Recurrence Relations Research Articles

Skew-Symmetric Generalized Normal and Generalized t Distributions

In this paper, we introduce the skew-symmetric generalized normal and the skew-symmetric generalized t distributions, which are skewed extensions of symmetric special cases of generalized skew-normal and generalized skew-t distributions, respectively. We derive key distributional properties for these new distributions, including a recurrence relation and an explicit form for the cumulative distribution function (cdf) of the skew-symmetric generalized t distribution. Numerical examples including a simulation study and a real data analysis are presented to illustrate the practical applicability of these distributions.

Third-Order Differential Subordination Features of Meromorphic Functions: Erdelyi–Kober Model Integral Operator Application

This study is concerned with the class of p-valent meromorphic functions, represented by the series f(ζ)=ζ−p+∑k=1−p∞dkζk, with the domain characterized by 0<|ζ|<1. We apply an Erdelyi–Kober-type integral operator to derive two recurrence relations. From this, we draw specific conclusions on differential subordination and differential superordination. By looking into suitable classes of permitted functions, we obtain various outcomes, including results analogous to sandwich-type theorems. The operator used can provide generalizations of previous operators through specific parameter choices, thus providing more corollaries.

Probabilistic degenerate Dowling polynomials associated with random variables

Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Whitney numbers of the second kind and those of the degenerate Dowling polynomials, namely, the probabilistic degenerate Whitney numbers of the second kind associated with and the probabilistic degenerate Dowling polynomials associated with . We derive some properties, explicit expressions, certain identities, recurrence relations and generating functions for those numbers and polynomials. In addition, we investigate their generalizations, namely, the probabilistic degenerate ‐Whitney numbers of the second kind associated with and the probabilistic degenerate ‐Dowling polynomials associated with , and get similar results to the aforementioned numbers and polynomials.

Recurrence Relations of Exceptional Laurent Biorthogonal Polynomials

Recurrence Relations of Exceptional Laurent Biorthogonal Polynomials

Bethe vectors and recurrence relations for twisted Yangian based models

We study Olshanski twisted Yangian based models, known as one-dimensional “soliton non-preserving” open spin chains, by means of algebraic Bethe Ansatz. The even case, when the bulk symmetry is \mathfrak{gl}_{2n}𝔤𝔩2n and the boundary symmetry is \mathfrak{sp}_{2n}𝔰𝔭2n or \mathfrak{so}_{2n}𝔰𝔬2n, was studied in [Ann. Henri Poincaré 20, 339 (2018)]. In the present work, we focus on the odd case, when the bulk symmetry is \mathfrak{gl}_{2n+1}𝔤𝔩2n+1 and the boundary symmetry is \mathfrak{so}_{2n+1}𝔰𝔬2n+1. We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and Y(\mathfrak{gl}_n)Y(𝔤𝔩n)-type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.

Dominion on Grids

The domination (number) of a graph G=(V,E), denoted by γ(G), is the size of the minimum dominating sets of V(G), also known as γ-sets. As such, the dominion of G, denoted by ζ(G), counts all its γ-sets. We proved a conjecture from one of the authors on the dominion of cycles C3k−1 and C3k−2, k≥2. Further, we found the formulae and recurrence relations for the dominions of several grids, Gm,n, with 2≤m≤4 and other results when m≤9 and n≤20. In general, domination and dominion play important roles in assessing certain vulnerabilities of any given network system.

Holographic stress tensor correlators on higher genus Riemann surfaces

In this work, we present a comprehensive study of holographic stress tensor correlators on general Riemann surfaces, extending beyond the previously well-studied torus cases to explore higher genus conformal field theories (CFTs) within the framework of the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We develop a methodological approach to compute holographic stress tensor correlators, employing the Schottky uniformization technique to address the handlebody solutions for higher genus Riemann surfaces. Through rigorous calculations, we derive four-point stress tensor correlators, alongside recurrence relations for higher-point correlators, within the AdS3/CFT2 context. Additionally, our research delves into the holography of cutoff AdS3 spaces, offering novel insights into the lower-point correlators of the TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}-deformed theories on higher genus Riemann surfaces up to the first deformation order.

Orthogonal Laurent Polynomials of Two Real Variables

ABSTRACTIn this paper, we consider an appropriate ordering of the Laurent monomials , that allows us to study sequences of orthogonal Laurent polynomials of the real variables and with respect to a positive Borel measure defined on such that . This ordering is suitable for considering the multiplication plus inverse multiplication operator on each variable and , and as a result we obtain five‐term recurrence relations, Christoffel–Darboux and confluent formulas for the reproducing kernel and a related Favard's theorem. A connection with the one‐variable case is also presented, along with some applications for future research.

