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We give a domain-theoretic semantics to a statistical programming language, using the plain old category of dcpos, in contrast to some more sophisticated recent proposals. Remarkably, our monad of minimal valuations is commutative, which allows for program transformations that permute the order of independent random draws, as one would expect. A similar property is not known for Jones and Plotkin's monad of continuous valuations. Instead of working with true real numbers, we work with exact real arithmetic, providing a bridge towards possible implementations (implementations by themselves are not addressed here). Rather remarkably, we show that restricting ourselves to minimal valuations does not restrict us much: All measures on the real line can be modeled by minimal valuations on the domain IR⊥ of exact real arithmetic. We give three operational semantics for our language, and we show that they are all adequate with respect to the denotational semantics. We also explore quite a few examples to demonstrate that our semantics computes exactly as one would expect and to debunk the myth that a semantics based on continuous maps would not be expressive enough to encode measures with non-compact support using only measures with compact support, or to encode measures via non-continuous density functions, for instance. Our examples also include some useful, non-trivial cases of distributions on higher-order objects. [ABSTRACT FROM AUTHOR]

AbstractThe hyperbolic secant distribution has several generalizations with applications in, for example, finance. In this study, we explore the dual geometric structure of one such generalization: the beta-logistic distribution. Within this family, two special cases of random variables, as examples, are of particular interests: their moments, by some recent results, give the Bernoulli and Euler polynomials, which are important objects in many areas of mathematics. This current study also uncovers that the beta-logistic distribution admits a α-parallel prior for any real number α, that has the potential for application in geometric statistical inference. [ABSTRACT FROM AUTHOR]

We study the existence of positive weak solutions for the following problem: −Δu+α(x)u=∫ΩF(y,u)|x−y|μ1dyf(x,u)inΩ,∂u∂η+βu=∫∂ΩG(y,u)|x−y|μ2dνg(x,u)on∂Ω,$$\begin{equation*} \begin{aligned} \hspace*{65pt}-\Delta u + \alpha (x) u &= {\left(\int \limits _{\Omega }\frac{F(y,u)}{|x-y|^{{\mu _1}}}\;dy\right)}f(x,u) \;\;\text{in} \; \Omega,\\ \hspace*{65pt} \frac{\partial u}{\partial \eta } + \beta u &= {\left(\int \limits _{\partial \Omega }\frac{G(y,u)}{|x-y|^{{\mu _2}}}\;d\nu \right)}g(x,u)\;\; \text{on} \; \partial \Omega, \end{aligned} \end{equation*}$$where Ω$\Omega$ is a bounded domain in R2$\mathbb {R}^2$ with smooth boundary, α(x)$\alpha (x)$ is a bounded measurable function on Ω$\Omega$, β$\beta$ is nonnegative real number, η$\eta$ is the unit outer normal to ∂Ω$\partial \Omega$, μ1∈(0,2)${\mu _1} \in (0, 2)$, and μ2∈(0,1)${\mu _2} \in (0, 1)$. The functions f$f$ and g$g$ have critical exponential growth, while F$F$ and G$G$ are their primitives. The proofs combine the constrained minimization method with energy methods and topological tools. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the α -fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its α -fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an α -fractal function is an attractor of an iterated function system. [ABSTRACT FROM AUTHOR]

In this paper we show that the following four-dimensional system of difference equations x n + 1 = y n α z n − 1 β , y n + 1 = z n γ t n − 1 δ , z n + 1 = t n ϵ x n − 1 μ , t n + 1 = x n ξ y n − 1 ρ , n ∈ N 0 , where the parameters α, β, γ, δ, ϵ, μ, ξ, ρ ∈ ℤ and the initial values x–i, y–i, z–i, t–i, i ∈ {0, 1}, are real numbers, can be solved in closed forms, extending further some results in literature. [ABSTRACT FROM AUTHOR]

Grüss inequality is subject of interest for many authors due to its effectiveness in predicting bounds in several quadrature problems. In the present article, we give weighted treatment of the discrete Čebyšev and Grüss type inequalities pertaining two n-tuples of real numbers in which the bounding constants are mobilised with bounding sequences of real numbers. As an application estimations of discrete Ostrowski type inequalities are provided. Finally, by practicing obtained results along with Jensen's difference, a wide range of estimations are formalised by considering Jensen-Grüss differences. [ABSTRACT FROM AUTHOR]

This is the first step in an attempt at a deformation theory for G2-instantons with 1-dimensional conic singularities. Under a set of model data, the linearization yields a Dirac operator P on a certain bundle over 핊5, called the link operator. As a dimension reduction, the link operator also arises from Hermitian Yang-Mills connections with isolated conic singularities on a Calabi-Yau 3-fold. Using the quaternion structure in the Sasakian geometry of 핊5, we describe the set of all eigen-values of P, denoted by SpecP. We show that SpecP consists of finitely many integers induced by certain sheaf cohomologies on ℙ², and infinitely many real numbers induced by the spectrum of the rough Laplacian on the pullback endomorphism bundle over 핊5. The multiplicities and the form of an eigensection can be described fairly explicitly. In particular, there is a relation between the spectrum on 핊5 to certain sheaf cohomologies on ℙ². Moreover, on a Calabi-Yau 3-fold, the index of the linearized operator for admissible singular Hermitian Yang-Mills connections is also calculated, in terms of these sheaf cohomologies. Using the representation theory of SU(3) and the subgroup S[U(1) x U(2)], we show an example in which SpecP and the multiplicities can be completely determined. [ABSTRACT FROM AUTHOR]

