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In this paper we study a family of doubled and quadrupled Fibonacci type sequences obtained by distance generalization of Fibonacci sequence. In particular we obtain doubled Fibonacci sequence, doubled and quadrupled Padovan sequence and quadrupled Narayana's sequence. We give a binomial direct formula for these sequences using graph methods, and also we derive a number of identities. Moreover, we study matrix generators of these sequences and determine connections with the Pascal's triangle. [ABSTRACT FROM AUTHOR]

We study a variant of the color-avoiding percolation model introduced by Krause et al. , namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős–Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if, after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size, as well as the fraction of vertices contained in the giant color-avoidin g connected component. It is known that these quantities converge, and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the fraction of vertices contained in the giant color-avoiding connected component, and we give a simpler asymptotic expression for it in the barely supercritical regime. In addition, in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we attempt to use the Dunkl analog to study the convergence properties of q-Phillips operators by using the q-Appell polynomials. By applying the new sequences of continuous functions ν s , q ( z ) = ( z − 1 2 [ s ] q ) ϱ $\nu _{s,q}(z)=\Bigl(z- \frac{1}{2[s]_{q}}\Bigr)_{\varrho}$ on [ 0 , ∞ ) $[0, \infty )$ , we construct an improved version of the q-Phillips operators. We calculate the qualitative outcomes in weighted Korovkin spaces to better understand the Phillips operators’ uniform convergence results. We obtain the approximation properties by use of the modulus of continuity and functions belonging to the Lipschitz class. Moreover, we give some direct theorems for the function belonging to Peetre’s K-functional.

This work focuses on the study of approximation properties of functions by Szász type operators involving Frobenius–Euler–Simsek-type polynomials, which have become more popular recently because of their special characteristics and functional organization. The convergence properties such as uniformly convergence and pointwise convergence in terms of modulus of continuity and Peetre-$ \it K $ functional are investigated with the help of these sequences of operators in depth. This paper also includes the estimation of the error of the approximation of these sequences of operators to some particular class of functions. The estimates are depicted using the Maple scientific computing program and presented in tables.

This work represents a natural coalescence of two important lines of work — learning mixtures of Gaussians and algorithmic robust statistics. In particular, we give the first provably robust algorithm for learning mixtures of any constant number of Gaussians. We require only mild assumptions on the mixing weights and that the total variation distance between components is bounded away from zero. At the heart of our algorithm is a new method for proving a type of dimension-independent polynomial identifiability — which we call robust identifiability — through applying a carefully chosen sequence of differential operations to certain generating functions that not only encode the parameters we would like to learn but also the system of polynomial equations we would like to solve. We show how the symbolic identities we derive can be directly used to analyze a natural sum-of-squares relaxation. [ABSTRACT FROM AUTHOR]

Let F be the finite field of order q and M (n , r , F) be the set of n × n matrices of rank r over the field F. For α ∈ F and A ∈ M (n , F) , let Z A , r α = { X ∈ M (n , r , F) | Tr (A X) = α }. In this article, we solve the problem of determining the cardinality of Z A , r α. We also solve the generalization of the problem to rectangular matrices. [ABSTRACT FROM AUTHOR]

Transposable elements (TEs) provide a prime example of genetic conflict because they can proliferate in genomes and populations even if they harm the host. However, numerous studies have shown that TEs, though typically harmful, can also provide fuel for adaptation. This is because they code functional sequences that can be useful for the host in which they reside. In this review, I summarize the "how" and "why" of adaptation enabled by the genetic conflict between TEs and hosts. In addition, focusing on mechanisms of TE control by small piwi-interacting RNAs (piRNAs), I highlight an indirect form of adaptation enabled by conflict. In this case, mechanisms of host defense that regulate TEs have been redeployed for endogenous gene regulation. I propose that the genetic conflict released by meiosis in early eukaryotes may have been important because, among other reasons, it spurred evolutionary innovation on multiple interwoven trajectories - on the part of hosts and also embedded genetic parasites. This form of evolution may function as a complexity generating engine that was a critical player in eukaryotic evolution. [ABSTRACT FROM AUTHOR]

