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We present a simple implicit solution for the time-dependent trajectory of a thin Asay foil ejecta diagnostic for the general case where the impinging ejecta cloud is generated by a source function characterized by an arbitrary (sustained) time dependence and a time-independent (stationary) particle velocity distribution. In the limit that the source function time dependence becomes a delta function, this solution—which is amenable to rapid numerical calculations of arbitrary accuracy—exactly recovers a previously published solution for the special case of instantaneous ejecta production. We also derive simple expressions for the free-surface arrival (catch-up) time as well as the true ejecta areal mass accumulation on the accelerating foil and place bounds on the level of error incurred when applying instant-production mass solutions to a sustained-production trajectory. We demonstrate these solutions with example calculations for hypothetical source functions spanning a wide range of ejecta production durations, velocity distributions, and temporal behaviors. These calculations demonstrate how the foil trajectory is often insensitive to the temporal dependence of the source function, instead being dominated by the velocity distribution. We quantify this insensitivity using a "compatibility score" metric. Under certain conditions, one may capitalize upon this insensitivity to obtain a good approximation of the second integral of the velocity distribution from the observed foil trajectory. [ABSTRACT FROM AUTHOR]

The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for every graph H, every H-minor-free graph can be obtained by clique-sums of "almost embeddable" graphs. Here a graph is "almost embeddable" if it can be obtained from a graph of bounded Euler-genus by pasting graphs of bounded pathwidth in an "orderly fashion" into a bounded number of faces, called the vortices, and then adding a bounded number of additional vertices, called apices, with arbitrary neighborhoods. Our main result is a full classification of all graphs H for which the use of vortices in the theorem above can be avoided. To this end, we identify a (parametric) graph \(\mathscr{S}_{t}\) and prove that all \(\mathscr{S}_{t}\) -minor-free graphs can be obtained by clique-sums of graphs embeddable in a surface of bounded Euler-genus after deleting a bounded number of vertices. We show that this result is tight in the sense that the appearance of vortices cannot be avoided for H-minor-free graphs, whenever H is not a minor of \(\mathscr{S}_{t}\) for some \(t\in \mathbb {N}\). Using our new structure theorem, we design an algorithm that, given an \(\mathscr{S}_{t}\) -minor-free graph G, computes the generating function of all perfect matchings of G in polynomial time. Our results, combined with known complexity results, imply a complete characterization of minor-closed graph classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every \(\mathscr{S}_{t}\) as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes. [ABSTRACT FROM AUTHOR]

This paper is about the positive part U q + of the q -deformed enveloping algebra U q ( sl ˆ 2). The algebra U q + admits an embedding, due to Rosso, into a q -shuffle algebra ▪. The underlying vector space of ▪ is the free algebra on two generators x , y. Therefore, the algebra ▪ has a basis consisting of the words in x , y. Let U denote the image of U q + under the Rosso embedding. In our first main result, we find all the words in x , y that are contained in U. One type of solution is called alternating. The alternating words have been studied by Terwilliger. There is another type of solution, which we call doubly alternating. In our second main result, we display many commutator relations involving the doubly alternating words. In our third main result, we describe how the doubly alternating words are related to the alternating words. [ABSTRACT FROM AUTHOR]

We study “partition Eisenstein series”, extensions of the Eisenstein series G 2 ⁢ k ⁢ ( τ ) {G_{2k}(\tau)} , defined by λ = ( 1 m 1 , 2 m 2 , … , k m k ) ⊢ k ↦ G λ ⁢ ( τ ) := G 2 ⁢ ( τ ) m 1 ⁢ G 4 ⁢ ( τ ) m 2 ⁢ ⋯ ⁢ G 2 ⁢ k ⁢ ( τ ) m k . \lambda=(1^{m_{1}},2^{m_{2}},\dots,k^{m_{k}})\vdash k\mapsto G_{\lambda}(\tau)% :=G_{2}(\tau)^{m_{1}}G_{4}(\tau)^{m_{2}}\cdots G_{2k}(\tau)^{m_{k}}. For functions ϕ : 풫 ↦ ℂ {\phi:\mathcal{P}\mapsto\mathbb{C}} on partitions, the weight 2 ⁢ k {2k} partition Eisenstein trace is the quasimodular form Tr k ⁡ ( ϕ ; τ ) := ∑ λ ⊢ k ϕ ⁢ ( λ ) ⁢ G λ ⁢ ( τ ) . \operatorname{Tr}_{k}(\phi;\tau):=\sum_{\lambda\vdash k}\phi(\lambda)G_{% \lambda}(\tau). These traces give explicit formulas for some well-known generating functions, such as the kth elementary symmetric functions of the inverse points of 2-dimensional complex lattices ℤ ⊕ ℤ ⁢ τ {\mathbb{Z}\oplus\mathbb{Z}\tau} , as well as the 2 ⁢ k {2k} th power moments of the Andrews–Garvan crank function. To underscore the ubiquity of such traces, we show that their generalizations give the Taylor coefficients of generic Jacobi forms with torsional divisor. [ABSTRACT FROM AUTHOR]

