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Many numerical expressions contain a separator such as a comma (e.g., 1,234) or a decimal point (e.g., 12.34) that divides the number into left and right parts. This research examines how individuals compare such numbers as a function of psychological distance. Integrating insights from research on diagnosticity, numerical cognition, and mental construal, the authors hypothesize that as psychological distance increases, the perceived diagnosticity of digits to the right of a separator (hereinafter "right digits") decreases faster than that of the digits to the left of a separator (hereinafter "left digits"), which leads individuals to assign less weight to the right digits in comparative judgment. Four studies offer triangulating support for this theorizing and the underlying mechanism. In addition, the authors show that this effect is attenuated or suppressed when individuals perceive that numerical ratings are more stable over time (thereby increasing the perceived diagnosticity of right digits; Study 3) and when the distinction between left and right digits is made less obvious (e.g., when removing the digit separator; Study 4). This research offers additional nuance to our understanding of numerical cognition and how psychological distance influences the processing of different types of information. [ABSTRACT FROM AUTHOR]

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]

This paper gives a new characterisation of completeness of real numbers, which provides a visual way to understand some concepts in mathematical analysis. [ABSTRACT FROM AUTHOR]

This study explores the L p {L^{p}} -boundedness and compactness characteristics of SG pseudo-differential operators within the Sobolev space framework. We analyze different attributes of minimal-maximal pseudo-differential operators using the SG symbol σ ⁢ ( x , λ ) {\sigma(x,\lambda)} within the class S m 1 , m 2 {S^{m_{1},m_{2}}} , where m 1 {m_{1}} and m 2 {m_{2}} are real numbers. The investigation delves into various aspects of these operators, and we derive a weak solution for the SG pseudo-differential equation by employing the Dunkl transform as part of the aforementioned theoretical framework. [ABSTRACT FROM AUTHOR]

In this article, the complete convergence for the partial sum of the moving average process { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } is established under some mild conditions, where { Y i , − ∞ < i < ∞ } is a sequence of extended negatively dependent and identically distributed random variables that is stochastically dominated by a random variable Y under a sub-linear expectation (Ω , H , E ̂) , and { a i , − ∞ < i < ∞ } is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results of moving the average process under extended negatively dependent random variables from the traditional probability space to the sub-linear expectation space. [ABSTRACT FROM AUTHOR]

Recently, we formulated an effective scalar electrodynamics model of relativistic type-II superconductivity. We discussed possible electromagnetic behaviors of superconductivity along the lines of the nonrelativistic Ginzburg–Landau theory. This brief work considers a 2 D condensate subjected not only to a homogeneous magnetic field but also to a static radial electric field (polar coordinates). The electric field is regarded to have two components: a central constant field and a Coulomb-like field with "charge" g, where g can take only evenly spaced real numbers. We provide the single-vortex supercurrent density together with its associated magnetic flux density for the ground state of the system. Feasible extensions of this example in 3 D curvilinear geometries could explain some superconductivity phenomena which do not occur in planar 2 D geometries. [ABSTRACT FROM AUTHOR]

We obtain a full asymptotic for the sum of ζ(n)(ρ)Xρ, where ζ(n)(s) denotes the nth derivative of the Riemann zeta function, X is a positive real number, and ρ denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height T, as T →∞.We also specify what the asymptotic formula becomes when X is a positive integer, highlighting the differences in the asymptotic expansions as X changes its arithmetic nature. [ABSTRACT FROM AUTHOR]

We study Hilbert Poincaré series associated to general seed functions and construct Cohen’s kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin–Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier’s kernel formula to totally real number fields. Finally, we show that the Rankin–Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant. [ABSTRACT FROM AUTHOR]