The deneutrosophication process is a process transforming from neutrosophic values to crisp output values. It is the final step for the operations within a neutrosophic set and system. Neutrosophic set theories are a generalization of intuitionistic fuzzy and fuzzy set theories, focusing on truth, indeterminacy, and falsity memberships independently. However, it isn't easy to generate a geometrical model such as a Bézier surface by using neutrosophic set theory through the deneutrosophication process. Therefore, this paper used an average of triangular footprint method in the deneutrosophication process to construct the neutrosophic Bézier surface (NBS) models by using approximation methods. Before generating the model, the neutrosophic control net (NCN) must first be introduced using the deneutrosophication process. After that, the NCN will be blended with the Bernstein basis function to generate the NBS approximation model. Next, some numerical examples of NBS will be provided. Finally, the deneutrosophication of NBS approximation models will be visualized, and its algorithm will be shown. [ABSTRACT FROM AUTHOR]

In this paper, we continue our investigation of the triple Hodge integrals satisfying the Calabi–Yau condition. For the tau-functions, which generate these integrals, we derive the complete families of the Heisenberg–Virasoro constraints. We also construct several equivalent versions of the cut-and-join operators. These operators describe the algebraic version of topological recursion. For the specific values of parameters associated with the KdV reduction, we prove that these tau-functions are equal to the generating functions of intersection numbers of ψ and κ classes. We interpret this relation as a symplectic invariance of the Chekhov–Eynard–Orantin topological recursion and prove this recursion for the general Θ -case. [ABSTRACT FROM AUTHOR]

The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the n -th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the n -th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every n -th production matrix can be factorized into a product of n matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the n -th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the n -th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases. [ABSTRACT FROM AUTHOR]

Let U (n , F q 2 ) denote the subgroup of unitary matrices of the general linear group GL (n , F q 2 ) which fix a Hermitian form and let M ≥ 2 be an integer. In earlier work, elements of the groups GL (n , F q) , Sp (2 n , F q ) , O ± (2 n , F q ) and O (2 n + 1 , F q ) with an M -th root have been described. Here we will describe the M -th powers in unitary groups for the regular semisimple, semisimple and cyclic elements, under the assumption that M and q are coprime. We will further describe the generating functions in the corresponding cases. [ABSTRACT FROM AUTHOR]

We consider one variant of stable assignment problems in a bipartite graph endowed with nonnegative capacities on the edges and quotas on the vertices. It can be viewed as a generalization of the stable allocation problem introduced by Baïou and Balinsky, which arises when strong linear orders of preferences on the vertices in the latter are replaced by weak ones. At the same time, our stability problem can be stated in the framework of a theory by Alkan and Gale on stable schedule matchings generated by choice functions of a wide scope. In our case, the choice functions are of a special, so-called mixed , type. The main content of this paper is devoted to a study of rotations in our mixed model, functions on the edges determining "elementary" transformations between close stable assignments. These look more sophisticated compared with rotations in the stable allocation problem (which are generated by simple cycles). We efficiently construct a poset of rotations and show that the stable assignments are in bijection with the so-called closed functions for this poset; this gives rise to a "compact" affine representation for the lattice of stable assignments and leads to an efficient method to find a stable assignment of minimum cost. [ABSTRACT FROM AUTHOR]

This paper presents exact derivations of the first two moments of the total order picking time in a warehouse for three routing heuristics, under the assumption of random storage. The analysis is done for general order size distributions and provides exact closed expressions in terms of the probability generating function of the order size distribution. We also indicate how the methods and insights in this paper extend to different storage policies and multi-block warehouses. The exact results derived in this paper are used to investigate effects of routing heuristics, order size distributions and layouts on warehouse efficiency. As illustration, we model the warehouse as a queueing system. By using approximations of the average order-lead time in terms of the first two moments of the order picking time, we are able to find optimal warehouse layouts and batch pick sizes. [ABSTRACT FROM AUTHOR]