This research article addresses the performance analysis of Markovian retrial queueing system with two types of customers, unreliable server, and Bernoulli feedback. Both regular customers (RC) and prime customers (PC) may either join, or balk from the system based on the trade-off between service profit and delay cost. When the system is busy, the regular customers have to choose whether to join a retrial orbit and make re-attempts or leave the system. Furthermore, due to congestion among regular customers, the server may discontinue the service during breakdown. Due to the unavailability of the service process, customers may experience dissatisfaction. Therefore, our objective is to introduce a Bernoulli feedback service process to enhance service quality, ensuring that customers are successfully served with a certain probability. To analyze the proposed model mathematically, Chapman-Kolmogorov (C-K) inflow-outflow balanced equations have been framed. Then, the probability generating function (PGF) method employed to explicitly derive the queue size distribution, throughput, and other performance metrics. These performance measures provide critical insights into system behavior, which are then incorporated to determine the equilibrium strategies for two types of joining strategies: (i) non-cooperative strategies and (ii) cooperative strategies. Finally, optimization approaches are employed to determine the optimum cost and make tactical decisions regarding the quality of service (QoS) in an integrated manner. The cost optimization is done using metaheuristic optimization techniques such as PSO and GWO. The analytic results established are validated by numerical simulation. The effect of various parameters on the performance indices are examined by cost optimization and sensitivity analysis. The comparison of both algorithms, including average fitness, standard deviation, and convergence analysis, were used and combined with Wilcoxon rank-sum test. [ABSTRACT FROM AUTHOR]

We define the generalized vertex operators expressed by charged fermions and state vectors corresponding to the n-partition. The action of the generalized vertex operators on state vectors and the generalized universal character (UC) plane partitions have been investigated by the fermion calculus techniques. Furthermore, we construct the infinite scalar product and the generating function associated with the generalized UC plane partitions. [ABSTRACT FROM AUTHOR]

General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan's result, in turn, is a generalization of the modular transformation of Eisenstein series E k (z) on SL 2 (Z) , where z → - 1 / z , z ∈ H . The proof of our result involves delicate analysis containing Cauchy Principal Value integrals. A simpler proof of a recent result of ours with Kesarwani giving a non-modular transformation for ∑ n = 1 ∞ σ 2 m (n) e - n y is also derived using the Lipschitz summation formula. In the pursuit of obtaining this transformation, we naturally encounter a new generalization of Raabe's cosine transform whose several properties are also demonstrated. As an application of our results, we get a generalization of Wright's asymptotic estimate for the generating function of the number of plane partitions of a positive integer n. [ABSTRACT FROM AUTHOR]

The almost-Riordan arrays and their inverses are investigating by the generating functions of the row sum, the alternating row sum, and the weighted row sum. The A , Z , and ω -sequences of the almost-Riordan arrays are characterized by the generating functions of these row sums. Additionally, using the generating functions of these row sums, the product of two almost-Riordan arrays is obtained. [ABSTRACT FROM AUTHOR]

We propose a systematic approach to construct a new family of stochastic symplectic schemes for the strong approximation of the solution of stochastic Hamiltonian systems. Our approach is based both on B-series and generating functions. The proposed schemes are a generalization of the implicit midpoint rule, they require derivatives of the Hamiltonian functions of at most order two, and are constructed by defining a generating function. We construct some schemes with strong convergence order one and a half, and we illustrate numerically their long term performance. [ABSTRACT FROM AUTHOR]

The search for one-step alternatives to the generalized method of moment (GMM) has identified broad classes of potential estimators such as generalized empirical likelihoods (GEL), empirical cressie-read (ECR), exponentially tilted empirical likelihood (ETEL), and minimum discrepancy (MD) estimators. While empirical likelihood (EL) dominates other ECR estimators in terms of higher-order asymptotics, it lacks robustness to model misspecification. ETEL was shown to combine higher-order efficiency and robustness to misspecification but demands strong moment generating function existence conditions. We show, both theoretically and via simulations, how to achieve the same goal under weaker moment existence conditions within the class of MD estimators. [ABSTRACT FROM AUTHOR]

We produce congruences modulo a prime p > 3 for sums ∑ k 3 k k x k over ranges 0 ≤ k < q and 0 ≤ k < q / 3 , where q is a power of p. Here x equals either c 2 / (1 − c) 3 , or 4 s 2 / (2 7 (s 2 − 1)) , where c and s are indeterminates. In the former case, we deal more generally with shifted binomial coefficients 3 k + e k . Our method derives such congruences directly from closed forms for the corresponding series. [ABSTRACT FROM AUTHOR]