Let m and N be positive integers and k be a negative real number. Let χ be a multiplier system of weight k on Γ 0 (N). In this paper, we give an exact formula for the Fourier coefficients of the holomorphic part of a harmonic Maass–Jacobi form of weight k, index m, and multiplier system χ on Γ 0 (N). As an application, we investigate the number of ratios of the Fourier coefficients of the holomorphic parts of two harmonic Maass–Jacobi forms. Moreover, we obtain the equivalent conditions for the principal part of a harmonic Maass–Jacobi form to be the principal part of a weak Jacobi form with poles only at infinity. [ABSTRACT FROM AUTHOR]

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit lower bound for the rank of universal quadratic forms over a given number field K, under the assumption that the codifferent of K is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields. [ABSTRACT FROM AUTHOR]

Sexual selection plays a crucial role in modern evolutionary theory, offering valuable insight into evolutionary patterns and species diversity. Recently, a comprehensive definition of sexual selection has been proposed, defining it as any selection that arises from fitness differences associated with nonrandom success in the competition for access to gametes for fertilization. Previous research on discrete traits demonstrated that non-random mating can be effectively quantified using Jeffreys (or symmetrized Kullback-Leibler) divergence, capturing information acquired through mating influenced by mutual mating propensities instead of random occurrences. This novel theoretical framework allows for detecting and assessing the strength of sexual selection and assortative mating. In this study, we aim to achieve two primary objectives. Firstly, we demonstrate the seamless alignment of the previous theoretical development, rooted in information theory and mutual mating propensity, with the aforementioned definition of sexual selection. Secondly, we extend the theory to encompass quantitative traits. Our findings reveal that sexual selection and assortative mating can be quantified effectively for quantitative traits by measuring the information gain relative to the random mating pattern. The connection of the information indices of sexual selection with the classical measures of sexual selection is established. Additionally, if mating traits are normally distributed, the measure capturing the underlying information of assortative mating is a function of the square of the correlation coefficient, taking values within the non-negative real number set [0, +∞). It is worth noting that the same divergence measure captures information acquired through mating for both discrete and quantitative traits. This is interesting as it provides a common context and can help simplify the study of sexual selection patterns. [ABSTRACT FROM AUTHOR]

In multi-criteria decision-making, the Best-worst method (BWM) enables decision-makers to determine the best and worst criteria, compare the relationship between the best criterion with other criteria and the worst criterion with other criteria, and then obtain the corresponding weight of each criterion based on the two sets of evaluation results obtained in the previous step by constructing a model. However, due to the diversity of decision-making criteria, it is difficult for decision-makers to assign a specific weight to multiple criteria, and the BWM is commonly employed to weight multi-criteria decision-making problems. In previous research, the BWM was extended to a fuzzy environment. In this study, the BWM is extended to neutrosophic sets. By describing language phrases and converting them into intermediate wisdom, decision-makers can determine the reference comparison of the best criterion to other criteria and the worst criterion to other criteria. Next, all the intelligent numbers are converted into real numbers using the calculation formula of the scoring function. Subsequently, based on the BWM, a nonlinear constraint optimization problem is formulated to determine the criteria weights and alternatives relative to different criteria. In this approach, different criterion weights can be obtained directly from the proposed BWM without further transformation. Meanwhile, the agreement ratio of the BWM is proposed to verify the reliability of the preference comparison. Finally, a case study is conducted to fully demonstrate the effectiveness and feasibility of the proposed neutrosophic BWM. The results indicate that the proposed neutrosophic BWM can obtain a reasonable preference ranking of alternatives and has a higher comparative consistency than the BWM. [ABSTRACT FROM AUTHOR]

Vector normalization is an important process in several algorithms. It is used in classical physical calculations, mathematical techniques, and machine learning, which has witnessed significant advancements in recent years. Normalization and regularization ensure the stability of solutions and play an important role in algorithm convergence. Normalization typically refers to the division of elements by their norm. Division should not be used in algorithmic implementations because its computational cost is considerably higher than that of multiply–add operations. Based on this, there is a well-known method referred to as the fast inverse square root (FISR) algorithm in floating-point calculations (IEEE754). In deeper-level embedded systems that require fast responses or power efficiency, integer instead of real number arithmetic (floating-point number arithmetic) should be used to increase speed. Conversely, in deeper-level embedded systems that require fast responses or power efficiency, integer arithmetic should be used instead of real number arithmetic (floating-point number arithmetic) to increase speed. Therefore, embedded engineers encounter problems in instances in which they use integer arithmetic for implementation, but real number arithmetic is required to compute vectors and other higher-dimensional algebra. There is no conventional normalization algorithm similar to the FISR algorithm for integer arithmetic; however, the proposed pseudo-normalization achieves vector normalization within a restricted domain using only multiply–add operations and bit shifts. This allows for fast and robust operations, even for low-performance MCUs that do not have power-efficient FPUs. As an example, this study demonstrates the computation of the arctangent (Arctan2 function; atan2(y, x)) with high precision using only integer multiply–add operations. In this study, we proposed a method of vector normalization using only integer arithmetic for embedded systems and confirmed its effectiveness by simulation using Verilog. The research results can contribute to various fields such as signal processing of IMU sensor data, faster artificial intelligence training, and efficient rendering of computer graphics. [ABSTRACT FROM AUTHOR]