The expected spacing, or average difference between consecutive order statistics, is known for uniform and exponential random variates. For other distributions, we can estimate it using the derivative of the inverse cumulative density (quantile) function, since passing a uniformly drawn value, whose spacing we know, through this function generates a random value from the distribution, and the difference between two such uniform values approximates the derivative. We calculate the spacing for two new distributions, the logistic and Gumbel, and show the estimator is exact for the first and approximate for the second. Comparing the estimators for six other distributions to numeric simulations shows they are also approximations, best in the middle of the order statistics with an error that goes inversely with the square of the sample size, but degrading in the tails. [ABSTRACT FROM AUTHOR]

In this paper, we derive some nonlinear differential equations from generating function of generalized harmonic numbers and give some identities involving generalized harmonic numbers and special numbers by using these differential equations. For example, for any positive integers N, n, r, a and any integer m ≥2, ... where S1 (n, k) is Stirling number of the first kind. [ABSTRACT FROM AUTHOR]

Given an integer n ≥ 4, how many inequivalent quadrilaterals with ordered integer sides and perimeter n are there? Denoting such number by Q(n), the answer is given by the following closed formula: ... . [ABSTRACT FROM AUTHOR]

In this paper, a multi-objective optimization strategy for efficient design of turbo machinery blades using sparse active subspaces is implemented for a turbofan stage design. The proposed strategy utilized sparse polynomial chaos expansion on a limited dataset to generate a function from which the differential and the covariance matrix can be obtained. Active subspace was used to compute the active variables via singular value decomposition and a hybrid polynomial correlated function expansion was used to construct a surrogate model on the active subspace. Coupled with freeform method and multi-objective genetic algorithm, an automated optimization loop was run at a single operating condition. An improvement in stage efficiency and total pressure ratio of 2.97% and 1.15% was achieved for the optimum design compared with the baseline. Additionally, total pressure loss coefficient decreased by 5.88%, exit flow angle by 34.65% and shock strength by 5.32%. The coupled effect of change in stagger angle, forward sweep, forward lean, and chord length reduced the recirculation at the hub, and blockage at the shroud due the tip leakage flow by decreasing the blade loading. The threshold value hyperparameter was found to be the most influential and must be accurately determined. [ABSTRACT FROM AUTHOR]

In the 1970s, Nicolas proved that the partition function p(n) is log-concave for n > 25 . In Heim et al. (Ann Comb 27(1):87–108, 2023), a precise conjecture on the log-concavity for the plane partition function pp (n) for n > 11 was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences { g d (n) } d , n with g d (1) = 1 and 0 ≤ g d n - n d ≤ g 1 n n - 1 d - 1 , polynomials { P n g d (x) } d , n given by ∑ n = 0 ∞ P n g d (x) q n : = exp x ∑ n = 1 ∞ g d (n) q n n = ∏ n = 1 ∞ 1 - q n - x f d (n) . We recover p (n) = P n σ 1 (1) and pp n = P n σ 2 (1) , where σ d (n) : = ∑ ℓ ∣ n ℓ d and f d (n) = n d - 1 . Let n ≥ 6 . Then the sequence { P n σ d (1) } d is log-concave for almost all d if and only if n is divisible by 3. Let id (n) = n . Then P n id (x) = x n L n - 1 (1) (- x) , where L n α x denotes the α -associated Laguerre polynomial. In this paper, we investigate Turán inequalities Δ n g d (x) : = P n g d (x) 2 - P n - 1 g d (x) P n + 1 g d (x) ≥ 0. Let n ≥ 6 and 0 ≤ x < 2 - 12 n + 4 . Then n is divisible by 3 if and only if Δ n g d (x) ≥ 0 for almost all d. Let n ≥ 6 and n ≢ 2 (mod 3) . Then the condition on x can be reduced to x ≥ 0 . We determine explicit bounds. As an analogue to Nicolas' result, we have for g 1 = id that Δ n id (x) ≥ 0 for all x ≥ 0 and all n. [ABSTRACT FROM AUTHOR]

In this article, we define three-dimensional q-Riordan arrays and q-Riordan representations for these arrays. Also, we give four cases of infinite multiplication three-dimensional matrices of these arrays. As applications, we obtain three-dimensional qq-Pascal-like matrix and its inverse matrix by Heine’s binomial formula, using combinatorial identities. Finally, we consider the generalization of three-dimensional qq-Pascal-like matrix and give some identities involving qq-binomial coefficients.