Experimental results obtained from shake-table tests on complete unreinforced masonry buildings (URM) play a crucial role in enhancing modeling techniques and reducing uncertainties in seismic assessments for this type of building. This paper presents an open-access database comprising 69 shake-table tests on URM buildings with a scale equal to or larger than 1:4 that have been documented in the literature. The database summarizes important geometric parameters, structural characteristics, and experimental results, including maximum shake-table accelerations, story drifts, and evolution of dynamic properties, as well as qualitative information such as observed failure modes and damage progression throughout the experiments. This database is meant to serve the research community as a resource for planning future experimental campaigns, benchmarking numerical models, validating analysis methods, and improving predictive capabilities. As an example application, we generate vulnerability functions (fragility curves) from the experimental data. Our findings reveal that buildings tested in experiments are less vulnerable compared to real buildings, particularly when severe damage is observed and when the ground motion intensity is low to moderate. The ultimate objective is to facilitate the exchange of experimental data and boost collaborative approaches in the seismic assessment of URM buildings. [ABSTRACT FROM AUTHOR]

In this paper, we focus on providing the exact bounds for the maximum number of limit cycles Z (3 , n) that planar piecewise linear differential systems with two zones separated by the curve y = x 3 under perturbation of arbitrary polynomials of x , y with degree n can have, where n ∈ N. By the first two order Melnikov functions, we achieve that Z (3 , 2) = 12 , Z (3 , n) = 2 n + 1 for 3 ≤ n ≤ 88 and Z (3 , n) ≥ 2 n + 1 for any n. The results are novel and improve the previous results in the literature. • We achieved a result about the coexistence of small amplitude and big amplitude limit cycles in Lemma 3.6 (see P 18), which yields system (1.1) ϵ can have 2 n small amplitude limit cycles simultaneously a large one for n odd. • We got a good way to avoid finding the zero points of the Wronskians in applying the useful algebraic method developed in [20]. Specifically, although [26] proved that (F 3) (which is the generating functions of M 1 (u)) is an ECT-system with accuracy 1 for n = 1 , 3 , 5 , it is impossible to find the zero points of the Wronskians for general n. We constructed an auxiliary ordered set of functions (F 2) ⊃ (F 3) (see (3.38) on P 19 and Lemma 3.7) so that we avoided these troubles. In fact, we have checked each Wronskians of (F 2) is non-vanishing since they have the form of M × P (v) v m / 2 (v > 0) , where M , m ≠ 0 are some constants and P (v) are some polynomials of v 2 with positive coefficients. Thus, we achieved Z (3 , n) ≤ 2 n + 1 for n ≤ 88 odd. • We obtained the expression of M 2 (u) without any restriction on perturbation parameters in Lemma 4.1, see P 25. So we can obtain the exact value of Z (3 , 2) = 12 , which improves the result Z (3 , 2) ≥ 5 in [9]. • We performed a transformation in Lemma 3.7 on P 19 so that the calculations of Wronskians are reduced greatly, and we chose clearly the perturbation parameters in Lemma 3.8(ii) (see P 20) such that the corresponding system (1.1) ϵ has at least 2 n + 1 limit cycles. [ABSTRACT FROM AUTHOR]

Generalized Hecke operators, originating from the replication formula in Monstrous Moonshine, were extended in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] to apply to harmonic Maass functions on modular curves of higher genera, building on works in [M. Koike, On replication formula and Hecke operators, Nagoya University, preprint]. Their action was further applied to weakly holomorphic modular functions, deriving numerous arithmetic properties of Fourier coefficients. In this paper, we extend these operators to weakly holomorphic modular forms of arbitrary even non-positive weights. In the process, we show [Proposition 3.1 of L. Beneish and M. H. Mertens, On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras, Math. Z. 297(1–2) (2021) 59–80], which is used to obtain dimension formulas for certain vertex operator algebras, can be derived from our results. Additionally, we identify the conditions under which the action of the generalized Hecke operator preserves holomorphicity. Moreover, we show that this action can be expressed as a linear combination of same-level or lower-level forms, refining the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717]. Finally, we establish more general congruence relations on Fourier coefficients, with the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] emerging as a special case. [ABSTRACT FROM AUTHOR]

The presence of big data, characterized by exceptionally large sample size, often brings the challenge of outliers and data distributions that exhibit heavy tails. An online learning estimation that incorporates anti-outlier capabilities while not relying on historical data is therefore urgently required to achieve robust and efficient estimators. In this paper, we introduce an innovative online learning approach based on a mode kernel-based objective function, specifically designed to address outliers and heavy-tailed distributions in the context of big data. The developed approach leverages mode regression within an online learning framework that operates on data subsets, which enables the continuous updating of historical data using pertinent information extracted from a new data subset. By amalgamating the asymptotic distribution functions generated by local mode estimators from all data subsets, the newly suggested estimator is efficiently updated by minimizing a weighted least squares-type loss function. To facilitate this process, we suggest a modified mode expectation-maximization algorithm for numerical optimization, ensuring both storage friendliness and computational efficiency. We demonstrate that the resulting estimator is asymptotically equivalent to the mode estimator calculated using the entire dataset, provided that the covariates in each data subset are homogeneous and the errors are homoskedastic. Monte Carlo simulations and an empirical study are presented to illustrate the finite sample performance of the proposed estimator. Supplemental materials for this paper are available online. [ABSTRACT FROM AUTHOR]