This paper introduced novel multidimensional Hardy-type inequalities with general kernels on time scales, extending existing results in the literature. We established generalized inequalities involving a general Hardy operator with multiple variables and kernels on arbitrary time scales. Our findings not only encompassed known results in the realm of real numbers (T = R), but also provided refinements and generalizations thereof. The proposed inequalities offered versatile applications in mathematical analysis and beyond, contributing to the ongoing exploration of inequalities on diverse time scales. [ABSTRACT FROM AUTHOR]

In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number s > 2 , we prove that every graph on n vertices with average degree d ≥ s contains a subgraph of average degree at least s on at most n d - s s - 2 (log d) O s (1) vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with n vertices and average degree at least n 1 - 2 s + ε contains a subgraph of average degree at least s on O ε , s (1) vertices, which is also optimal up to the constant hidden in the O(.) notation, and resolves a conjecture of Verstraëte. [ABSTRACT FROM AUTHOR]

The aim of this article is to find a geometric and physical interpretation of fractional order derivatives for a general class of functions defined over a bounded or unbounded domain. We show theoretically and geometrically that the absolute value of the fractional derivative value of a function is inversely proportional to the area of the triangle. Further, we prove geometrically that the fractional derivatives are inversely proportional to the classical integration in some sense. The established results are verified numerically for non‐monotonic, trigonometric, and power functions. Further, this article establishes a significant connection between the area of the projected fence and the area of triangles. As the area of triangles decreases, the area of the projected fence increases, and vice versa. We calculate the turning points of the fractional derivative values of different functions with respect to order α$$ \alpha $$, including non‐monotonic, trigonometric, and power functions. In particular, we demonstrate that for the power function xβ$$ {x}^{\beta } $$, with β$$ \beta $$ being a positive real number, the value α=0.5$$ \alpha =0.5 $$ is a turning point when x=β$$ x=\beta $$. However, for x>β$$ x>\beta $$, the turning point shifts to the left of point (0.5,0)$$ \left(0.5,0\right) $$ and shifts to the right of point (0.5,0)$$ \left(0.5,0\right) $$ for x<β.$$ x<\beta. $$ We discuss the physical interpretation of fractional order derivatives in terms of fractional divergence. We present some applications of fractional tangent lines in the field of numerical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field 풦 = ℚ(ς8) over ℚ. We employ binary codes to generate a lattice over ℚ(ς8), which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels. [ABSTRACT FROM AUTHOR]

In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable. [ABSTRACT FROM AUTHOR]

This article studies the breaking of the Lorentz symmetry at the Planck length in quantum mechanics. We use three-dimensional ℘ -adic vectors as position variables, while the time remains a real number. In this setting, the Planck length is 1 / ℘ , where ℘ is a prime number, and the Lorentz symmetry is naturally broken. In the framework of the Dirac-von Neumann formalism for quantum mechanics, we introduce a new ℘ -adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The discreteness of the ℘ -adic space imposes substantial restrictions on the solutions of the new equation. This equation admits localized solutions, which is impossible in the standard case. We show that an isolated quantum system whose evolution is controlled by the ℘ -adic Dirac equation does not satisfy the Einstein causality, which means that the speed of light is not the upper limit for the speed at which conventional matter or energy can travel through space. The new ℘ -adic Dirac equation is not intended to replace the standard one; it should be understood as a new version (or a limit) of the classical equation at the Planck length scale. [ABSTRACT FROM AUTHOR]

An intuitionistic fuzzy set is a more generalised tool than a fuzzy set for handling unpredictability as, in an intuitionistic fuzzy set, there is scope for considering a grade of non-membership, independent of the grade of membership, only satisfying the condition that their sum is less or equal to 1. The motivation of this paper is to introduce the notion of intuitionistic type-2 fuzzy normed linear space (IT2FNLS). Here, to each vector x, we assign two fuzzy real number valued grades, one for its norm and the other for the negation of its norm. A theorem of the decomposition of the intuitionistic type-2 fuzzy norm into a family of pairs of Felbin-type fuzzy norms is established. Also, we deal with Cauchyness and convergence of sequences in the IT2FNLS. Later on, in the finite-dimensional IT2FNLS, the completeness property and compactness property are explored. Finally, we define two types of intuitionistic type-2 fuzzy continuity and examine the relations between them. [ABSTRACT FROM AUTHOR]

