of governance, at some point, such antidemocratic practices require the support of a significant part of the population to be implemented. The present research investigates how and whether the COVID‐19 pandemic might have favoured greater support for antidemocratic governmental practices, on the premise of regaining control and security. Using data from 17 countries (N = 4364) and national‐level indicators (i.e., real number of contagions and deaths, and sociopolitical indicators), we test how the risk of contagion and death from COVID‐19, along with personal orientations (i.e., social dominance orientation [SDO], right‐wing authoritarianism [RWA], and perceived anomie) motivate authoritarian and antidemocratic practices. Results from multilevel models indicate that risk perception and perceptions of political instability predict a wish for stronger leadership, agreement with martial law, and support for a controlling government especially when SDO and RWA are high, while more egalitarian and less conservative people agree less with these authoritarian measures in spite of the levels of risk perception. We discuss the implications for these findings for future research on similar but also dissimilar external events (natural disasters, war, or terror incidents) and the consequences for societies with higher authoritarian tendencies. [ABSTRACT FROM AUTHOR]

Denote by τ k (n) , ω (n) and μ 2 (n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [ t ] be the integral part of real number t. For f = ω , 2 ω , μ 2 , τ k , we prove that ∑ n ≤ x f x n = x ∑ d ≥ 1 f (d) d (d + 1) + O (x f + ) for x → ∞ , where ω = 5 3 1 1 0 , 2 ω = 9 1 9 , μ 2 = 2 5 , τ k = 5 k − 1 1 0 k − 1 and > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès. [ABSTRACT FROM AUTHOR]

Based on an extension of Riemann sums, Moore and Yang have defined an integral notion for the context of continuous inclusion monotonic interval functions in which the limits of integration are real numbers. This integral notion generalizes the usual one for real-valued functions based on Riemann sums. In this paper we extend this approach by considering intervals as limits of integration and abolishing the inclusion monotonic restriction of the integrable interval functions. Also, such a new integration notion is used to define interval probability density functions and use it in interval probability distribution functions. [ABSTRACT FROM AUTHOR]

Trapezoidal positive/negative fuzzy numbers have no single definition; instead, various authors define them in relation to different concepts. This means that arithmetic operations for trapezoidal fuzzy numbers also differ. For the operations of addition, subtraction, and scalar multiplication, there are not many differences; for multiplication, however, there are many differences. In general, multiplication is divided into various cases. For the inverse operation, there is not much to define; in general, for any trapezoidal fuzzy number u ~ ,   u ~ ⊗ 1 u ~ = i ~ = (1 , 1 , 0 , 0) does not necessarily apply. As a result of the different arithmetic operations for multiplication and division employed by various authors, several researchers have tackled the same problem and reached different solutions, meaning that the application will also produce different results. To date, many authors have proposed various alternatives for the algebra of the trapezoidal fuzzy number. In this paper, using the parametric form approach to trapezoidal fuzzy numbers, an alternative to multiplication with only one formula is constructed for various cases. Furthermore, based on the definition of multiplication for any trapezoidal fuzzy number, u ~ is constructed 1 u ~ so that u ~ ⊗ 1 u ~ = i ~ = (1 , 1 , 0 , 0) . Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. Furthermore, we modify the elementary row operational steps for the trapezoidal fuzzy number matrix, which can be used to determine the inverse of a trapezoidal fuzzy number matrix with the order m × m . We also give the steps and examples necessary to determine the general inverse for a trapezoidal fuzzy number matrix of the order m × n with m   ≠ n . This ability to easily determine the inverse and general inverse of a trapezoidal fuzzy number matrix has a number of applications, such as solving fully trapezoidal fuzzy number linear systems and fuzzy transportation problems, especially in applications in fields outside of mathematics; for example, the application of triangular fuzzy numbers in medical problems is a topic currently receiving a significant amount of attention. [ABSTRACT FROM AUTHOR]