The synchronization of Plasmodium parasites as they replicate within red blood cells of their vertebrate host remains largely unexplored. Understanding this synchronization could reveal how parasites optimize their lifecycle to maximize transmission, evade the immune response and maximize energy acquisition. Rhythmic replication fulfils some criteria of an endogenous oscillator with time of day cues potentially provided by temperature, oxygen levels, hormones and/or nutrient availability. Recent research on a rodent malaria model has highlighted that rhythms associated with the host's feeding/fasting cycle are a crucial factor influencing the synchronization of the erythrocytic stages of Plasmodium to the host's circadian cycle. Innate immune responses are also rhythmic and can regulate host metabolism, suggesting that the innate immune response triggered by Plasmodium contributes to its rhythmic replication. Here, we outline how the interplay between immune responses and metabolism could influence the timing and synchronization of Plasmodium's replication rhythm, focusing on the roles of the cytokine tumour necrosis factor, mitochondrial function and metabolites generated by the tricarboxylic acid cycle in highly activated monocytes. These processes are pivotal in controlling parasitemia and determining disease outcome, suggesting that a better understanding of energy metabolism on rhythmic host–parasite interactions may provide new insights for therapeutic interventions against malaria. This article is part of the Theo Murphy meeting issue 'Circadian rhythms in infection and immunity'. [ABSTRACT FROM AUTHOR]

Abstract.For the Sparre Andersen model of risk theory, we obtain some stochastic order results, in terms of the Laplace transform order and the moment generating function order, for random variables associated with the event of ruin. These include an extension of the deficit at ruin. We also derive some reliability properties for zero-modified compound geometric distributions. Finally, we give a general result comparing the maximal aggregate losses of two (classical) risk processes perturbed by diffusion, which is shown to be valid for any order satisfying the convolution closure property. [ABSTRACT FROM AUTHOR]

We consider two specific families of binomial trees and forests: simply generated binomial d-ary trees and forests versus their increasing phylogenetic version, with tree nodes in increasing order from the root to any of its leaves. The analysis (both pre-asymptotic and asymptotic) consists of some of the main statistical features of their total progenies. We take advantage of the fact that the random distribution of those trees are obtained while weighting the counts of the underlying combinatorial trees. We finally briefly stress a rich alternative randomization of combinatorial trees and forests, based on the ratio of favorable count outcomes to the total number of possible ones. [ABSTRACT FROM AUTHOR]

The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions of the idea of a Riordan array have included almost Riordan arrays, double Riordan arrays, and their generalizations. After giving a brief overview of the Riordan group, we define two further extensions of the notion of Riordan arrays, and we give a number of applications for these extensions. The relevance of these applications indicates that these new extensions are worthy of study. The first extension is that of the reverse symmetrization of a Riordan array, for which we give two applications. The first application of this symmetrization is to the study of a family of Riordan arrays whose symmetrizations lead to the famous Robbins numbers as well as to numbers associated with the 20 vertex model of mathematical physics. We provide closed-form expressions for the elements of these arrays, and we also give a canonical Catalan factorization for them. We also describe an alternative family of Riordan arrays whose symmetrizations lead to the same integer sequences. The second application of this symmetrization process is to the area of the enumeration of lattice paths. We remain with the applications to lattice paths for the second extension of Riordan arrays that we introduce, which is the interleaved Riordan array. The methods used include generating functions, linear algebra, weighted compositions, and linear recurrences. In the case of the symmetrization process applied to Riordan arrays, we focus on the principal minor sequences of the resulting square matrices in the context of integrable lattice models. [ABSTRACT FROM AUTHOR]

In this paper, we define double almost-Riordan arrays and find that the set of all double almost-Riordan arrays forms a group, called the double almost-Riordan group. We also obtain the sequence characteristics of double almost-Riordan arrays and give the production matrices of two types for double almost-Riordan arrays. In addition, we discuss the algebraic properties of the double almost-Riordan group, and finally give the compression of double almost-Riordan arrays and their sequence characteristics. [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]