In cooperative games with transferable utilities defined on a variable set of players, we characterize the family of values satisfying efficiency, linearity, the equal treatment property, and the weak null player out property. The last property weakens the usual null player out property, and together with efficiency, it is interpreted as considering equal support from others for null players. Together with the fact that efficiency, linearity, and the equal treatment property characterize the Shapley value along with the usual null player out property, our result reveals how weakening the null player out property can expand the possibilities of solutions. The characterized family contains well-known values in the literature, such as the Shapley value, equal division value, equal surplus division value, and the egalitarian non-separable contributions value, etc. In addition, each value in the characterized family is determined by an infinite sequence of real numbers. Furthermore, the equal treatment property in our characterization can be replaced by the balanced contributions property for symmetric players. Comparing this result with the existing one also shows that how weakening the balanced contributions property more can expand the possibilities of solutions. [ABSTRACT FROM AUTHOR]

Let $ p_1 \lt p_2 \lt \cdots \lt p_n $ p 1 < p 2 < ⋯ < p n be positive numbers and r a non-negative real number. The Loewner matrix associated with the function $ x^{r+1} $ x r + 1 given by $ L_{r+1}=\begin {bmatrix}\frac {p_i^{r+1}-p_j^{r+1}}{p_i-p_j}\end {bmatrix} $ L r + 1 = [ p i r + 1 − p j r + 1 p i − p j ] and matrix $ P_r=[\begin {smallmatrix}{(p_i+p_j)^r}\end {smallmatrix}] $ P r = [ (p i + p j) r ] (the Hadamard inverse of rth Hadamard power of well-known Cauchy matrix) have same inertia. A question was left open in Inertia of Loewner matrices. Indiana Univ Math J. 2016;65(4):1251–1261 by Bhatia, Friedland and Jain to find a connection between these two matrix families. We aim to answer this question firmly in terms of a congruence relation between $ L_{r+1} $ L r + 1 and $ P_r $ P r . Indeed, a non-singular matrix X over $ \mathbb {R} $ R is explicitly obtained such that $ X'P_rX=L_{r+1} $ X ′ P r X = L r + 1 in this paper. [ABSTRACT FROM AUTHOR]

The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel class of real number codes termed as ℝ-spectral codes, derived from real symmetric operators. Our focus is on addressing the challenges associated with error correction in communication systems utilizing real number codes. Our study initiates with a meticulous definition and analysis of the fundamental properties of ℝ-spectral codes. These properties serve as the foundation for comprehending the distinctive characteristics that ℝ-spectral codes offer, ensuring the integrity and reliability of transmitted data. Our exploration extends to the construction of maximum distance separable (MDS) ℝ-spectral codes, which boast the maximum error correction capability. As a culmination of our research, we present an efficient decoding technique tailored for ℝ-spectral codes. Leveraging an MDS ℝ-spectral code and our decoding technique, we achieve optimal error correction and ensure good numerical stability. This innovative approach opens new avenues for enhancing the performance of real-number codes in practical applications. [ABSTRACT FROM AUTHOR]

The early quantification of the catchment water balance in the 17th century has been well documented. But there is one book, called L’Hydrographe or La Science des Eaux, that was published by a Jesuit priest, Père Jean François, in Rennes in 1653 (20 years before the publication of Pierre Perrault’s De l’Origine des Fontaines) that has been largely overlooked. The book is split into four parts that deal with the formation, movement and mixing of waters and the origin of springs. Further parts were published with La Science des Eaux that deal with the arts of surveying and drawing of maps, and the construction of canals and fountains, with a final part on doing arithmetic with integers and real numbers “with pen and counters.” François was convinced, much more than Perrault, that the waters of springs were the result of rainfall and snowmelt. [ABSTRACT FROM AUTHOR]

Let a 0 , a 1 , ... , a n − 1 be real numbers and let A = C i r c (a 0 , a 1 , ... , a n − 1) be a circulant matrix with f (x) = Σ j = 0 n − 1 a j x j . First, we prove that C i r c (a 0 , a 1 , ... , a n − 1) must be invertible if the sequence a 0 , a 1 , ... , a n − 1 is a strictly monotonic sequence and a 0 + a 1 + ⋯ + a n − 1 ≠ 0. Next, we reduce the calculation of f (ε 0) f (ε) ... f (ε n − 1) for a prime n by using the techniques on finite fields, where ε is a primitive n -th root of unity. Finally, we provide two examples to explain how to use the obtained results to calculate the determinant of a circulant matrix. [ABSTRACT FROM AUTHOR]