Psychometrics conceptualizes a person's proficiency (or ability , or competence), in a cognitive or educational domain, as a latent numerical quantity. Yet both conceptual and empirical studies have shown that the assumption of quantitative structure for such phenomena is unlikely to be tenable. A reason why most applications of psychometrics nevertheless continue to treat them as if they were numerical quantities may be that quantification is thought to be necessary to enable measurement. This is indeed true if one regards the task of measurement as the location of a measurand at a point on the real number line (the viewpoint adopted by, for example, the representational theory of measurement, the realist theory of measurement as the discovery of ratios, and Rasch measurement theory). But this is not the only philosophically respectable way of defining the notion of measurement. This paper suggests that van Fraassen's more expansive view of measurement as, in general, location in a logical space (which could be the real continuum, as in metrological applications in the physical sciences, but could be a different mathematical structure), provides a more appropriate conceptual framework for psychometrics. Taking educational measurement as a case study, it explores what that could look like in practice, drawing on fuzzy logic and mathematical order theory. It suggests that applying this approach to the assessment of intersubjectively constructed phenomena, such as a learner's proficiency in an inherently fuzzily-defined subject area, entails recognizing the theory-dependent nature of valid representations of such phenomena, which need not be conceived of structurally as values of quantities. Finally, some connections are made between this "qualitative mathematical" theorization of educational assessment, and the application of techniques from machine learning and artificial intelligence in this area. [ABSTRACT FROM AUTHOR]

Usually, it is not direct to translate theoretical results to practice. One of the challenges is the real computation of solutions to proposed problems, when infinite numbers (such as real numbers or the unit interval) are considered. The same is true when consequences (information) from a data set modeled by logical rules need to be obtained. In this framework, this information is obtained from the immediate consequences operator and ensuring its continuity is fundamental for making sure the computation of this information in a countable number of iterations. Recently, a versatile operator using generalized quantifiers has been introduced. This note will advance in the knowledge introducing necessary and sufficient conditions to ensure the continuity of this new flexible operator. [ABSTRACT FROM AUTHOR]

We study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be characterized by any variant of the 290‐Theorem of Bhargava–Hanke. [ABSTRACT FROM AUTHOR]

In the current investigation, we offer the generalized version of q 1 ⁢ q 2 -Simpson's type inequalities via (α , m) -coordinated convex functions. To validate their generalized behavior, we demonstrate the link between our outcomes and the already derived ones. Moreover, we provide some application to special means of positive real numbers to support our findings. The principal outcomes raised in this investigation are extensions and generalizations of the comparable results in the history on Simpson's inequalities for coordinated convex functions. [ABSTRACT FROM AUTHOR]

In the early 1980s, the theory of didactic situations put the analysis of mathematical activity at the core of the didactic analysis, thus initiating a new research programme in mathematics education: the epistemological programme. This paper describes and interprets some of the contributions of the anthropological theory of the didactic (ATD) to the development of this programme. These contributions come from the new institutional and ecological perspective provided by the ATD to approach the didactic problem and from the significance it assigns to macrodidactic phenomena characterised by didactic transposition. They are illustrated with research works involving different fields of school mathematics in which the ATD team led by Spanish researchers played a central role: research in school algebra and its relations with functional modelling, differential calculus, and real numbers. The paper concludes by briefly describing the contributions of the ATD to the dialogue between didactic theories, another domain that has notable Spanish contributions. [ABSTRACT FROM AUTHOR]

In recent years, there has been a surge in new chaos-based cryptographic algorithms, many of which claim to have unusually large keyspaces. Although cryptographic primitives such as symmetric-key ciphers should have a secret keyspace large enough to resist brute force attacks, simply increasing the size of a secret key may not lead to improved security margins. An n-bit key may not necessarily have a keyspace of 2 n - 1 due to the key scheduling algorithm or how the key is used. In this paper, we cryptanalyse several chaos-based algorithms from the perspective of their key schedules. Our numerical analysis is based on the known-plaintext attack model, Kerckhoff's principle and considers the number representations used for real number computation. Our analysis reveals that the actual security margins for these ciphers are significantly lower, some by a factor of over 2 100 than what was claimed. We then provide accurate keyspace estimates for these ciphers. Finally, we highlight alternative solutions for how secret keys can be used in the context of chaos-based cryptography and propose a simple key schedule as a proof of concept. Despite its simplicity, the proposed key schedule not only ensures that the keyspace matches the key length but also passes both the NIST and ENT statistical test suites, making it a viable option for generating secure cryptographic keys. Our work contributes towards addressing one of the fundamental problems in chaos-based cryptography that limits its real-world impact and reputation within the cryptographic community. [ABSTRACT FROM AUTHOR]