In this article, we discuss a bivariate distribution whose conditionals are univariate binomial distributions, but the marginals are not binomial that exhibits both positive and negative correlation. Some useful structural properties of this distribution namely marginals, moments, generating functions, stochastic ordering are investigated. Simple proofs of both positive and negative correlation, marginal over-dispersion, the distribution of the maximum and minimum order statistics for a random sample of size 2 are also derived. The distribution is shown to be a member of the multi-parameter exponential family, and some natural and useful consequences are also outlined. The proposed distribution reduces to another bivariate discrete distribution, namely the bivariate Poisson conditionals recently studied by Ghosh, Marques, and Chakraborty (2021) under certain parametric conditions. Finally, the distribution is fitted to two bivariate count data sets having positive and negative correlation separately to illustrate its' suitability. [ABSTRACT FROM AUTHOR]

The multiplication of cells leads to consider the cell division model with mutation. Large cell counts therefore appear, implying that the distribution of the final number of mutant cells is a heavy tailed distribution. This distribution can be interpreted as a compound Poisson distribution, which depends on two parameters: the average number of mutations and the fitness parameter (heavy tail index). In this work, we specify some conditions that ensure a stability property of heavy-tailed distributions, namely, the distribution of random sum of random variables is a heavy-tailed distribution. We apply the obtained result to the compound Poisson distribution. To estimate the tail index, the Hill estimator and the generating function method are used. A comparative study is performed using these two estimators. [ABSTRACT FROM AUTHOR]

In online services, password-based authentication, a prevalent method for user verification, is inherently vulnerable to keyboard input data attacks. To mitigate these vulnerabilities, image-based authentication methods have been introduced. However, these approaches also face significant security challenges due to the potential exposure of mouse input data. To address these threats, a protective technique that leverages the SetCursorPos() function to generate artificial mouse input data has been developed, thereby concealing genuine user inputs. Nevertheless, adversaries employing advanced machine learning techniques can distinguish between authentic and synthetic mouse data, leaving the security of mouse input data insufficiently robust. This study proposes an enhanced countermeasure utilizing Generative Adversarial Networks (GANs) to produce synthetic mouse data that closely emulate real user input. This approach effectively reduces the efficacy of machine learning-based adversarial attacks. Furthermore, to counteract real-time threats, the proposed method dynamically generates synthetic data based on historical user mouse sequences and integrates it with real-time inputs. Experimental evaluations demonstrate that the proposed method reduces the classification accuracy of mouse input data by adversaries to approximately 62%, thereby validating its efficacy in strengthening the security of mouse data. [ABSTRACT FROM AUTHOR]

AbstractThe moment generating function and moments of a real-valued random variable can be expressed in terms of the survival function or inverse survival function. In this article, we give the representation of characteristic function via the generalized inverse survival function or generalized inverse distribution function for arbitrary distributions. We extend existing studies, both univariate and multivariate, by giving formulae of the expectation of a more general function of a random variable. These results have potential applications in a variety of fields. To illustrate their applications, we calculate the characteristic functions of several distributions appear in distortion riskmetrics and weighted entropies. [ABSTRACT FROM AUTHOR]

The article discusses the Aristotelian Phronesis Model (APM) and the need for further research and alternative forms of measurement to fully explore its value and validity. The authors emphasize the importance of additional assessments and qualitative research to deepen the understanding of phronesis. They suggest incorporating the work of other scholars to enrich the assessment of the blueprint function, a key aspect of the APM. The article encourages researchers to explore different models and measures of wisdom for comparison and empirical evaluation. [Extracted from the article]

This study explores the impact of stochastic resetting on the random walk dynamics within scale-free (u , v) -flowers. Utilizing the generating function technique, we develop a recursive relationship for the generating function of the first passage time and establish a connection between the mean first passage time with and without resetting. Our investigation spans multiple scenarios, with the random walker starting from various positions and aiming to reach different target nodes, allowing us to identify the optimal resetting probability that minimizes the mean first passage time for each case. We demonstrate that stochastic resetting significantly improves search efficiency, especially in larger networks. These findings underscore the effectiveness of stochastic resetting as a strategy for optimizing search algorithms in complex networks, offering valuable applications in domains such as biological transport, data networks, and search processes where rapid and efficient exploration is vital. [ABSTRACT FROM AUTHOR]

Quadratic forms with random kernel matrices are ubiquitous in applications of multivariate statistics, ranging from signal processing to time series analysis, biomedical systems design, wireless communications performance analysis, and other fields. Their statistical characterization is crucial to both design guideline formulation and efficient computation of performance indices. To this end, random matrix theory can be successfully exploited. In particular, recent advancements in spectral characterization of finite-dimensional random matrices from the so-called polynomial ensembles allow for the analysis of several scenarios of interest in wireless communications and signal processing. In this work, we focus on the characterization of quadratic forms in unit-norm vectors, with unitarily invariant random kernel matrices, and we also provide some approximate but numerically accurate results concerning a non-unitarily invariant kernel matrix. Simulations are run with reference to a peculiar application scenario, the so-called spectrum sensing for wireless communications. Closed-form expressions for the moment generating function of the quadratic forms of interest are provided; this will pave the way to an analytical performance analysis of some spectrum sensing schemes, and will potentially assist in the rate analysis of some multi-antenna systems. [ABSTRACT FROM AUTHOR]