Consider the polynomial differential system of degree m of the form x ˙ = − y (1 + μ (a 2 x − a 1 y)) + x (ν (a 1 x + a 2 y) + Ω m − 1 (x , y)) , y ˙ = x (1 + μ (a 2 x − a 1 y)) + y (ν (a 1 x + a 2 y) + Ω m − 1 (x , y)) , where μ and ν are real numbers such that (μ 2 + ν 2) (μ + ν (m − 2)) (a 1 2 + a 2 2) ≠ 0 , m > 2 and Ωm−1(x,y) is a homogenous polynomial of degree m − 1. A conjecture, stated in J. Differential Equations 2019, suggests that when ν = 1, this differential system has a weak center at the origin if and only if after a convenient linear change of variable (x,y) → (X,Y) the system is invariant under the transformation (X,Y,t) → (−X,Y, −t). For every degree m we prove the extension of this conjecture to any value of ν except for a finite set of values of μ. [ABSTRACT FROM AUTHOR]

As a leading economic center in China and an international metropolis, Shenzhen has great significance in promoting sustainable urban development. To predict its amount of domestic waste clearance, a new multivariable grey prediction model with combinatorial optimization of parameters is established in this paper. Firstly, the new model expands the value range of the order r of a grey accumulation generation operator from positive real numbers (R +) to all real numbers (R), which enlarges the optimization space of parameter and has positive significance for improving model performance. Secondly, the dynamic background-value coefficient λ is introduced into the new model to improve the smoothing effect of the nearest neighbor generated sequences. Thirdly, with the objective function of minimizing the mean absolute percentage error (MAPE), the particle swarm optimization (PSO) is employed to optimize parameters r and λ to improve the overall performance of the new model. The new model is used to simulate and predict the amount of domestic waste clearance in Shenzhen, and the MAPE of the new model is only 0.27%, which is far superior to several other similar models. Lastly, the new model is applied to predict the amount of domestic waste clearance in Shenzhen. The results indicate the amount of domestic waste clearance in 2028 could be 9.96 million tons, an increase of 20.58% compared to 2021.This highlights the significant challenge that Shenzhen faces in terms of urban domestic waste treatment. Therefore, some targeted countermeasures and suggestions have been proposed to ensure the sustainable development of Shenzhen's economy and society. [ABSTRACT FROM AUTHOR]

A complex Polytopic fuzzy set (CPoFS) extends a Polytopic fuzzy set (PoFS) by handling vagueness with degrees that range from real numbers to complex numbers within the unit disc. This extension allows for a more nuanced representation of uncertainty. In this research, we develop Complex Polytopic Fuzzy Sets (CPoFS) and establish basic operational laws of CPoFS. Leveraging these laws, we introduce new operators under a confidence level, including the confidence complex Polytopic fuzzy Einstein weighted geometric aggregation (CCPoFEWGA) operator, the confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (CCPoFEOWGA) operator, the confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (CCPoFEHGA) operator, the induced confidence complex Polytopic fuzzy Einstein ordered weighted geometric aggregation (I-CCPoFEOWGA) operator and the induced confidence complex Polytopic fuzzy Einstein hybrid geometric aggregation (I-CCPoFEHGA) operator, enhancing decision-making precision in uncertain environments. We also investigate key properties of these operators, including monotonicity, boundedness, and idempotency. With these operators, we create an algorithm designed to solve multiattribute decision-making problems in a Polytopic fuzzy environment. To demonstrate the effectiveness of our proposed method, we apply it to a numerical example and compare its flexibility with existing methods. This comparison will underscore the advantages and enhancements of our approach, showing its efficiency in managing complex decision-making scenarios. Through this, we aim to demonstrate how our method provides superior performance and adaptability across different situations. [ABSTRACT FROM AUTHOR]

We applied an algebraic approach to solve a system consisting of four objective functions such that practitioners can absorb operational studies with two completed supply chains under service competition and relative price conditions. We extended the application of algebraic methods to a new environment that will arouse interested researchers to work on this research trend. Moreover, we checked a related inventory model to point out that the previously proposed solution procedure only considered the case where the numerator and denominator of the objective function both are positive real numbers. Hence, we provie a revision for their solution process to cover the overlooked four conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a geometric circulant matrix with geometric sequence i.e. a geometric circulant matrix whose first row is $ (g,gq,gq^{2},\ldots,gq^{n-1}) $ (g , gq , g q 2 , ... , g q n − 1) , where g is a nonzero complex number and q is a nonzero real number. The determinant, the Euclidean norm and bounds for the spectral norm of such matrix are obtained. Also, in the case when such matrix is singular, we obtain the Moore-Penrose inverse of such matrix and show that the group inverse of such matrix does not exist (in most cases). The obtained results are illustrated by examples. [ABSTRACT FROM AUTHOR]

The harmonic polynomial was defined in order to understand better the harmonic topological index. Here, we obtain several properties of this polynomial, and we prove that several properties of a graph can be deduced from its harmonic polynomial. Also, we prove that graphs with the same harmonic polynomial share many properties although they are not necessarily isomorphic. [ABSTRACT FROM AUTHOR]

In this article we define and study a zeta function ζ G —similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζ G (k) - 1 at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζ G . We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro- C groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζ G are rational functions in p - s if G is virtually abelian. For finite groups G we calculate ζ G using the rational representation theory of G. [ABSTRACT FROM AUTHOR]

