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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]

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It is known that the following five counting problems lead to the same integer sequence f t (n) : the number of nonequivalent compact Huffman codes of length n over an alphabet of t letters, the number of "nonequivalent" complete rooted t-ary trees (level-greedy trees) with n leaves, the number of "proper" words (in the sense of Even and Lempel), the number of bounded degree sequences (in the sense of Komlós, Moser, and Nemetz), and the number of ways of writing 1 = 1 t x 1 + ⋯ + 1 t x n with integers 0 ≤ x 1 ≤ x 2 ≤ ⋯ ≤ x n . In this work, we show that one can compute this sequence for all n < N with essentially one power series division. In total we need at most N 1 + ε additions and multiplications of integers of cN bits (for a positive constant c < 1 depending on t only) or N 2 + ε bit operations, respectively, for any ε > 0 . This improves an earlier bound by Even and Lempel who needed O (N 3) operations in the integer ring or O (N 4) bit operations, respectively. [ABSTRACT FROM AUTHOR]

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form. [ABSTRACT FROM AUTHOR]

In this article, we study the M/M/1 queueing systems with negative customers and unreliable repairers. The arrival of a negative customer causes the failure of the server, and the server is repaired immediately. The repairers may successfully repair the server with probability p, and the server continues to serve customers in the system. Otherwise, once the repair fails, the system will be cleared and all customers are forced to leave the system. By analyzing the quasi-birth-and-death process and using the probability generating function method, we obtain the steady-state probability and some performance measures of the system. Then, based on the reward-cost structure, we discuss the strategic behavior of customers and consider the equilibrium strategy. Finally, the effects of system parameters on the performance measures, individual benefit, and social benefit are analyzed by numerical examples. [ABSTRACT FROM AUTHOR]

We prove that for any graph G of maximum degree at most Δ, the zeros of its chromatic polynomial χ G (x) (in C) lie inside the disc of radius 5.94Δ centered at 0. This improves on the previously best known bound of approximately 6.91Δ. We also obtain improved bounds for graphs of high girth. We prove that for every g there is a constant K g such that for any graph G of maximum degree at most Δ and girth at least g , the zeros of its chromatic polynomial χ G (x) lie inside the disc of radius K g Δ centered at 0, where K g is the solution to a certain optimization problem. In particular, K g < 5 when g ≥ 5 and K g < 4 when g ≥ 25 and K g tends to approximately 3.86 as g → ∞. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph G to the generating function of so-called broken-circuit-free forests in G. We also establish a zero-free disc for the generating function of all forests in G (aka the partition function of the arboreal gas) which may be of independent interest. [ABSTRACT FROM AUTHOR]

In this article, we study hook lengths of ordinary partitions and t -regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in 2-regular partitions. For a positive integer k , let p (k) (n) denote the number of hooks of length k in all the partitions of n. We prove that p (k) (n) ≥ p (k + 1) (n) for all n ≥ 0 and n ≠ k + 1 ; and p (k) (k + 1) − p (k + 1) (k + 1) = − 1 for k ≥ 2. For integers t ≥ 2 and k ≥ 1 , let b t , k (n) denote the number of hooks of length k in all the t -regular partitions of n. We find generating functions of b t , k (n) for certain values of t and k. Exploring hook length biases for b t , k (n) , we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that b 2 , 2 (n) ≥ b 2 , 1 (n) for all n > 4 , whereas b 2 , 2 (n) ≥ b 2 , 3 (n) for all n ≥ 0. We also propose some conjectures on biases among b t , k (n). [ABSTRACT FROM AUTHOR]

In this paper, we describe a systematic way of obtaining the exact generating functions for p ¯ (2 n) , p ¯ (4 n) (first proved by Fortin et al.), p ¯ (8 n) , p ¯ (16 n) , etc. where p ¯ (n) denotes the number of overpartitions of n. We further establish several new infinite families of congruences modulo 2 4 and 2 5 for p ¯ (n) . For example, we prove that for all n , α , β ≥ 0 and primes p ≥ 5 , p ¯ 3 4 α + 1 p 2 β + 1 24 p n + 24 j + 7 p ≡ 0 (mod 2 5) and p ¯ 3 2 α + 1 (24 n + 23) ≡ 0 (mod 2 5) , where ( - 6 p) = - 1 and 1 ≤ j ≤ p - 1 . The last congruence was proved by Xiong (Int J Number Theory 12:1195–1208, 2016) for modulo 2 4 . [ABSTRACT FROM AUTHOR]