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui–Hà–Jayanthan–Thomas [Collect. Math. 72 (2021), pp. 605–614]. Complete solutions are delivered for two conjectures proposed by these authors. For given real numbers a and b, we consider the set Res(a,b) of possible values of the resurgence of I+J where I and J are ideals in disjoint sets of variables having resurgence a and b, respectively. Some questions and partial results about Res(a,b) are discussed. [ABSTRACT FROM AUTHOR]

Let (t (m)) m ≥ 0 be Thue-Morse sequence and b > 2 be an integer. In this paper, we prove that the real numbers 1 , ∑ m = 0 ∞ t (m 2) b m + 1 and ∑ m = 0 ∞ t (m 3) b m + 1 are linearly independent over Q . [ABSTRACT FROM AUTHOR]

Let be an array of martingale difference random vectors and be an array of m × d matrices of real numbers. In this paper, the Marcinkiewicz–Zygmund type weak law of large numbers for maximal weighted sums of martingale difference random vectors is obtained with not necessarily finite p-th (1 < p < 2) moments. Moreover, the complete convergence and strong law of large numbers are established under some mild conditions. An application to multivariate simple linear regression model is also provided. [ABSTRACT FROM AUTHOR]

Context. In many areas of science and technology, the numerical solution of problems is not enough for the further development of the implementation of the obtained results. Among the existing information visualization approaches, the one that allows you to effectively reveal unstructured actionable ideas, generalize or simplify the analysis of the received data is chosen. The results of visualization of generalized structural diagrams based on the textual description of the algorithm clearly reflect the interaction of its parts, which is important at the system engineering stage of computer design. Objective of the study is the analysis and software implementation of structure visualization using the example of discrete harmonic transformation calculators obtained as a result of the synthesis of an algorithm based on cyclic convolutions with the possibility of extending the structure visualization to other computational algorithms. Method. The generalized scheme of the synthesis of algorithms of fast harmonic transformations in the form of a set of cyclic convolution operations on the combined sequences of input data and the coefficients of the harmonic transformation function with their visualization in the form of a generalized structural diagram of the calculator. The results. The result of the work is a software implementation of the visualization of generalized structural diagrams for the synthesized algorithms of cosine and Hartley transformations, which visually reflect the interaction of the main blocks of the computer. The software implementation of computer structure visualization is made in TypeScript using the Phaser 3 framework. Conclusions. The work considers and analyzes the developed software implementation of visualization of the general structure of the calculator for fast algorithms of discrete harmonic transformations in the domain of real numbers, obtained as a result of the synthesis of the algorithm based on cyclic convolutions. The results of visualization of variants of structural schemes of computers clearly and clearly reflect the interaction of its parts and allow to evaluate one or another variant of the computing algorithm in the design process [ABSTRACT FROM AUTHOR]

Ostrowski's iterative method is a classical method for solving systems of nonlinear equations. However, it is not stable enough. In order to obtain a more stable Ostrowski-type method, this paper presented a new family of fourth-order single-parameter Ostrowski-type methods for solving nonlinear systems. As a generalization of the Ostrowski's methods, the Ostrowski's methods are a special case of the new family. It was proved that the order of convergence of the new iterative family was always fourth-order when the parameters take any real number. Finally, the dynamical behavior of the family was briefly analyzed using real dynamical tools. The new iterative method can be applied to solve a wide range of nonlinear equations, and it was used in numerical experiments to solve the Hammerstein equation, boundary value problem, and nonlinear system. These numerical results supported the theoretical results. [ABSTRACT FROM AUTHOR]

Let N denote a sufficiently large real number. In this paper, we prove that for 1 < c < 104349/77419, c ≠ 43, for almost all real numbers T ∈ (N, 2N] (in the sense of Lebesgue measure), the Diophantine inequality |p1c + p2c - T| < T-9/10c (104349/77419 -c) is solvable in primes p1, p2. In addition, it is proved that the Diophantine inequality |p1c + p2c + p3c + p4c - N| < N-9/10c(104349/77419 -c) is solvable in primes p1, p2, p3, p4. This result constitutes a refinement upon that of Li and Cai. [ABSTRACT FROM AUTHOR]