We produce congruences modulo a prime p > 3 for sums ∑k3k k xk over ranges 0 ≤ k < q and 0 ≤ k < q/3, where q is a power of p. Here x equals either c2/(1 − c)3, or 4s2/(27(s2 − 1)), where c and s are indeterminates. In the former case, we deal more generally with shifted binomial coefficients 3k+e k. Our method derives such congruences directly from closed forms for the corresponding series. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a spinor algebra of k-balancing numbers referred to as the k-balancing and k-Lucas-balancing spinors. First, we give k-balancing quaternions and their some algebraic properties. Then we introduce a spinor family of k-balancing numbers by defining a linear and injective correspondence between k-balancing and k-Lucas-balancing quaternions to spinors. Here, we give various algebraic properties of this spinor such as the Binet form, Catalan’s identity, d’Ocagne identity, generating and exponential generating functions. Moreover, we obtain various partial sum formulae for these sequences in closed form and also give matrix representations. [ABSTRACT FROM AUTHOR]

In this work, we study the properties of the cumulative residual entropy generating function (CREGF). We also discuss the non parametric estimation of CREGF. We introduce dynamic cumulative residual entropy generating function (DCREGF). It is shown that the DCREGF determines the distribution uniquely. We study some characterization results using the relationship between DCREGF, hazard rate, and mean residual life function. A new class of life distributions based on decreasing DCREGF is introduced. Finally, we develop a test for decreasing DCREGF and study its performance. [ABSTRACT FROM AUTHOR]

We study the T -system of type A ∞ , also known as the octahedron recurrence/equation, viewed as a 2 + 1 -dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with 'flat' initial data, we consider initial data along parallel 'slanted' planes perpendicular to an arbitrary admissible direction (r , s , t) ∈ Z + 3 . The corresponding solutions of the T -system are interpreted as partition functions of dimer models on some suitable 'pinecone' graphs introduced by Bousquet–Mélou, Propp, and West in 2009. The T -system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate Hermite-based Peters-type Simsek polynomials with generating functions. By using generating function methods, we determine some of the properties of these polynomials. By applying the derivative operator to the generating functions of these polynomials, we also determine many of the identities and relations that encompass these polynomials and special numbers and polynomials. Moreover, using integral techniques, we obtain some formulas covering the Cauchy numbers, the Peters-type Simsek numbers and polynomials of the first kind, the two-variable Hermite polynomials, and the Hermite-based Peters-type Simsek polynomials. [ABSTRACT FROM AUTHOR]

This paper presents a novel framework for introducing generalized 1-parameter 3-variable Hermite polynomials. These polynomials are characterized through generating functions and series definitions, elucidating their fundamental properties. Moreover, utilising a factorisation method, this study establishes recurrence relations, shift operators, and various differential equations, including differential, integro-differential, and partial differential equations. [ABSTRACT FROM AUTHOR]

We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work of Panholzer and Prodinger, we apply this method to study the depth distributions of individual nodes or leaves in rooted binary trees, plane trees, noncrossing trees, and increasing trees; the height distributions of individual vertices and individual steps in Dyck paths; and the number of diagonals separating two fixed sides of a convex polygon in a triangulation or dissection. We obtain both exact and asymptotic results, which sometimes refine known formulas or provide combinatorial proofs of results from the probability literature. [ABSTRACT FROM AUTHOR]

Based on the extensive application of the q$$ q $$‐series and q$$ q $$‐polynomials including (for example) the q$$ q $$‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the q$$ q $$‐Laguerre polynomials. The mission of this paper is to find the general q$$ q $$‐operational equation together with the expansion issue of the bivariate q$$ q $$‐Laguerre polynomials from the perspective of q$$ q $$‐partial differential equations. We also give some applications including some q$$ q $$‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the U(n+1)$$ U\left(n&#x0002B;1\right) $$‐type generating functions for the bivariate q$$ q $$‐Laguerre polynomials by the technique based upon q$$ q $$‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate q$$ q $$‐Laguerre polynomials by applying q$$ q $$‐operational equations. U(n+1)$$ U\left(n&#x0002B;1\right) $$ [ABSTRACT FROM AUTHOR]