On the Incomplete Edouard and Incomplete Edouard–Lucas Numbers

This study introduces two new sequences: the incomplete Edouard and the incomplete Edouard–Lucas numbers. In addition, we establish some of the properties, identities, and recurrence relations of these sequences. The relations of these new sequences with balancing, Lucas-balancing, incomplete balancing, and incomplete Lucas-balancing numbers are explored, and some new identities are provided. Moreover, the functions for generating the incomplete Edouard and incomplete Edouard–Lucas numbers are stated.

Features of Structural Hardware Transformation of Information in Cryptosystems

The article considers types, functions and some features of cryptosystems, as well as circuit design options for expanding their functionality. Options for generating keys and ciphers used in cryptosystems with a typical structural organization, and technologies for creating encryption chains are presented. An encryption scheme is described, in the recurrent formula of the algorithm of which the previous blocks of both encryption and plaintext are used. This scheme reliably protects against any unauthorized modification of the encrypted text. Structural diagrams of the organization of symmetric and asymmetric cryptosystems are given. A variant of implementing a decoding procedure in a threshold MIMA cryptomodule for sharing a secret with a masking transformation is proposed, in which the necessary time and hardware costs for performing the procedure of reconstructing the original secret are minimized. The presented material can be the part of the original sections of a necessary and sufficiently provided in mathematical terms textbook on the basics and modern problems of cryptography.

Features of Structural Hardware Transformation of Information in Cryptosystems

The article considers types, functions and some features of cryptosystems, as well as circuit design options for expanding their functionality. Options for generating keys and ciphers used in cryptosystems with a typical structural organization, and technologies for creating encryption chains are presented. An encryption scheme is described, in the recurrent formula of the algorithm of which the previous blocks of both encryption and plaintext are used. This scheme reliably protects against any unauthorized modification of the encrypted text. Structural diagrams of the organization of symmetric and asymmetric cryptosystems are given. A variant of implementing a decoding procedure in a threshold MIMA cryptomodule for sharing a secret with a masking transformation is proposed, in which the necessary time and hardware costs for performing the procedure of reconstructing the original secret are minimized. The presented material can be the part of the original sections of a necessary and sufficiently provided in mathematical terms textbook on the basics and modern problems of cryptography.

In search of a higher Bochner theorem

We initiate the study of a natural generalisation of the classical Bochner–Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

(q, γ)-Bernstein Basis Functions on the Interval [a ; b

In this paper, we present a novel set of (q, γ)-Bernstein basis functions that are parameterized by q and and defined over an interval. We give detailed proofs for several key properties of these functions, including the partition of unity, recurrence relations, degree elevation, and the Marsden identity. Additionally, we introduce and validate the (q, γ)-De Casteljau algorithm, providing comprehensive examples to illustrate its implementation. These results are analyzed to highlight the theoretical and practical implications of these (q, γ)-Bernstein basis functions in various fields, such as polynomial approximation, numerical methods, and Computer Aided Geometric Design (CAGD). Furthermore, we discuss potential extensions and applications of these functions, considering their impact on future research and developments in the domain. By exploring these aspects, we aim to offer a robust framework for understanding and utilizing (q, γ)-Bernstein basis functions.

The maximum magnitude of natural and induced earthquakes

A key element in the assessment of seismic hazard is estimation of the maximum possible earthquake magnitude, Mmax. A great deal of effort has been invested in developing approaches to estimate Mmax for natural (tectonic) earthquakes, especially in regions of relatively low seismicity where it is difficult to associate observed seismicity with known geological faults. In probabilistic seismic hazard analysis, there has been a tendency to assign a narrow range of large values to Mmax. This results in the impression that hazard results are insensitive to this parameter, which is not the case when the Mmax distribution captures the full range of possible values. For induced seismicity, Mmax estimates can have far-reaching implications both in terms of quantitative assessments of the resulting seismic hazard and risk, and in terms of the public and regulatory perception of this risk. Estimates of Mmax for induced seismicity need to distinguish between driven earthquakes, for which magnitudes are largely controlled by operational parameters, and triggered tectonic earthquakes, together with estimates of the likelihood of such triggering. Distributions of triggered Mmax may be limited to smaller magnitudes than distributions for natural seismicity due to the shallow depth of most injection/extraction wells. For the management of induced seismic risk, the expected largest event magnitude (which may be influenced by a Traffic Light Scheme in operation) may be more relevant than any physical upper bound truncating the recurrence relationship.