In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: (− Δ) α / 2 u (x) = f (u (x) , v (x)) , x ∈ R n , (− Δ) α / 2 v (x) = g (u (x) , v (x)) , x ∈ R n. Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang ("A direct method of moving spheres on fractional order equations," J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu ("Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space," Potential Anal., vol. 46, pp. 569–588, 2017). [ABSTRACT FROM AUTHOR]

When the adaptive filter length is increased, the calculation complexity increases rapidly because the relationship between the calculation and the adaptive filter length N contains a power function with no secondary path identification algorithm. Under the basic premise of unreduced noise reduction, herein, a simplified frequency-domain feedback active noise control algorithm is proposed. To reduce the computation complexity, the total delay is adopted as the estimated secondary path; the filtered reference signal is produced in the frequency domain by using multiplication to replace convolution calculation in the time domain and then updating the adaptive filter coefficients in the frequency domain. Therefore, the computational complexity has a logarithmic function with the increased adaptive filter length in the proposed algorithm. If the adaptive filter length is 512, the existing WSMANC algorithm's calculation is 271,360 real number multiplications, while that of the proposed algorithm is only 38,912 real number multiplications. To verify the proposed algorithm's stability, convergence speed, and noise reduction, the single-frequency noise, narrowband white noise, and narrowband pink noise, respectively, are used as the primary noise types in the simulations. The results show that (1) the proposed SFDFBANC algorithm can obtain similar noise reduction performance to existing algorithm, (2) the convergence rate is faster than existing algorithm, and (3) if the adaptive filter length is more than 64, the proposed algorithm exhibits a lower computational complexity. [ABSTRACT FROM AUTHOR]

Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials. [ABSTRACT FROM AUTHOR]

We study some properties of spaces obtained by interpolation of the Sobolev spaces Wk,1(Rd) and Wl,∞(Rd), where l and r are nonnegative integers, and d≥2. We are concerned with the standard real and complex methods of interpolation. In the case of the real method, an old result of De Vore and Scherer (1979) gives that (Wl,1 (Rd),Wl,∞(Rd))θ,pθ​​=Wl,pθ​(Rd), where θ∈(0,1) and 1/pθ​=1−θ. We complement this result by considering the case l ≠ r. We prove that, when l ≠ r, (Wl,1(Rd),Wr,∞(Rd))θ,q​=Bqσ,q, ​(Rd),(⋆) where σ:=(1−θ)l+θr and 1/q=1−θ, if and only if l−r∈R∖[1,d]. Also, we prove a similar fact when Wl,¹ is replaced in (⋆) by a space Ws,p where s ≠ r is a real number and p∈(1,∞). Several other problems like the boundedness of the Riesz transforms on interpolation spaces are also considered. In the case of the complex method, it was proved by M. Milman (1983) that, for any 1l,1(Rd),Wl,p(Rd))θ​=Wl,pθ​(Rd),(⋆⋆) where 1/pθ​=(1−θ)+θ/p. We show by simple arguments that (⋆⋆) fails when p=∞ and l≥1, answering a question of P. W. Jones (1984). As an immediate consequence of these arguments, we show that certain closed subspaces of (C(Td))N (with N∈N∗) that are described by Fourier multipliers are not complemented in (C(Td))N. [ABSTRACT FROM AUTHOR]

TT□C is a generic family of effectful, extensional type theories with a forcing interpretation parameterized by modalities. This paper identifies a subclass of TT□C theories that internally realizes continuity principles through stateful computations, such as reference cells. The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. In this paper, the modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled internally in the theory. [ABSTRACT FROM AUTHOR]