To meet the needs of society, the next generation of communication systems will have to learn to juggle multiple responsibilities effectively. This is leading to exponential growth in both circuit complexity and error rates. It is possible to solve these issues by using the intelligent reflecting surface (IRS)-Assisted space shift keying (SSK) modulation technique. IRS is a cutting-edge technology that shows promise for future networks due to its low mistake rates and low-energy consumption. Similarly, SSK modulation is favored due to its preference for simpler hardware. Therefore, integrating IRS with SSK is preferable for minimizing both error rates and circuit complexity. In this article, a new framework is developed to characterize the average bit error rate (ABER) with the help of the moment generating function (MGF) over Nakagami-m fading channels. The ABER is derived by factoring the direct path and IRS reflected path at the receiver and also comparing the performance analysis over the Rayleigh fading channel. The analytical results are validated through Monte-Carlo simulations. [ABSTRACT FROM AUTHOR]

We introduce a new flexible distribution four-parameter which is called [0,1]Truncated Exponentiated Exponential Burr type X distribution ([0,1] TEE-BX). Using binomial series expansion and exponential expansion, the new distribution is expanded by four parameters. We derive moments, moment generating function, quantile function, order statistics and the Rényi entropy. The maximum likelihood estimation method is used to estimate the parameters of the proposed TEE-BX model [0,1]. Finally, using two real-world data sets, the performance of the [0,1] TEE-BX distribution is explored. Based on the certain goodness of fit criteria, we conclude that the [0,1] TEE-BX distribution has a better fit than the other distributions. [ABSTRACT FROM AUTHOR]

In the realm of specialized functions, the allure of q -calculus beckons to many scholars, captivating them with its prowess in shaping models of quantum computing, noncommutative probability, combinatorics, functional analysis, mathematical physics, approximation theory, and beyond. This study explores a new idea called the multidimensional q -Hermite polynomials, using different q -calculus techniques. Numerous properties and novel findings regarding these polynomials are derived, encompassing their generating function, series representations, recurrence relations, q -differential formula, and operational principles. Further, we proved that these polynomials are quasi-monomial in q -aspect. As the applications, these findings are subsequently employed to address connection between the multidimensional q -Hermite polynomials and the two-variable q -Legendre polynomials for the first time. Various characterizations are examined, as well the graphical representations of the two-variable q -Legendre polynomials are provided by the surface plots and graphs of distribution of zeros for the q -Legendre polynomials with some specific set of parameters are shown using Mathematica. Our investigations shed light on the intricate nature of these polynomials, elucidating their behaviour and facilitating deeper understanding within the realm of q -calculus. [ABSTRACT FROM AUTHOR]

The present note is an addition to the author's recent paper [44], concerning general multiplicative systems of random variables. Using some lemmas and the methodology of [13], we obtain a general extremal inequality, with corollaries involving Rademacher chaos sums and those analogues for multiplicative systems. In particular we prove that a system of functions generated by bounded products of a multiplicative system is a convergence system. [ABSTRACT FROM AUTHOR]

Summary We introduce WatFinder , a tool designed to identify and visualize protein–water interactions (water bridges, water-mediated associations, or water channels, fluxes, and clusters) relevant to protein stability, dynamics, and function. WatFinder is integrated into ProDy , a Python API broadly used for structure-based prediction of protein dynamics. WatFinder provides a suite of functions for generating raw data as well as outputs from statistical analyses. The ProDy framework facilitates comprehensive automation and efficient analysis of the ensembles of structures resolved for a given protein or the time-evolved conformations from simulations in explicit water, as illustrated in five case studies presented in the Supplementary Material. Availability and implementation ProDy is open-source and freely available under MIT License from https://github.com/ProDy/ProDy. [ABSTRACT FROM AUTHOR]