We consider a one-dimensional variant of a recently introduced settlement planning problem in which houses can be built on finite portions of the rectangular integer lattice subject to certain requirements on the amount of insolation they receive. In our model, each house occupies a unit square on a 1 × n strip, with the restriction that at least one of the neighboring squares must be free. We are interested mostly in situations in which no further building is possible, i.e. in maximal configurations of houses in the strip. We reinterpret the problem as a problem of restricted packing of vertices in a path graph and then apply the transfer matrix method in order to compute the bivariate generating functions for the sequences enumerating all maximal configurations of a given length with respect to the number of houses. This allows us to determine the asymptotic behavior of the enumerating sequences and to compute some interesting statistics. Along the way, we establish close connections between our maximal configurations and several other types of combinatorial objects, including restricted permutations and walks on certain small oriented graphs. We then generalize our results in several directions by considering multi-story houses, by varying the insolation restrictions, and, finally, by considering strips of width 2 and 3. At the end we comment on several possible directions of future research. [ABSTRACT FROM AUTHOR]

The Priority Arbiter-based Physical Unclonable Function (PA-PUF) is a one-way function implemented in hardware, where output or response bits, corresponding to a given set of challenge bits, are influenced by the device-specific parameters. An n -length PA-PUF can be represented as an n -variable Boolean function. The research we conducted demonstrates the exploration of Boolean functions obtained by implementing a PA-PUF construction with randomly selected delay parameters. We observe that the set of Boolean functions generated from PA-PUF is significantly larger than the set of Boolean functions generated from classical Arbiter-based PUF. Further, we look into the bias estimation of the response bit generated from PA-PUF towards the impact of changing specific bits in the challenge input. Finally, we perform a comparative analysis on PA-PUF to study its cryptographic properties and a detailed analysis of the machine learning attacks. [ABSTRACT FROM AUTHOR]

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 on [ 0 , ∞) , 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. [ABSTRACT FROM AUTHOR]

We introduce a new flexible count distribution by combining Bernoulli and Poisson moment exponential (PMEx) distributions. The new model named the Bernoulli PMEx distribution. Some mathematical properties are studied, including the hazard rate function, moments, moment generating function, probability generating function, and dispersion index. A count regression model is also proposed based on this distribution. The maximum likelihood estimation method is used to estimate the model parameters. In the end, three datasets from different fields are utilized for application purposes. The findings show that the new model efficiently analyzed these datasets as compared to Poisson, discrete Pareto, discrete Rayleigh, discrete Burr‐Hatke, and discrete inverted Topp–Leone distributions. [ABSTRACT FROM AUTHOR]

We examine the diffusive dynamics of a lattice random walk subject to resetting in a one-dimensional spatially heterogeneous environment composed of two media separated by an interface. At random times the walker may reset its position to the interface, but only when in the left medium. In addition the spatial heterogeneity results from having unequal diffusivities and biases in the two media. We construct the Master equation for the dynamics of the walker occupation probability in unbounded space, solve it exactly in terms of generating functions, and analyse the dynamics of the first and second moment. Making use of the closed form solution in the unbounded case, we build the analytic solution of the Master equation in finite and semi-infinite domains. By bounding the space on the right with a reflecting boundary we study the first-passage dynamics to a single fully absorbing target placed in the left medium away from the interface. As reset strongly increases the time to reach the target, we find that the first-passage dynamics enter the motion-limited regime even for relative small resetting probability. We also identify a surprising non-monotonic dependence of the first-passage probability mode as a function of the bias. By deriving an analytic expression for the mean first-passage time, we show when its value is independent of the diffusivity and bias in the left medium, uncovering another example of the so-called mean disorder indifference phenomenon. [ABSTRACT FROM AUTHOR]

Stochastic population models with mild catastrophes have gained much attention in recent years due to their wide application in a variety of areas including computer-communications systems. This article considers a population model in which both the arrival process of individuals and catastrophes occur as per the Markovian arrival process (MAP) and are independent of each other. The killing mechanism takes place according to the binomial distribution. The steady-state analysis of the model is carried out using the vector generating function approach and the population size distributions at an arbitrary, post-catastrophe, and pre-arrival epochs are presented in terms of the roots of the characteristic equation. Several performance measures of the system are studied in detail, and the impact of critical parameters is duly investigated. [ABSTRACT FROM AUTHOR]

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In Grindrod et al. [13] , the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed (unweighted or weighted) graphs, showing that it depends on the number of cycles in the undirectization of the graph. We also consider backtrack-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case. Finally, for weighted directed graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound, and we also prove a version of Ihara's theorem. [ABSTRACT FROM AUTHOR]