Casimirs of the Virasoro algebra

We explicitly solve a recurrence relation due to Feigin and Fuchs to obtain the Casimirs of the Virasoro algebra in terms of the inverse of the Shapovalov form. Combined with our recent result for the inverse Shapovalov form, this allows us to write the Casimir operators as linear combinations of products of singular vectors.

Understanding admixture fractions: theory and estimation of gene-flow.

Estimation of admixture proportions has become one of the most commonly used computational tools in population genomics. However, there is remarkably little population genetic theory on statistical properties of these variables. We develop theoretical results that can accurately predict means and variances of admixture proportions within a population using models with recombination and genetic drift. Based on established theory on measures of multilocus disequilibrium, we show that there is a set of recurrence relations that can be used to derive expectations for higher moments of the admixture proportions distribution. We obtain closed form solutions for some special cases. Using these results, we develop a method for estimating admixture parameters from estimated admixture proportions obtained from programs such as Structure or Admixture. We apply this method to HapMap 3 data and find that the population history of African Americans, as expected, is not best explained by a single admixture event between people of European and African ancestry. The model of constant gene flow starting at 8 generations and ending at 2 generations before present gives the best fit.

A recursive algorithm for dynamics of planar multibody systems with frictional unilateral constraints

The contact impact problem in planar multibody systems can be efficiently solved by formulating it as the linear complementarity problem, which requires a complex modeling process. To simplify the process, a recursive algorithm for the dynamics of planar multibody systems with frictional unilateral constraints is proposed based on the reduced multibody system transfer matrix method. Firstly, the contact forces of frictional unilateral constraints are integrated into the recurrence relations of system components using Lagrange multipliers. Subsequently, the relative motion equations of the contact positions are discretized in time, which are then utilized to describe the linear complementarity problem for systems. The dynamics of the system is solved by the Moreau time-stepping method with the recursive method. Finally, the proposed algorithm was validated using the woodpecker toy and used to model a slider-crank mechanism with clearance, which shows its characteristics of facilitating modeling, universal, and highly programmable. This recursive algorithm provides an effective tool for solving non-smooth planar multibody systems while extending the application of the multibody system transfer matrix method.

Linearithmic Mean and Standard Deviation Sorting Method (LMDSM): Efficient and Stable Algorithm

Sorting algorithms have significance in the field of big data and high-performance computing, where effective data processing is of major significance. This paper presents the Linearithmic Mean and Standard Deviation Sorting Method (LMDSM), a new sorting method designed to handle enormous datasets and complex computational environments. Further this paper present a comprehensive comparative analysis of LMDSM with conventional sorting algorithms like Merge Sort and Quick Sort, emphasizing its efficiency and stability in different scenarios. LMDSM consistently outperforms other sorting algorithms in terms of sorting time when tested on arrays of 850,292 data in various orders. Furthermore, paper provide a rigorous theoretical analysis of LMDSM, encompassing its recurrence relation and time complexity derivation. Our research highlights the importance of LMDSM as an effective option for organizing jobs in current computing environments, providing significant importance in speed and reliability.

Asymptotic limits of transient patterns in a continuous-space interacting particle system

We study a discrete-time interacting particle system with continuous state space, which is motivated by a mathematical model for turnover through branching in actin filament networks. It gives rise to transient clusters reminiscent of actin filament assemblies in the cortex of living cells. We reformulate the process in terms of the inter-particle distances and characterize their marginal and joint distributions. We construct a recurrence relation for the associated characteristic functions and pass to the large population limit, reminiscent of the Fleming–Viot super processes. The precise characterization of all marginal distributions established in this work opens the way to a detailed analysis of cluster dynamics. We also obtain a recurrence relation that enables us to compute the moments of the asymptotic single particle distribution characterizing the transient aggregates. Our results indicate that aggregates have a fat-tailed distribution.

Regularized dynamic mode decomposition algorithm for time sequence predictions

Dynamic Mode Decomposition (DMD) aims at extracting intrinsic mechanisms in a time sequence via linear recurrence relation of its observables, thereby predicting later terms in the sequence. Stability is a major concern in DMD predictions. We adopt a regularized form and propose a Regularized DMD (ReDMD) algorithm to determine the regularization parameter. This leverages stability and accuracy. Numerical tests for Burgers’ equation demonstrate that ReDMD effectively stabilizes the DMD prediction while maintaining accuracy. Comparisons are made with the truncated DMD algorithm.