We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let S be a real symmetric n × n -matrix with zeros on the diagonal and let θ be a real number. We construct nonzero solutions (S , θ) of the set of quadratic equations ∑ k S i , k = 0 and ∑ k S i , k S k , j + S i , j 2 = θ S i , j for i < j. Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields. [ABSTRACT FROM AUTHOR]

For a connected graph G with order n , let e (G) be the number of its distinct eigenvalues and d be the diameter. We denote by m G (μ) the eigenvalue multiplicity of μ in G. It is well known that e (G) ≥ d + 1 , which shows m G (μ) ≤ n − d for any real number μ. A graph is called m i n i m a l if e (G) = d + 1. In 2013, Wong et al. characterize all minimal graphs with m G (0) = n − d. In this paper, by applying the star complement theory, we prove that if G is not a path and m G (μ) = n − d , then μ ∈ { 0 , − 1 }. Furthermore, we completely characterize all minimal graphs with m G (− 1) = n − d. [ABSTRACT FROM AUTHOR]

Let G be a connected graph and let DQ(G) be the distance signless Laplacian matrix of G with eigenvalues ρ1 ≥ ρ2 ≥ · · · ≥ ρn. The spread of the matrix DQ(G) is defined as s(DQ(G)) := maxi,j |ρi - ρj | = ρ1 - ρn. We derive new bounds for the distance signless Laplacian spectral radius ρ1 of G. We establish a relationship between the distance signless Laplacian energy and the spread of DQ(G). For a real number α ≠ 0, the graph invariant mα(G) is the sum of the α-th power of the distance signless Laplacian eigenvalues of G. Finally, we obtain various bounds for the graph invariant mα(G). [ABSTRACT FROM AUTHOR]

Given a real number a ≠ 0, we consider the set of homeomorphisms f : R \ {0} → R \ {a} where {(x, y) : x = 0} is a vertical asymtote, {(x, y) : y = a} is a horizontal asymtote and f is strictly increasing in each connected component (-∞, 0) and (0,+∞). In this context, similar to circle homeomorphisms, all possible dynamics are shown. It is established the relationship between existence of periodic orbits and the limit sets. Also, whenever f-n(0) ≠ a for all n ∈ N, then the non-existence of periodic orbits leads to a non-trivial limit set, which is either the whole line R or perfect and nowhere dense. It is shown a notion of separation of points that leads to transitivity. [ABSTRACT FROM AUTHOR]

To construct Caputo fractional-order time-delayed cellar neural networks (FOTDCNNs) that characterize real environments, this article introduces partial uncertainties, fuzzy operators, and nonlinear activation functions into the network models. Specifically, both the fuzzy AND operator and the fuzzy OR operator are contemplated in the master–slave systems. In response to the properties of the considered cellar neural networks (NNs), this article designs a new class of mixed control protocols that utilize both the error feedback information of systems and the sampling information of impulse moments to achieve network synchronization tasks. This approach overcomes the interference of time delays and uncertainties on network stability. By integrating the fractional-order comparison principle, fractional-order stability theory, and hybrid control schemes, readily verifiable asymptotic synchronization conditions for the studied fuzzy cellar NNs are established, and the range of system parameters is determined. Unlike previous results, the impulse gain spectrum considered in this study is no longer confined to a local interval (−2, 0) and can be extended to almost the entire real number domain. This spectrum extension relaxes the synchronization conditions, ensuring a broader applicability of the proposed control schemes. [ABSTRACT FROM AUTHOR]