We propose a simple imperative programming language, ERC, that features arbitrary real numbers as primitive data type, exactly. Equipped with a denotational semantics, ERC provides a formal programming language-theoretic foundation to the algorithmic processing of real numbers. In order to capture multi-valuedness, which is well-known to be essential to real number computation, we use a Plotkin powerdomain and make our programming language semantics computable and complete: all and only real functions computable in computable analysis can be realized in ERC. The base programming language supports real arithmetic as well as implicit limits; expansions support additional primitive operations (such as a user-defined exponential function). By restricting integers to Presburger arithmetic and real coercion to the ‘precision’ embedding Z ∋ p 7→ 2p ∈ R, we arrive at a first-order theory which we prove to be decidable and model-complete. Based on said logic as specification language for preconditions and postconditions, we extend Hoare logic to a sound (w.r.t. the denotational semantics) and expressive system for deriving correct total correctness specifications. Various examples demonstrate the practicality and convenience of our language and the extended Hoare logic. [ABSTRACT FROM AUTHOR]

We consider the Vilenkin orthonormal system on a Vilenkin group G and the Vilenkin-Fourier coefficients f̂(n), n ∈ N, of functions f∈ LP(G) for some 1 < p ⩽ 2 We obtain certain sufficient conditions for the finiteness of the series sum n = 1∑∞ an |f̂(n)|r where {an} is a given sequence of positive real numbers satisfying a mild and 0 < r < 2 We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of f and give multiplicative analogue of some results due to Móricz (2010), Móricz and Veres (2011), Golubov and Volosivets (2012), and Volosivets and Kuznetsova (2020). [ABSTRACT FROM AUTHOR]

In this paper, we focus on obtaining the closed-form solution for the following bidimensional system of higher-order rational difference equations: u(1) n+1 = 1 3 - 2u(2) n-m, u(2) n+1 = 1 3 - 2u(1) n-m, n,m N0, where the initial values u(1) -j and u(2) -j, j {0, 1, ...,m} are real numbers not equal to 3/2. We show that the solutions of this system are associated with Mersenne numbers and/or Mersenne-Lucas numbers. It is shown that the global stability of positive solutions of this system holds. Finally, we provide numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

We define and study the notion of the golden ratio of order k = 0, denoted ϕk, as a generalized form of the golden ratio ϕ for any real number k = 0. We show that similar to the special case of ϕ and its conjugate ϕ, ϕk and ϕk are the two distinct roots of a quadratic polynomial for any fixed real k = 0. We express some numerical and algebraic properties of ϕk and ϕk and write their relations to ϕ and ϕ, respectively, with some examples for some special values of k. In particular, it is shown that ϕk = ϕ and ϕk = ϕ if and only if k = 0. We show that Z[(k + 1)ϕk] is a subring of the ring Z[ϕ] for any nonnegative integer k. We will define the golden rectangle of order k (or k-golden rectangle for short) with a class of examples for all k = 0. We also discuss some cases of two Fibonacci numbers in connection to the golden ratio. We will show that the ratio of height to width of the pages of the Gutenberg Bible is the golden ratio of order k ϕ= 0. Actually, some erroneous ideas and examples of disputed observations related to the golden ratio are good reasons to apply ϕk to improve the measurements regarding ?ϕ for some k ϕ= 0. Finally, we end the paper by posing a question related to the Penrose tiling and quasicrystals in connection to the golden ratio of order k > 0. [ABSTRACT FROM AUTHOR]

The general sum-connectivity and Platt indices of a graph G are defined by SCa(G) = Σxy∈E(G)(dx +dy)a and Pla(G) = Σxy∈E(G)(dx +dy -2)a, respectively, where a is a real number, E(G) indicates the edge set of G, and dv indicates the degree of a vertex v of G. The cyclomatic number of G is the least number of edges required to be deleted from G to make it acyclic. If the maximum degree of G is less than 5, then G is referred to as a molecular graph. In this paper, the problem of determining graphs possessing the minimum values of the indices SCa and Pla among all connected (molecular) graphs of order n and cyclomatic number t is solved for 0 < a < 1 and n ≥ 2(t - 1) ≥ 2 with n ≤ 4. It is proved that the difference between the maximum and minimum degrees of the aforementioned extremal graphs is at most 1. [ABSTRACT FROM AUTHOR]