The present study introduces fractal interpolation function as an attractor of the iterated function system involving a finite number of Reich contractions on a complete metric space. Some amendments are made concerning the results of Reich contractions in the existing literature, with respect to Hausdorff metric, by imposing certain additional constraints on the Reich contractivity factors. Moreover, a non-smooth version of classical interpolant is constructed with Reich contractions as an analogue to the α -fractal function. A fractal operator is proposed to approximate any given continuous function using the newly generated Reich-type fractal function. Apart from the construction, Reich-fractal functions are employed to reconstruct the Lorenz attractor, and the results are graphically compared with fractal functions generated using other contraction mappings. [ABSTRACT FROM AUTHOR]

In this paper we consider a number triangle known as the Rascal triangle and place it within an infinite sequence of number triangles we call Iterated Rascal triangles. Initially, the name "Rascal triangle" was a clever play of words. However, we show a surprising and direct connection between Iterated Rascal triangles and Pascal's triangle. We prove an unexpected and elegant recursive formula generates the entries in any Iterated Rascal triangle. We establish formulas, generating functions, and combinatorial identities for them. From these principals we prove a version of Lucas' Theorem for the Iterated Rascal triangles. Moreover, we consider an identity from the Rascal triangle that has a well-known analogous identity in Pascal's triangle and extend it to all Iterated Rascal triangles. Although many can be made, we conclude with one conjecture. [ABSTRACT FROM AUTHOR]

We extend the notion of bargraphs as first quadrant, semi-perimeter defined lattice paths to that of a bargraph prefix where we relax the bargraph defining parameters and allow the bargraph to extend into negative territory. A bargraph prefix is any initial section of a bargraph. Generating functions for the prefixes which separately track the number of up, down and horizontal steps are found. The asymptotics for the average height of the last prefix step in the different bargraph extensions is given. [ABSTRACT FROM AUTHOR]

Recently, a new dispersion index, as a measures of information, has been introduced and called varentropy. In this article, we introduce new measures of variability based on two measures of uncertainty, namely, the Kerridge inaccuracy measure and the Kullback-Leibler divergence. Their generating functions are considered and their infinite series representations are given. These new measures and associated properties, bounds and illustrative examples are all presented in detail. Finally, an application of Kullback-Leibler divergence and its dispersion index is illustrated by using the mean-variance rule. [ABSTRACT FROM AUTHOR]

We propose a method to obtain a new class of copulas using a probability generating function (PGF) of positive-integer valued random variable. Some existing copulas in the literature are sub-families of the proposed copulas. Various dependence measures and invariant property of the tail dependence coefficient under PGF transformation are also discussed. We propose an algorithm for generating random numbers from the PGF copula. The bivariate concavity properties, such as Schur concavity and quasi-concavity, associated with the PGF copula are studied. Two new generalized FGM copulas are introduced using PGFs of geometric and discrete Mittag-Leffler distributions. The proposed two copulas improved the Spearman's rho of FGM copula by (−0.3333, 0.4751) and (−0.3333, 0.9573). Finally, we analyse a real dataset to illustrate the practical application of the proposed copulas. [ABSTRACT FROM AUTHOR]

The whimsical Las Vegas/Monte Carlo cubic dice are generalized to construct the combinatorial problem of enumerating all n-dimensional hypercube dice and dice of other shapes that exhibit cubic, icosahedral, and higher symmetries. By utilizing powerful generating function techniques for various irreducible representations, we derive the combinatorial enumerations of all possible dice in n-dimensional space with hyperoctahedral symmetries. Likewise, a number of shapes that exhibit icosahedral symmetries such as a truncated dodecahedron and a truncated icosahedron are considered for the combinatorial problem of dice enumerations with the corresponding shapes. We consider several dice with cubic symmetries such as the truncated octahedron, dodecahedron, and Rubik's cube shapes. It is shown that all enumerated dice are chiral, and we provide the counts of chiral pairs of dice in the n-dimensional space. During the combinatorial enumeration, it was discovered that two different shapes of dice exist with the same chiral pair count culminating to the novel concept of isochiral polyhedra. The combinatorial problem of dice enumeration is generalized to multi-coloring partitions. Applications to chirality in n-dimension, molecular clusters, zeolites, mesoporous materials, cryptography, and biology are also pointed out. Applications to the nonlinear n-dimensional hypercube and other dicey encryptions are exemplified with romantic, clandestine messages: "I love U" and "V Elope at 2". [ABSTRACT FROM AUTHOR]