Extracellular vesicles (EVs) are valuable targets for liquid biopsy. However, attempts to introduce EV‐based biomarkers into clinical practice have not been successful to the extent expected. One of the reasons for this failure is the lack of reliable methods for EV baseline purification from complex biofluids, such as cell‐free plasma or serum. Because available one‐step approaches for EV isolation are insufficient to purify EVs, the majority of studies on clinical samples were performed either on a mixture of EVs and lipoproteins, whilst the real number of EVs and their individual specific biomarker content remained elusive, or on a low number of samples of sufficient volume to allow elaborate 2‐step EV separation by size and density, resulting in a high purity but utmost low recovery. Here we introduce Fast Protein Liquid Chromatography (FPLC) using Superose 6 as a matrix to obtain small EVs from biofluids that are almost free of soluble proteins and lipoproteins. Along with the estimation of a realistic number of small EVs in human samples, we show temporal resolution of the effect of the duration of postprandial phase on the proportion of lipoproteins in purified EVs, suggesting acceptable time frames additionally to the recommendation to use fasting samples for human studies. Furthermore, we assessed a potential value of pure EVs for liquid biopsy, exemplarily examining EV‐ and tumour‐biomarkers in pure FPLC‐derived fractions isolated from the serum of patients with pancreatic cancer. Consistent among different techniques, showed the presence of diseases‐associated biomarkers in pure EVs, supporting the feasibility of using single‐vesicle analysis for liquid biopsy. [ABSTRACT FROM AUTHOR]

A vertex-degree-based topological index φ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by m G , k . Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index φ. This number was denoted by m k G , φ and called the k-matchings of G with respect to the topological index φ. It turns out that m 1 G , φ = φ G , and so for k ≥ 2 , the k-matching numbers m k G , φ can be viewed as kth order topological indices which involve both the topological index φ and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices SC α , over the set T n of trees with n vertices, when α is a real number in the interval − 1 , 0. [ABSTRACT FROM AUTHOR]

In this study, we investigated how secondary students interpret algebraic expressions that contain literal symbols to stand for variables. We hypothesized that the natural number bias (i.e., the tendency to over-rely on knowledge and experiences based on natural numbers) would affect students to think that the literal symbols stand for natural numbers only rather than for any rational or real number (integrity effect); and that the arithmetical values of the algebraic expressions are of the same sign as the expressions' phenomenal sign (phenomenal sign effect). The participants (138 8th and 9th graders) were asked to evaluate 48 statements about numbers that can or cannot be assigned to six algebraic expressions that contained literal symbols (e.g., a, -d-4). The results supported the main hypothesis of the study with respect to the integrity as well as the phenomenal sign effect and also indicated that the former was stronger than the latter. Additionally, the most salient characteristics of the form of each expression, such as its sign, appeared to affect students' responses regarding the arithmetical values they may represent. Theoretical and educational implications are discussed. [ABSTRACT FROM AUTHOR]

Let p , q ≥ 1 be two real numbers such that 1 p + 1 q = 1 , and let a , b ∈ R be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form | f (a p + b q) | ≤ | f (a) | 1 p | f (b) | 1 q , and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities. [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]

Denoting by F2$F_2$ the free group over a two‐element alphabet, we show in set‐theory without the axiom of choice ZF$\mathsf {ZF}$ that the existence of a (2, 2)‐paradoxical decomposition of free F2$F_2$‐sets follows from the conjunction of a weakened consequence of the Hahn‐Banach axiom and a weakened consequence of the axiom of choice for pairs. The existence in ZF$\mathsf {ZF}$ of a paradoxical decomposition with 4 pieces of the sphere in the 3‐dimensional euclidean space follows from the same two statements restricted to the set R$\mathbb {R}$ of real numbers. Our result is linked to the (m,n)$(m,n)$‐paradoxical decompositions of free F2$F_2$‐sets previously obtained by Pawlikowski (m=n=3$m=n=3$, cf. [11]) and then by Sato and Shioya (m=3$m=3$ and n=2$n=2$, cf. [13]) with the sole Hahn‐Banach axiom. [ABSTRACT FROM AUTHOR]

Liouville proved the existence of a set L of transcendental real numbers now known as Liouville numbers. Erdős proved that while L is a small set in that its Lebesgue measure is zero, and even its s-dimensional Hausdorff measure, for each s > 0, equals zero, it has the Erdős property, that is, every real number is the sum of two numbers in L . He proved L is a dense G δ -subset of R and every dense G δ -subset of R has the Erdős property. While being a dense G δ -subset of R is a purely topological property, all such sets contain c Liouville numbers. Each dense G δ -subset of R , including L , is homeomorphic to the product N ℵ 0 of copies of the discrete space N of all natural numbers. Also this product space is homeomorphic to the space P of all irrational real numbers and the space T of all transcendental real numbers. Hence every dense G δ -subset of R has cardinality c . Indeed, any dense G δ -subset of R has a chain Xm, m ∈ (0 , ∞) of homeomorphic dense G δ -subsets such that X m ⊂ X n , for n < m, and X n ∖ X m has cardinality c . Finally, every real number r ≠ 1 is equal to ab, for some a , b ∈ L . [ABSTRACT FROM AUTHOR]