This paper examines a discrete-time retrial queuing system that incorporates negative customers, system breakdowns, and repairs. In this model, an arriving customer has the option to go directly to the server, pushing the currently served customer, if any, to the front of the orbit queue, or to join the orbit based on a First-Come-First-Served (FCFS) discipline. The study also considers negative customers who not only remove the customer currently being served but also cause a server breakdown. An in-depth analysis of the model is conducted using a generating function approach, leading to the determination of the distribution and expected values of the number of customers in the orbit and the entire system. The paper explores the stochastic decomposition law and provides bounds for the difference between the steady-state distribution of this system and a comparable standard system. Recursive formulas for the steady-state distributions of the orbit and the system are developed. Additionally, it is shown that the studied discrete-time system can approximate the M/G/1 continuous-time version of the model. The research includes a detailed examination of the customer's sojourn time distribution in the orbit and the system, utilizing the busy period of an auxiliary system. The paper concludes with numerical examples that highlight how different system parameters affect various performance characteristics, and a section summarizing the key research contributions. [ABSTRACT FROM AUTHOR]

Enumerating the extremal submodular functions defined on subsets of a fixed base set has only been done for base sets up to five elements. This paper reports the results of attempting to generate all such functions on a six-element base set. Using improved tools from polyhedral geometry, we have computed 360 billion of them, and provide the first reasonable estimate of their total number, which is expected to be between 1000 and 10,000 times this number. The applied Double Description and Adjacency Decomposition methods require an insertion order of the defining inequalities. We introduce two novel orders, which speed up the computations significantly, and provide additional insight into the highly symmetric structure of submodular functions. We also present an improvement to the combinatorial test used as part of the Double Description method, and use statistical analyses to estimate the degeneracy of the polyhedral cone used to describe these functions. The statistical results also highlight the limitations of the applied methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider complete homogeneous symmetric polynomials evaluated for variables repeated with given multiplicities; in other words, we consider polynomials obtained from complete homogeneous polynomials by identifying some subsets of their variables. We represent such polynomials as linear combinations of the powers of the variables, where all exponents are equal to the degree of the original polynomial. We give two proofs for the proposed formulas: the first proof uses the decomposition of the generating function into partial fractions, and the second involves the inverse of the confluent Vandermonde matrix. We also discuss the computational feasibility of the proposed formulas. [ABSTRACT FROM AUTHOR]

When modeling real-world data, we face the challenge of determining which probability distribution best represents the data. To address this intricate problem, we rely on goodness-of-fit tests. However, when the data come from a bivariate negative binomial distribution, the literature reveals no existing goodness-of-fit test for this distribution. For this reason, in this article, we propose and study a computationally convenient goodness-of-fit test for the bivariate negative binomial distribution. This test is based on a bootstrap approximation and a parallelization strategy. To this end, we use a reparameterization technique based on the probability generating function and a Cramér-von Mises-type statistic. From the simulation studies, we conclude that the results converge to the established nominal levels as the sample size increases, and in all cases considered, the parametric bootstrap method provides an accurate approximation of the null distribution of the statistic we propose. Additionally, we verify the power of the proposed test, as well as its application to five real datasets. To accelerate the massive computational work, we employ the parallelization strategy that, according to Novoa-Muñoz (2024), was the most efficient among the techniques he analyzed. [ABSTRACT FROM AUTHOR]

In this study, we introduce generalized k-order Fibonacci and Lucas (F&L) polynomials that allow the derivation of well-known polynomial and integer sequences such as the sequences of k-order Pell polynomials, k-order Jacobsthal polynomials and k-order Jacobsthal F&L numbers. Within the scope of this research, a generalization of hybrid polynomials is given by moving them to the k-order. Hybrid polynomials defined by this generalization are called k-order F&L hybrinomials. A key aspect of our research is the establishment of the recurrence relations for generalized k-order F&L hybrinomials. After we give the recurrence relations for these hybrinomials, we obtain the generating functions of hybrinomials, shedding light on some of their important properties. Finally, we introduce the matrix representations of the generalized k-order F&L hybrinomials and give some properties of the matrix representations. [ABSTRACT FROM AUTHOR]