Let b ≥ 2 be an integer and A = (a n) n = 1 ∞ be a strictly increasing subsequence of positive integers with η : = lim sup n → ∞ a n + 1 a n < + ∞. For each irrational real number ξ , we denote by v ˆ b , A (ξ) the supremum of the real numbers v ˆ for which, for every sufficiently large integer N , the equation ‖ b a n ξ ‖ < (b a N ) − v ˆ has a solution n with 1 ≤ n ≤ N. For every v ˆ ∈ [ 0 , η ] , let V ˆ b , A (v ˆ) ( V ˆ b , A ⁎ (v ˆ)) be the set of all real numbers ξ such that v ˆ b , A (ξ) ≥ v ˆ ( v ˆ b , A (ξ) = v ˆ) respectively. In this paper, we give some results of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ). When η = 1 , we prove that the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) are equal to (1 − v ˆ 1 + v ˆ ) 2 for any v ˆ ∈ [ 0 , 1 ]. When η > 1 and lim n → ∞ ⁡ a n + 1 a n exists, we show that the Hausdorfff dimension of V ˆ b , A (v ˆ) is strictly less than (η − v ˆ η + v ˆ ) 2 for some v ˆ , which is different with the case η = 1 , and we give a lower bound of the Hausdorfff dimensions of V ˆ b , A (v ˆ) and V ˆ b , A ⁎ (v ˆ) for any v ˆ ∈ [ 0 , η ]. Furthermore, we show that this lower bound can be reached for some v ˆ. [ABSTRACT FROM AUTHOR]

We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3$\hskip.001pt 3$–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3$\hskip.001pt 3$. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of Z 4 Z 4 Z 4 -additive cyclic codes generated by 3-tuples of polynomials. We discuss their algebraic structure and show that generator matrices can be constructed for all codes in this class. We study asymptotic properties of this class of codes by using a Bernoulli random variable. Moreover, let 0 < δ < 1 be a real number such that the entropy h 4 ((k + l + t) δ 6) < 1 4 , we show that the relative minimum distance converges to δ and the rate of the random codes converges to 1 k + l + t , where k, l, and t are pairwise co-prime positive odd integers. Finally, we conclude that the Z 4 Z 4 Z 4 -additive cyclic codes are asymptotically good. [ABSTRACT FROM AUTHOR]

Let w=(w1,⋯,wd)$w=(w_1,\dots,w_d)$ be a d$d$‐tuple of positive real numbers such that ∑iwi=1$\sum _{i}w_i =1$ and w1⩾⋯⩾wd$w_1\geqslant \cdots \geqslant w_d$. A d$d$‐dimensional vector x=(x1,⋯,xd)∈Rd$x=(x_1,\dots,x_d)\in \mathbb {R}^d$ is said to be w$w$‐singular if for every ε>0$\epsilon >0$, there exists T0>1$T_0>1$ such that for all T>T0$T>T_0$, the system of inequalities max1⩽i⩽d|qxi−pi|1wi<εTand0

Let Aj, Bj, Pj, and Qj ∈ Mn(C), where j = 1, 2, . . ., m. For a real number c ∈ [0, 1], we prove the following interpolation inequality: |||∑mj=1AjPjQj∗Bj∗|||² ≤ (max {L, M})4|||Kc||| |||K1-c|||, where L= ||∑m j=1|Aj∗|²||1/2, M = ||∑m j=1|Bj∗|²||1/2, and Kc = (c|P1|² + (1 − c) |Q1|²) ⊕ · · · ⊕ (c|Pm|² + (1 − c) |Qm|²). Many other related interpolation inequalities are also obtained. [ABSTRACT FROM AUTHOR]

This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let g(x) and h(x) have the same box dimension t (1 < t ≤ 2), the box dimension of g(x) + h(x) may or may not exist. If it exists, it can take an arbitrary real number γ satisfying 1 < γ ≤ t. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers t1 and t2 that satisfy 1 < t1 < t2 < t ≤ 2. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function Wt(x), we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals. [ABSTRACT FROM AUTHOR]

Uniform designs are widely used for experiments with mixtures. The uniformity of the design points is usually evaluated with a discrepancy criterion. In this paper, we propose a new criterion to measure the deviation between the design point distribution and a Dirichlet distribution. The support of the Dirichlet distribution, is defined by the set of d-dimensional vectors whose entries are real numbers in the interval [0,1] such that the sum of the coordinates is equal to 1. This support is suitable for mixture experiments. Depending on its parameters, the Dirichlet distribution allows symmetric or asymmetric, uniform or more concentrated point distribution. The difference between the empirical and the target distributions is evaluated with the Kullback–Leibler divergence. We use two methods to estimate the divergence: the plug-in estimate and the nearest-neighbor estimate. The resulting two criteria are used to build space-filling designs for mixture experiments. In the particular case of the flat Dirichlet distribution, both criteria lead to uniform designs. They are compared to existing uniformity criteria. The advantage of the new criteria is that they allow other distributions than uniformity and they are fast to compute. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this study, we developed a new method of topology optimization for truss structures by quantum annealing. To perform quantum annealing analysis with real variables, representation of real numbers as a sum of random number combinations is employed. The nodal displacement is expressed with binary variables. The Hamiltonian H is formulated on the basis of the elastic strain energy and position energy of a truss structure. It is confirmed that truss deformation analysis is possible by quantum annealing. For the analysis of the optimization method for the truss structure, the cross-sectional area of the truss is expressed with binary variables. The iterative calculation for the changes in displacement and cross-sectional area leads to the optimal structure under the prescribed boundary conditions. [ABSTRACT FROM AUTHOR]