The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. Special attention is given to modelling such functions by systems of functional equations. [ABSTRACT FROM AUTHOR]

In The traveling salesman problem (TSP) is an essential and the most popular conventional combinatorial optimization network problem in operations research, in which the TSP evaluates the shortest route or path in a network. In TSP, every node has been visited only once, excluding the starting node. In TSP, edge lengths are usually expressed to indicate journey time and expenses instead of distance from a location. The exact arc length can't be predicted because journey times and expenses vary depending on the amount of payload, climate, highway conditions, and so on. As a result, the Neutrosophic numbers introduce a new tool for dealing with unpredictability in TSP. The present article addresses TSP on a neutrosophic network where the edge weight is a neutrosophic number rather than a real number. For solving the Neutrosophic TSP, an algorithmic technique based on the genetic algorithm (GA) is proposed. We created a new mutation and crossover for our suggested GA. We used mathematical examples to show the usefulness of the algorithm that we suggested. The results of experiments suggest that the proposed GA can find the shortest path in a TSP within a neutrosophic framework. This provides valuable insights for decision-makers dealing with real-world situations characterized by imprecise and incomplete data. [ABSTRACT FROM AUTHOR]

Let k be a positive integer. A k‐cycle‐factor of an oriented graph is a set of disjoint cycles of length k that covers all vertices of the graph. In this paper, we prove that there exists a positive constant c such that for n sufficiently large, any oriented graph on n vertices with both minimum out‐degree and minimum in‐degree at least (1∕2−c)n contains a k‐cycle‐factor for any k≥4. Additionally, under the same hypotheses, we also show that for any sequence n1,...,nt with ∑i=1tni=n and the number of the ni equal to 3 is αn, where α is any real number with 0<α<1∕3, the oriented graph contains t disjoint cycles of lengths n1,...,nt. This conclusion is the best possible in some sense and refines a result of Keevash and Sudakov. [ABSTRACT FROM AUTHOR]

In real-life multi-criteria decision-making problems, decision-making data often exhibit ambiguity due to incomplete information. Additionally, qualitative judgments by decision-makers can introduce fallacies and inaccuracies. Consequently, these problems cannot be resolved using precise values alone. To address this, the present study enhances the Best-Worst Method(BWM) by incorporating interval neutrosophic sets, thereby improving its applicability to real-life multicriteria decision-making issues. In the modified BWM approach detailed in this study, decision-makers express preferences using linguistic terms, which are then converted into interval neutrosophic numbers. These numbers facilitate the comparative assessment between the best and other criteria, as well as between the other criteria and the worst criterion. All interval neutrosophic numbers are subsequently converted into real numbers using the score function s(a). Furthermore, a new nonlinear constrained optimization model concerning interval neutrosophic numbers is formulated according to the BWM framework. The resultant data, representing the weights of different criteria, do not require further transformation. A consistency ratio for BWM is also introduced to evaluate the reliability of preference comparisons. Comparative analysis of three methods using the same case study confirms the efficacy and viability of the proposed method, namely the interval neutrosophic set based BWM. [ABSTRACT FROM AUTHOR]