In this study, we establish some properties of Bronze Fibonacci and Bronze Lucas sequences. Then we find the relationships between the roots of the characteristic equation of these sequences with these sequences. What is interesting here is that even though the roots change, equality is still maintained. Also, we derive the special relations between the terms of these sequences. We give the important relations among these sequences, positive and negative index terms, with the sum of the squares of two consecutive terms being related to these sequences. In addition, we present the application of generalized Bronze Fibonacci sequences to hyperbolic quaternions. For these hyperbolic quaternions, we give the summation formulas, generating functions, etc. Moreover, we obtain the Binet formulas in two different ways. The first is in the known classical way and the second is with the help of the sequence's generating functions. In addition, we calculate the special identities of these hyperbolic quaternions. Furthermore, we examine the relationships between the hyperbolic Bronze Fibonacci and Bronze Lucas quaternions. Finally, the terms of the generalized Bronze Fibonacci sequences are associated with their hyperbolic quaternion values. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of sequences related to Horadam-type sequences. Specifically, we consider the repunit sequence { r n } n ≥ 0 , which is defined by the initial terms r 0 = 0 and r 1 = 1 and follows the Horadam recurrence relation given by r n = 11 r n − 1 − 10 r n − 2 for n ≥ 2 . Many studies have explored generalizations of integer sequences in different directions: some by preserving the initial terms, some by preserving the recurrence relation, and some by considering different numerical sets beyond positive integers. In this article, we take the third approach. Specifically, we study these sequences in the context of the tricomplex ring T. We define the Tricomplex Repunit sequence { t r n } n ≥ 0 , with initial terms t r 0 = (0 , 1 , 11) and t r 1 = (1 , 11 , 111) , and governed by the recurrence relation t r n = 11 t r n − 1 − 10 t r n − 2 , for n ≥ 2 . This sequence is also a Horadam-type sequence but defined in the tricomplex ring T. In this paper, we establish the properties of the Tricomplex Repunit sequence and establish several new as well as well-known identities associated with it, including Binet's formula, Tagiuri–Vajda's identity, d'Ocagne's identity, and Catalan's identity. We also derive the generating function for this sequence. Furthermore, we study various additional properties of these generalized sequences and establish results concerning the summation of terms related to the Tricomplex Repunit sequence, and one of our main goals is to determine analogous or symmetrical properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence. [ABSTRACT FROM AUTHOR]

In this paper, p , q -analogues of the generalized harmonic polynomials and the generalized hyperharmonic polynomials of order ν are defined. We found new sums of these numbers by using combinatorial operations. We also obtained the generating function and used it to reproduce new equations. The applications of matrices are made in the end. [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 goal of this research is to present and investigate a novel class of 2D-Laguerre polynomials, which contains some well-known 2D-polynomials as special cases.We discuss its numerous features, including integral representations, operational identities, the monomiality principle, generating functions, and linkages to Appll functions of two variables and Lauricella functions of r-variables. Finally, we raise relevant issues for further research. [ABSTRACT FROM AUTHOR]

In the paper, the authors present the positivity and decreasing property of the normalized tails of the series expansion of the generating function of the classical Bernoulli numbers and prove the increasing property of the ratio between two normalized tails of the series expansion of the generating function of the classical Bernoulli numbers by showing the increasing property of the ratio between two Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

Purpose: In this paper, we propose an efficient spectral method for solving the two-dimensional Benjamin–Bona–Mahony–Burgers equation. The new basis functions align well with the problem, the discrete system is sparse and can be efficiently inverted, and the numerical solutions exhibit spectral accuracy in space. Design/methodology/approach: To efficiently simulate the two-dimensional Benjamin–Bona–Mahony–Burgers equation, we utilize transformed generalized Jacobi polynomials and construct the basis functions using the tensor product of these newly introduced polynomials. We provide relevant approximation results. Subsequently, we propose a spectral scheme for the underlying problem, and prove the well-posedness of the scheme, along with the boundedness and energy dissipation of the numerical solutions. We analyze the generalized stability and convergence of the numerical solution of the proposed scheme. Some numerical simulations are presented to demonstrate the efficacy of this newly proposed method. Findings: The new basis functions generated by tensor product of the transformed Jacobi polynomial align well with the underlying problem and simplify the theoretical analysis. The spatial discrete system is sparse and can be efficiently inverted. The numerical solutions exhibit spectral accuracy in space. Originality/value: We introduce transformed generalized Jacobi polynomials to construct basis functions and present relevant approximation results. We propose an efficient spectral scheme for the two-dimensional Benjamin–Bona–Mahony–Burgers equation, accompanied by optimal error analysis. This new approach achieves spectral accuracy. Moreover, the proposed method and the techniques developed in this work can be applied to simulate a wide range of other nonlinear problems. [ABSTRACT FROM AUTHOR]

Probability models play crucial role in modeling loss in the finance and actuarial sciences. In this article, a new family of loss distributions known as the Tan F-Loss family of distributions is proposed with the Tan Weibull Loss distribution as a special case. The density exhibits decreasing, right skewed, symmetric, and approximately symmetric shapes. The hazard rate function shows decreasing and modified bathtub shapes. The statistical properties of the Tan Weibull Loss distribution including the quantile function, moments, expansion of the general rate, moment generating function, incomplete moment, and order statistics are studied. The maximum likelihood estimators of the distribution are also studied. Simulations are carried out to examine the behavior of the estimators. The results show that the estimators are consistent. The usefulness of the proposed distribution is demonstrated with two insurance loss datasets. The results show that the proposed distribution gives a better parametric to the two datasets compared with the competing distributions. [ABSTRACT FROM AUTHOR]