We consider a system of two coupled parallel queues with infinite waiting rooms. The time setting is discrete. In either queue, the service of a customer requires exactly one discrete time slot. Arrivals of new customers occur independently from slot to slot, but the numbers of arrivals into both queues within a slot may be mutually dependent. Their joint probability generating function (pgf) is indicated as A (z 1 , z 2) and characterizes the whole model. In general, determining the steady-state joint probability mass function (pmf) u (m , n) , m , n ≥ 0 or the corresponding joint pgf U (z 1 , z 2) of the numbers of customers present in both queues is a formidable task. Only for very specific choices of the arrival pgf A (z 1 , z 2) are explicit results known. In this paper, we identify a multi-parameter, generic class of arrival pgfs A (z 1 , z 2) , for which we can explicitly determine the system-content pgf U (z 1 , z 2) . We find that, for arrival pgfs of this class, U (z 1 , z 2) has a denominator that is a product, say r 1 (z 1) r 2 (z 2) , of two univariate functions. This property allows a straightforward inversion of U (z 1 , z 2) , resulting in a pmf u (m , n) which can be expressed as a finite linear combination of bivariate geometric terms. We observe that our generic model encompasses most of the previously known results as special cases. [ABSTRACT FROM AUTHOR]

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics. [ABSTRACT FROM AUTHOR]

It is known that the generating function of a sequence of Toeplitz matrices may not describe the asymptotic distribution of the eigenvalues of the considered matrix sequence in the non‐Hermitian setting. In a recent work, under the assumption that the eigenvalues are real, admitting an asymptotic expansion whose first term is the distribution function, fast algorithms computing all the spectra were proposed in different settings. In the current work, we extend this idea to non‐Hermitian Toeplitz matrices with complex eigenvalues, in the case where the range of the generating function does not disconnect the complex field or the limiting set of the spectra, as the matrix‐size tends to infinity, has one nonclosed analytic arc. For a generating function having a power singularity, we prove the existence of an asymptotic expansion, that can be used as a theoretical base for the respective numerical algorithm. Different generating functions are explored, highlighting different numerical and theoretical aspects; for example, non‐Hermitian and complex symmetric matrix sequences, the reconstruction of the generating function, a consistent eigenvalue ordering, the requirements of high‐precision data types. Several numerical experiments are reported and critically discussed, and avenues of possible future research are presented. [ABSTRACT FROM AUTHOR]

A large number of association rules often minimizes the reliability of data mining results; hence, a dimensionality reduction technique is crucial for data analysis. When analyzing massive datasets, existing models take more time to scan the entire database because they discover unnecessary items and transactions that are not necessary for data analysis. For this purpose, the Fuzzy Rough Set-based Horse Herd Optimization (FRS-HHO) algorithm is proposed to be integrated with the Map Reduce algorithm to minimize query retrieval time and improve performance. The HHO algorithm minimizes the number of unnecessary items and transactions with minimal support value from the dataset to maximize fitness based on multiple objectives such as support, confidence, interestingness, and lift to evaluate the quality of association rules. The feature value of each item in the population is obtained by a Map Reduce-based fitness function to generate optimal frequent itemsets with minimum time. The Horse Herd Optimization (HHO) is employed to solve the high-dimensional optimization problems. The proposed FRS-HHO approach takes less time to execute for dimensions and has a space complexity of 38% for a total of 10 k transactions. Also, the FRS-HHO approach offers a speedup rate of 17% and a 12% decrease in input–output communication cost when compared to other approaches. The proposed FRS-HHO model enhances performance in terms of execution time, space complexity, and speed. [ABSTRACT FROM AUTHOR]