We propose a new machine learning formulation designed specifically for extrapolation. The textbook way to apply machine learning to drug design is to learn a univariate function that when a drug (structure) is input, the function outputs a real number (the activity): f(drug) → activity. However, experience in real-world drug design suggests that this formulation of the drug design problem is not quite correct. Specifically, what one is really interested in is extrapolation: predicting the activity of new drugs with higher activity than any existing ones. Our new formulation for extrapolation is based on learning a bivariate function that predicts the difference in activities of two drugs F(drug1, drug2) → difference in activity, followed by the use of ranking algorithms. This formulation is general and agnostic, suitable for finding samples with target values beyond the target value range of the training set. We applied the formulation to work with support vector machines , random forests , and Gradient Boosting Machines. We compared the formulation with standard regression on thousands of drug design datasets, gene expression datasets and material property datasets. The test set extrapolation metric was the identification of examples with greater values than the training set, and top-performing examples (within the top 10% of the whole dataset). On this metric our pairwise formulation vastly outperformed standard regression. Its proposed variations also showed a consistent outperformance. Its application in the stock selection problem further confirmed the advantage of this pairwise formulation. [ABSTRACT FROM AUTHOR]

Deep Convolutional Neural Networks (DCNNs) are very useful for image-based pattern classification problems because of their efficient feature extraction capabilities. Although DCNNs have good generalization performance, their applicability is limited due to slow learning speed, as they are based on iterative weight-update algorithms. This study presents a new noniterative DCNN that can be trained in real-time. The fundamental block of the proposed DCNN is fixed real number-based filters for convolution operations for multi-feature extraction. After a finite number of feature extraction layers, nonlinear kernel mapping along with pseudo-inverse is used for the classification of extracted feature vectors. The proposed DCNN, named Deep Convolutional Kernelized Classification (DCKC), is noniterative, as the mask coefficients of its convolution operations are fixed real numbers. The kernel function with predefined parameters of DCKC does a nonlinear mapping of extracted features, and pseudo-inverse is used to find its output weights. The proposed noniterative DCKC was evaluated on benchmark face recognition databases, achieving better results and establishing its superiority. [ABSTRACT FROM AUTHOR]

AbstractAn equation was proposed to obtain the full range of vapor pressures (VPs) for a substance at an arbitrary point of VP data. The basic model is a vapor pressure equation in the form of a reduced Antoine equation derived from the van der Waals equation. Here, Antoine constant C is defined as c by dividing it into gas constant and critical temperature, and is expressed as a polynomial function of reduced temperature. Previously, the exponents were integers, but have been improved to real numbers. Edmister’s formula is a special case in which the Antoine constant C divided by the critical temperature in Chen’s equation is 0. Since this is equivalent to polynomial of c being zero, the intercept can be set to zero. The results, estimated using the derived equation, were compared with VP data for 76 substances. The error rate of the equation using the acentric factor as a variable was 0.49%. This was lower than the 0.50% error rate of the Lee-Kesler method. Meanwhile, by using the correlation between the coefficients, the VP equation can be derived at any one point of VP data other than the acentric factor. For 76 substances, the equation derived from the corresponding VPs at reduced temperatures of 0.25 ∼ 0.95 in increments of 0.05 showed a mean error rate of 0.68%. This can be considered an accuracy equivalent to the Lee-Kesler method, which has the limitation of deriving a VP equation only at the reduced temperature of 0.7. [ABSTRACT FROM AUTHOR]

The topological index is just one of several very useful tools that graph theory has made available to chemists. Topological indices are invariants of real numbers under graph isomorphisms. Several topological indices have been defined. Some of them are used to model chemical, pharmaceutical and other properties of molecules. The eccentric connectivity index (eci) is also a topological index. The eci of G, denoted by 휀c(G), is defined as 휀c(G) =∑ v∈V (G)deg(v)e(v), where deg(v) represents the degree of a vertex v and e(v) is its eccentricity. In this paper, exact formula for the eci of complementary prisms is derived. [ABSTRACT FROM AUTHOR]

In this paper, we consider the viscoelastic Kirchhoff plate equation with memory and time delay. Under appropriate conditions on the real numbers μ1$$ {\mu}_1 $$ and μ2$$ {\mu}_2 $$, we prove the existence of a compact global attractor with finite fractal dimension through a stabilizability estimate. Additionally, we demonstrate the existence of a fractal exponential attractor. To the best of our knowledge, these findings are novel for for this system. [ABSTRACT FROM AUTHOR]