Riordan arrays, denoted by pairs of generating functions $ (g(z), f(z)) $ (g (z) , f (z)) , are infinite lower-triangular matrices that are used as combinatorial tools. In this paper, we present Riordan and stochastic Riordan arrays that have connections to the Fibonacci and modified Lucas numbers. Then, we present some pseudo-involutions in the Riordan group that are based on constructions starting with a certain generating function $ g(z) $ g (z). We also present a theorem that shows how to construct pseudo-involutions in the Riordan group starting with a certain generating function $ f(z) $ f (z) whose additive inverse has compositional order 2. The theorem is then used to construct more pseudo-involutions in the Riordan group where some arrays have connections to the Fibonacci and modified Lucas numbers. A MATLAB algorithm for constructing the pseudo-involutions is also given. [ABSTRACT FROM AUTHOR]

We mainly focus on the willow algorithm which can be used to solve the optimal dynamic multi-cycle investment and consumption problem under the jump-diffusion stochastic volatility model. First we obtain the moment generating function of the risky asset and then coalescing the Johnson-Curve transformation theory to generate the willow algorithm, which solving the multi-cycle dynamic investment and consumption problem is designed based on the two-dimensional willow framework. Moreover, through comparing our proposed solution with the optimal investment and consumption display solutions under the geometric Brownian motion model, we further discuss the analysis of the willow algorithm sensitivity. The willow algorithm for optimal investment and consumption proposed in this paper is able to extend the willow method which is effective from the field of option pricing to the field of investment portfolio, and it also provides a new idea for numerically solving the multi-period optimal investment and consumption decision. [ABSTRACT FROM AUTHOR]

In this paper, we improve the dynamic programming based reachable set computation method by replacing the constant size grid in the original method with a variable size grid. With this improvement, the computational time consumption can be significantly reduced while maintaining the accuracy. The proposed method represents the reachable set as a sublevel set of a discount cost-to-go function, which is generated by dynamic programming. In order to compute the discount cost-to-go function quickly and accurately, the proposed method consists of three steps: (1) Rough computation. This step uses a coarse grid to obtain an interpolation function that is close to the real discount cost-to-go function; (2) Upsampling. This step is for generating a fine grid; (3) Fine tuning. This step generates an interpolation function that exactly approximates the real discount cost-to-go function. This paper theoretically proves the correctness of the proposed method and verifies its effectiveness by some examples. • Improve dynamic programming-based reachable set computation by a variable size grid. • The proposed method can significantly reduce the time consumption. • Illustrate the effectiveness of our method by theoretical proofs and examples. [ABSTRACT FROM AUTHOR]

Let S n denote the symmetric group of permutations acting on n elements. We investigate the double sequence { N ℓ (n) } counting the number of ℓ tuples of elements of the symmetric group S n , where the components commute, normalized by the order of S n . Our focus lies on exploring log-concavity with respect to n: N ℓ (n) 2 - N ℓ (n - 1) N ℓ (n + 1) ≥ 0. We establish that this depends on n (mod 3) for sufficiently large ℓ . These numbers are studied by Bryan and Fulman as the nth orbifold characteristics, generalizing work by Macdonald and Hirzebruch–Hofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products. Notably, N 2 (n) represents the partition numbers p(n), while N 3 (n) represents the number of non-equivalent n-sheeted coverings of a torus studied by Liskovets and Medynkh. The numbers also appear in algebra since | S n | N ℓ (n) = Hom Z ℓ , S n . [ABSTRACT FROM AUTHOR]

A model of recognition algorithms for solving the problems of classifying the objects presented in a feature space of large dimensions is constructed. A new approach to developing such a model is proposed based on constructing a set of representative features and determining a suitable set of n/a three-dimensional threshold functions in the process of generating an extreme recognition algorithm. A structural description of the proposed model of recognition algorithms in the form of a sequence of computational procedures is given. Parameterization of these algorithms has been carried out, which makes it possible to set and solve the problem of determining an extreme recognition algorithm within the limits of the created model. The results of a comparative analysis of the proposed and known recognition algorithms are given. [ABSTRACT FROM AUTHOR]

The problem of calculating integrals of rapidly oscillating functions for a class of functions with continuous second and partially continuous third derivatives limited by Lipschitz conditions with a constant Lipschitz L is considered. The a priori information about the integrand function contains fixed values of the function and its first and second derivatives, which are given at N fixed nodes of an arbitrary grid approximately, with a specific error. This method of specifying a priori information narrows down the class of integrable functions to the so-called interpolation class of functions and allows generating a quadrature formula optimal in terms of accuracy for it and obtaining an optimal estimate of its error by applying the method of boundary functions. [ABSTRACT FROM AUTHOR]