Quantum annealing machines are next-generation computers for solving combinatorial optimization problems. Although physical simulations are one of the most promising applications of quantum annealing machines, a method how to embed the target problem into the machines has not been developed except for certain simple examples. In this study, we focus on a method of representing real numbers using binary variables, or quantum bits. One of the most important problems for conducting physical simulation by quantum annealing machines is how to represent the real number with quantum bits. The variables in physical simulations are often represented by real numbers but real numbers must be represented by a combination of binary variables in quantum annealing, such as quadratic unconstrained binary optimization (QUBO). Conventionally, real numbers have been represented by assigning each digit of their binary number representation to a binary variable. Considering the classical annealing point of view, we noticed that when real numbers are represented in binary numbers, there are numbers that can only be reached by inverting several bits simultaneously under the restriction of not increasing a given Hamiltonian, which makes the optimization very difficult. In this work, we propose three new types of real number representation and compared these representations under the problem of solving linear equations. As a result, we found experimentally that the accuracy of the solution varies significantly depending on how the real numbers are represented. We also found that the most appropriate representation depends on the size and difficulty of the problem to be solved and that these differences show a consistent trend for two annealing solvers. Finally, we explain the reasons for these differences using simple models, the minimum required number of simultaneous bit flips, one-way probabilistic bit-flip energy minimization, and simulation of ideal quantum annealing machine. [ABSTRACT FROM AUTHOR]

We study the Cantor real base numeration system which is a common generalization of two positional systems, namely the Cantor system with a sequence of integer bases and the Rényi system with one real base. We focus on the case of an alternate base B given by a purely periodic sequence (β n) n ≥ 1 of real numbers greater than 1. We answer an open question of Charlier et al. (J Number Theory 254:184–198, 2024, https://doi.org/10.1016/j.jnt.2023.07.008) on the set of numbers with eventually periodic B -expansions. We also investigate for which bases all sufficiently small rationals have a purely periodic B -expansion. We show that a necessary condition for this phenomenon is that δ = ∏ n = 1 p β n (where p is the period-length of B ) is a Pisot or a Salem unit. We also provide a sufficient condition. We thus generalize the results known for the Rényi numeration system, i.e. for the case when p = 1 . We provide a class of alternate bases in which all rational numbers in the interval [0, 1) have a purely periodic B -expansion. [ABSTRACT FROM AUTHOR]

Mixed-signal applications have emerged as a significant trend in the semiconductor industry, with considerable efforts directed towards developing fast and accurate designs that integrate both analog and digital components. However, mixed-signal verification presents a major challenge due to the slow verification time and limited robustness of traditional verification techniques. In this study, a verification architecture for a successive-approximation register (SAR) analog-to-digital converter (ADC) real number model (Real Number Modeling – RNM) using SystemVerilog is presented, which utilizes an efficient UVM-based methodology. The proposed approach combines the UVM capabilities with the RNM model of the SAR ADC to generate a reusable, fast, and robust verification environment with a reduced time-to-market. The testbench creation and simulation were carried out using Cadence Xcelium. The proposed verification architecture employs constrained-random stimulus generation, analog assertions, and coverage metrics to enhance verification effectiveness. Additionally, aim of this work is to emphasize on the RNM efficiency with SystemVerilog, and apply its modeling capabilities for a SAR ADC. The presented real number model was compared to a Verilog-AMS model. The conducted experiments provided evidence that the proposed RNM model exhibits a significant improvement in simulation efficiency compared to previous works documented in the literature (simulation time was at 0.5 s, compared to a Verilog-AMS reference model's simulation at 20 s). This improvement in efficiency is achieved without compromising on the accuracy of the simulation, ensuring that the model maintains a satisfactory level of precision. [ABSTRACT FROM AUTHOR]

A universal number (Unum) is a number representation format that can reduce the memory contention issues in multicore processors and parallel computing systems by optimizing the bit storage in the arithmetic operations. Given the safety-critical nature of applications of Unum format, there is a dire need to rigorously assess the correctness of the Unum based arithmetic operations. Unums are of three types, namely, Unum-I, Unum-II and Unum-III (commonly known as Posits). In this paper, we provide a higher-order-logic formalization of Unum-III (posits). In particular, we formally model a posit format (binary encoding of a posit), which is comprised of the sign, exponent, regime and fraction bits, using the HOL Light theorem prover. In order to prove the correctness of a posit format, we formally verify various properties regarding conversions of a real number to a posit and a posit to a real number and the scaling factors of the regime, exponent and fraction bits of a posit using HOL Light. [ABSTRACT FROM AUTHOR]