Signifiable computability aims to separate what is theoretically computable from what is computable through performable processes on computers with finite amounts of memory. Real numbers and sequences thereof, data types, and instances are treated as finite texts, and memory limitations are made explicit through a requirement that the texts be stored in the available memory on the devices that manipulate them. In Part I of our investigation, we define the concepts of signification and reference of real numbers. We extend signification to number tuples, data types, and data instances and show that data structures representable as tuples of discretely finite numbers are signifiable. From the signification of real tuples, we proceed to the constructive signification of multidimensional matrices and show that any data structure representable as a multidimensional matrix of discretely finite numbers is signifiable. [ABSTRACT FROM AUTHOR]

Motivated by certain results known from the literature, in this paper we give several new dynamic inequalities for B-monotone functions with respect to time scales nabla calculus. If the time scale represents the set of real numbers, our results reduce to integral inequalities known from the literature. On the other hand, in the setting of positive integers, we obtain new discrete inequalities for B-monotone sequences. [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]

AbstractThe advent of industrial big data has provided an unprecedented opportunity to achieve a data-driven monitoring of large-scale complex processes. When a process involves massive categorical variables each evaluated by attribute levels rather than real numbers, which is common in modern manufacturing and service applications, the existing process monitoring methods typically fail in modeling the joint distribution of these categorical variables due to the curse of high dimensionality. To fill this research gap, we propose a novel data mining–based framework—a nonparametric method—for High-Dimensional (HD) categorical process monitoring. Specifically, a series of multiscale frequent patterns are particularly defined and quickly extracted to characterize both the significant individual behaviors and the major collective behaviors of HD categorical variables. Then all these discovered multiscale patterns, serving as informative surrogates of the original HD categorical data, are monitored sequentially from low scale to high scale via a principled and powerful multiple hypotheses testing procedure embedded with an alpha spending function and a false discovery rate approach. The superiority of our proposed method is validated extensively by numerical simulations and real case studies. It is capable of maintaining a desired false alarm rate when the process is normal and becoming very sensitive to many different kinds of process shifts. [ABSTRACT FROM AUTHOR]

We introduce FdeSolver, an open-source Julia package designed to solve fractional-order differential equations efficiently. The available solutions are based on product-integration rules, predictor–corrector algorithms, and the Newton-Raphson method. The package covers solutions for one-dimensional equations with orders of positive real numbers. For higher-dimensional systems, it supports orders up to one. Incommensurate derivatives are allowed and defined in the Caputo sense. Here, we summarize the implementation for a representative class of problems and compare it with available alternatives in Julia and MATLAB. Moreover, FdeSolver leverages the power and flexibility of the Julia environment to offer enhanced computational performance, and our development emphasizes adherence to the best practices of open research software. To highlight its practical utility, we demonstrate its capability in simulating microbial community dynamics and modeling the spread of COVID-19. This latter application involves fitting the order of derivatives grounded on real-world epidemiological data. Overall, these results highlight the efficiency, reliability, and practicality of the FdeSolver Julia package. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to propose a mechanical procedure to generate fuzzy data from the observed real number data. The observed data may be imprecise owing to some unexpected situation. In this case, the decision-makers may treat the observed data to be fuzzy data by providing suitable membership functions. However, the determination of those membership functions is frequently subjective for the different decision-makers. To avoid this biased subjectivity raised by the decision-makers, this paper proposes a mechanical procedure to generate fuzzy data. The essence of this mechanical procedure is to propose an extended form to generate a fuzzy set from a pre-determined family. This extended form is based on two functions, such that the conventional form can be covered when these these two functions are taken to be the identity functions. One of these two functions is adopted to rearrange the pre-determined family to be a nested family. A more general form is also proposed by considering a countable subfamily. [ABSTRACT FROM AUTHOR]

Let ᴦ be a distance-regular graph with valency k and diameter D ≥ 3. It has been shown that for a fixed real number α > 2, if ᴦ has at most αk vertices, then there are only finitely many such graphs, except for the cases where (D = 3 and ᴦ is imprimitive) and (D = 4 and ᴦ is antipodal and bipartite). And there is a classification for α ≤3. In this paper, we further study such distance-regular graphs for α > 3. Let β ≥ 3 be an integer, and let ᴦ be a distance-regular graph with valency k, diameter D ≥ 3 and at most βk + 1 vertices. Note that if D ≥ β+1, then ᴦ must have at least βk + 2 vertices. Thus, the assumption that ᴦ has at most βk + 1 vertices implies that D ≤ β. We focus on the case where D = β and provide a classification of distance-regular graphs having at most Dk + 1 vertices. [ABSTRACT FROM AUTHOR]