We consider a very weak chain condition for a poset, that is the absence of subsets which are order isomorphic to the set of real numbers in their natural ordering; we study generalised radical groups in which this finiteness condition is set on the poset of subgroups which do not have certain properties which are generalizations of normality. This completes many previous results which considered (apparently) stronger chain conditions. [ABSTRACT FROM AUTHOR]

We parameterize countable locally standard Boolean measure algebras by pairs of a Steinitz number and a real number greater or equal to 1. This is an analog of the theorems of Dixmier and Baranov. [ABSTRACT FROM AUTHOR]

Ptolemy's inequality is a classic relationship between the distances among four points in Euclidean space. Another relationship between six distances is the 4-point condition, an inequality satisfied by the lengths of the six paths that join any four points of a metric (or weighted) tree. The 4-point condition also characterizes when a finite metric space can be embedded in such a tree. The curious observer might realize that these inequalities have similar forms: if one replaces addition and multiplication in Ptolemy's inequality with maximum and addition, respectively, one obtains the 4-point condition. We show that this similarity is more than a coincidence. We identify a family of Ptolemaic inequalities in CAT-spaces parametrized by a real number and show that a certain limit involving these inequalities, as the parameter goes to negative infinity, yields the 4-point condition, giving an elementary proof that the latter is the tropicalization of Ptolemy's inequality. [ABSTRACT FROM AUTHOR]

Let p be a configuration of n points in R d for some n and some d ≥ 2 . Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing p given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when p will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that p is generic. [ABSTRACT FROM AUTHOR]

Chipman contended, in stark contrast to the conventional view, that, utility is not a real number but a vector, and that it is inherently lexicographic in nature. On the other hand, in recent years continuous multi-utility representations of a preorder on a topological space, which proved to be the best kind of continuous representation, have been deeply studied. In this paper, we first state a general result, which guarantees, for every preordered topological space, the existence of a lexicographic order-embedding of the Chipman type. Then, we combine the Chipman approach and the continuous multi-utility approach, by stating a theorem that guarantees, under certain general conditions, the coexistence of these two kinds of continuous representations. [ABSTRACT FROM AUTHOR]

Recently, the growth of the products of consecutive partial quotients a i (x) in the continued fraction expansion of a real number x was studied in connections with improvements to Dirichlet's theorem. In this paper, for a non-decreasing positive measurable function F (x 1 , ... , x m) and a function ϕ : ℕ → ℝ > 0 , we consider the set ℰ F (ϕ) = { x ∈ [ 0 , 1 ] : F (a n (x) , ... , a n + m − 1 (x)) ≥ ϕ (n)  for infinitely many  n ∈ ℕ } , and obtain its Lebesgue measure ℒ (ℰ F (ϕ)). As an application of our result, we reprove a theorem of Bakhtawar–Hussain–Kleinbock–Wang. We also consider the case when F (x 1 , ... , x m) = x 1 + ⋯ + x m . [ABSTRACT FROM AUTHOR]

The emergence of quantum computers is having an impact on almost every field of engineering. Electrical drives, in particular, can benefit from the power of quantum computing. This paper presents a quantum version of the pulse-width modulation (PWM) algorithm, which is used widely to control electrical motors. The proposed algorithm is implemented using a real number comparator, which is novel to the authors' knowledge. Unlike other quantum comparators in the literature, which provide quantum versions of binary-based comparators, the suggested comparator compares real numbers ranging from 0 to 100%, making it more usable in practice for many industrial applications, particularly control. When applied to DC motor speed control, the quantum PWM controller achieved similar speed precision to the classical version, but with lower energy consumption and less switching. It could therefore significantly increase the lifetime of power devices while achieving similar control performance at lower energy consumption. [ABSTRACT FROM AUTHOR]