In this paper explicit necessary and sufficient conditions for the constrained Sylvester-observer equation are established, in order to have a solution over the field of real numbers. Furthermore, a procedure is given for the computation of the solution. Our approach is based on properties of real and polynomial matrices. Applications of the main results of this paper to linear control theory are discussed. [ABSTRACT FROM AUTHOR]

Let (a (n) : n ∈ N) denote a sequence of nonnegative integers. Let 0. a (1) a (2) ... denote the real number obtained by concatenating the digit expansions, in a fixed base, of consecutive entries of (a (n) : n ∈ N) . Research on digit expansions of this form has mainly to do with the normality of 0. a (1) a (2) ... for a given base. Famously, the Copeland-Erdős constant 0.2357111317 ... , for the case whereby a (n) equals the n th prime number p n , is normal in base 10. However, it seems that the "inverse" construction given by concatenating the decimal digits of (π (n) : n ∈ N) , where π denotes the prime-counting function, has not previously been considered. Exploring the distribution of sequences of digits in this new constant 0.0122 ... 9101011 ... would be comparatively difficult, since the number of times a fixed m ∈ N appears in (π (n) : n ∈ N) is equal to the prime gap g m = p m + 1 - p m , with the behaviour of prime gaps notoriously elusive. Using a combinatorial method due to Szüsz and Volkmann, we prove that Cramér's conjecture on prime gaps implies the normality of 0. a (1) a (2) ... in a given base g ≥ 2 , for a (n) = π (n) . [ABSTRACT FROM AUTHOR]

Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of Θ (n 3) triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them. For families of Θ (n 3 - α) triangles, with 0 ≤ α ≤ 1 , there might not be a point in more than Θ (n 3 - 2 α) of those triangles. An empty triangle of P is a triangle spanned by P not containing any point of P in its interior. Bárány conjectured that there exists an edge spanned by P that is incident to a super-constant number of empty triangles of P. The number of empty triangles of P might be as low as Θ (n 2) ; in such a case, on average, every edge spanned by P is incident to a constant number of empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number 0 ≤ α ≤ 1 there exists a point set of size n with Θ (n 3 - α) empty triangles such that any point of the plane is only in O (n 3 - 2 α) empty triangles. [ABSTRACT FROM AUTHOR]

Purpose: A stochastic technique for solving interval non-linear problems using generalized Hukuhara (GH)-difference is proposed. The non-linear programming problem in interval form is transformed into an equivalent non-linear programming problem with real coefficients by associating a Gaussian random variable to the interval and the six-sigma rule. The conceptualized idea eliminates the decision maker's instinctive selection of weight functions and provides an alternative to the order relation method, max-min criteria-based methods and bi-level approaches for representing intervals as real numbers. To demonstrate a coherent understanding, numerical examples have been used. Design/methodology/approach: A stochastic approach has been used to develop a solution technique for solving interval nonlinear programming problems which arise in the modeling of scientific and engineering problems under uncertain environments. Findings: The proposed idea eliminates the decision maker's instinctive selection of weight functions and provides an alternative to the order relation method, max-min criteria-based methods and bi-level approaches for representing intervals as real numbers. This method provides specific results rather than in the interval form, which are more practical and implementable by the decision maker. Originality/value: This is to certify, that the research paper submitted is an outcome of original work. I have duly acknowledged all the sources from which the ideas and extracts have been taken. This article has not been submitted elsewhere for publication. [ABSTRACT FROM AUTHOR]

In humanitarian aid scenarios, the model of cumulative capacitated vehicle routing problem can be used in vehicle scheduling, aiming at delivering materials to recipients as quickly as possible, thus minimizing their wait time. Traditional approaches focus on this metric, but practical implementations must also consider factors such as driver labor intensity and the capacity for on-site decision-making. To evaluate driver workload, the operation times of relief vehicles are typically used, and multi-objective modeling is employed to facilitate on-site decision-making. This paper introduces a multi-objective cumulative capacitated vehicle routing problem considering operation time (MO-CCVRP-OT). Our model is bi-objective, aiming to minimize both the cumulative wait time of disaster-affected areas and the extra expenditures incurred by the excess operation time of rescue vehicles. Based on the traditional grey wolf optimizer algorithm, this paper proposes a dynamic grey wolf optimizer algorithm with floating 2-opt (DGWO-F2OPT), which combines real number encoding with an equal-division random key and ROV rules for decoding; in addition, a dynamic non-dominated solution set update strategy is introduced. To solve MO-CCVRP-OT efficiently and increase the algorithm's convergence speed, a multi-objective improved floating 2-opt (F2OPT) local search strategy is proposed. The utopia optimum solution of DGWO-F2OPT has an average value of two fitness values that is 6.22% lower than that of DGWO-2OPT. DGWO-F2OPT's average fitness value in the algorithm comparison trials is 16.49% less than that of NS-2OPT. In the model comparison studies, MO-CCVRP-OT is 18.72% closer to the utopian point in Euclidean distance than CVRP-OT. [ABSTRACT FROM AUTHOR]