Symmetry plays a crucial role in the study of dual matrices and dual matrix group inverses. This paper is mainly divided into two parts. We present the definition of the spectral norm of a dual real matrix A ^ , (which is usually represented in the form A ^ = A + ε A 0 , A and A 0 are, respectively, the standard part and the infinitesimal part of A ^ ) and two matrix decompositions over dual rings. The group inverse has been extensively investigated and widely applied in the solution of singular linear systems and computations of various aspects of Markov chains. The forms of the dual group generalized inverse (DGGI for short) are given by using two matrix decompositions. The relationships among the range, the null space, and the DGGI of dual real matrices are also discussed under symmetric conditions. We use the above-mentioned facts to provide the symmetric expression of the perturbed dual real matrix and apply the dual spectral norm to discuss the perturbation of the DGGI. In the real field, we present the symmetric expression of the group inverse after the matrix perturbation under the rank condition. We also estimate the error between the group inverse and the DGGI with respect to the P-norm. Especially, we find that the error is the infinitesimal quantity of the square of a real number, which is small enough and not equal to 0. [ABSTRACT FROM AUTHOR]

The Erdős similarity conjecture asserted that an infinite set of real numbers cannot be affinely embedded into every measurable set of positive Lebesgue measure. The problem is still open, in particular for all fast decaying sequences. In this paper, we relax the problem to the bi-Lipschitz embedding and obtain some sharp criteria about the bi-Lipschitz Erdős similarity problem for strictly decreasing sequences. [ABSTRACT FROM AUTHOR]

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ -Fibonacci statistical convergence, strong Δ -Fibonacci summability, and Δ -Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers. [ABSTRACT FROM AUTHOR]

This document is from the College Mathematics Journal and contains a section with problems and solutions for college mathematics students and teachers. The editors encourage active participation by submitting solutions and proposing new and interesting problems. The document provides instructions for submitting problems and solutions, including the accepted file formats. It also includes a list of proposed problems and their corresponding solutions. Additionally, the document includes an editorial comment regarding a previously published incorrect solution and provides the intended solution for that problem. [Extracted from the article]

Let K be a totally real number field of degree n over ℚ , with discriminant and regulator Δ K , R K , respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by | Δ K | − 1 ∕ 2 γ K n exp (2 n ρ ∞ (Λ K)) , where Δ K is the discriminant of the field K, γ K is the additive Hermite–Humbert constant over positive-definite unary forms for K and ρ ∞ (Λ K) is the covering radius of the log-unit lattice. In particular, when K is Galois over ℚ and n is a prime number, the number of homothety classes of unary forms is upper bounded by (2 3 n) n exp (O (n 3 2 log (n) R K 1 n − 1 )) | Δ K | 1 ∕ 2 , where R K is the regulator of K. Moreover, if K is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by (2 3 n) n exp (O (n 3 2 log (n))) | Δ K | 1 ∕ 2 . [ABSTRACT FROM AUTHOR]

This paper is dedicated to defining and studying for the first time a two-fold algebra over the neutrosophic real number ring by merging the fuzzy set mapping with the algebraic operations of the neutrosophic real number ring. Also, we study the elementary algebraic properties of the defined two-fold algebra through its algebraic operations and substructures such as homomorphisms and ideals. [ABSTRACT FROM AUTHOR]

The paper investigates a new type of contraction called (M, K, ψ)-HR-Ćirić-Reich-Rus contractions, which extends the existing research on Ćirić-Reich-Rus contraction mappings. We prove fixed point theorems by enriching (M, K, ψ)-HR-Ćirić-Reich-Rus contractions with respect to the asymptotic regularity of generalized averaged mappings. These mappings are denoted as Tφ = (1 - φ) I + φT, where I is the identity map and φ is a function from a normed space to real numbers. [ABSTRACT FROM AUTHOR]