Background: Network meta-analysis is developed to compare all available treatments; therefore it enriches evidence for clinical decision-making, offering insights into treatment effectiveness and safety when faced with multiple options. However, the complexity and numerous treatment comparisons in network meta-analysis can challenge healthcare providers and patients. The purpose of this study aimed to introduce a graphic design to present complex rankings of multiple interventions comprehensively. Methods: Our team members developed a "beading plot" to summary probability of achieving the best treatment (P-best) and global metrics including surface under the cumulative ranking curve (SUCRA) and P-score. Implemented via the "rankinma" R package, this tool summarizes rankings across diverse outcomes in network meta-analyses, and the package received an official release on the Comprehensive R Archive Network (CRAN). It includes the 'PlotBead()' function for generating beading plots, which represent treatment rankings among various outcomes. Results: Beading plot has been designed based on number line plot, which effectively displays collective metrics for each treatment across various outcomes. Order on the -axis is derived from ranking metrics like P-best, SUCRA, and P-score. Continuous lines represent outcomes, and color-coded beads signify treatments. Conclusion: The beading plot is a valuable graphic that intuitively displays treatment rankings across diverse outcomes, enhancing reader-friendliness and aiding decision-making in complex network evidence scenarios. While empowering clinicians and patients to identify optimal treatments, it should be used cautiously, alongside an assessment of the overall evidence certainty. [ABSTRACT FROM AUTHOR]

We report an in-depth study of the reductive activation of O2 by the nonheme [FeII(L 25)(MeCN)]2+ complex carried out by cyclic voltammetry. Experimental evidence is obtained for the slow coordination of dioxygen to the ferrous center yielding an FeII/O2 adduct with a strong FeII–O2 character rather than an FeIII–superoxo one. Electron injection in the FeII–O2 species occurs at a potential of ca. −700 mV vs. SCE, i.e. 200 mV above the O2 to O2˙− reduction, leading to the formation of a FeIII–peroxo intermediate and then FeIII–hydroperoxo upon protonation by residual water. The experimental CVs recorded at variable scan rate or variable FeII concentration are well simulated taking into account a detailed mechanism initiated by the competitive reduction of O2 and the FeII–O2 adduct. Analysis of the concentration of the reaction intermediates generated as a function of the applied potential indicates that the FeIII–peroxo intermediate significantly accumulates at a potential of −650 mV. Oxidative bromination of anisole is assayed under electrolytic conditions at this potential to yield bromoanisole products. The low faradaic yields observed reveal that deleterious reactions such as direct reduction of reaction intermediates likely occur. Based on the detailed mechanism elucidated, a number of improvements to achieve more efficient catalytic reactions can be proposed. [ABSTRACT FROM AUTHOR]

The present article proposes a novel hybrid approach for addressing the synthesis problem of rectangular planar arrays under multiple constraints, through joint optimization of the weight function and the improved grey wolf optimization (IGWO) algorithm. Firstly, the grey wolf optimization (GWO) algorithm is improved by using tent chaotic mapping, nonlinear convergence factor, dominant wolf dynamic belief strategy, and opposition‐based learning strategy to increase the population diversity and the ability to jump out of the local optimum. Secondly, the array elements are weighted using the weight function to generate the position distribution matrix, which reduces the thinned matrix optimization time and improves the optimization efficiency. Finally, the position distribution matrix is used to generate the thinned array, and the IGWO algorithm is applied to perform the sparse optimization with multiple constraints. The effectiveness of the method is verified through numerical simulation and full‐wave simulation experiments, demonstrating its capability to enhance array antenna performance and reduce peak sidelobe level (PSLL). These experimental results hold significant engineering implications and provide valuable references for addressing the array distribution problem under multiple constraints. [ABSTRACT FROM AUTHOR]

Navigation safety is critical for many autonomous systems such as self-driving vehicles in an urban environment. It requires an explicit consideration of boundary constraints that describe the borders of any infeasible, non-navigable, or unsafe regions. We propose a principled boundary-aware safe stochastic planning framework with promising results. Our method generates a value function that can strictly distinguish the state values between free (safe) and non-navigable (boundary) spaces in the continuous state, naturally leading to a safe boundary-aware policy. At the core of our solution lies a seamless integration of finite elements and kernel-based functions, where the finite elements allow us to characterize safety-critical states' borders accurately, and the kernel-based function speeds up computation for the non-safety-critical states. The proposed method was evaluated through extensive simulations and demonstrated safe navigation behaviors in mobile navigation tasks. Additionally, we demonstrate that our approach can maneuver safely and efficiently in cluttered real-world environments using a ground vehicle with strong external disturbances, such as navigating on a slippery floor and against external human intervention. [ABSTRACT FROM AUTHOR]

Denote by N ℓ (n) the number of ℓ -tuples of elements in the symmetric group S n with commuting components, normalized by the order of S n . In this paper, we prove asymptotic formulas for N ℓ (n) . In addition, general criteria for log-concavity are shown, which can be applied to N ℓ (n) among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form c (a) c (b) > c (a + b) for certain families of sequences c(n). [ABSTRACT FROM AUTHOR]

In this paper, we introduce an R package EATME, which is known as Exponentially weighted moving average (EWMA) control chart with Adjustments To Measurement Error. The main purpose of this package is to correct for measurement error effects in continuous or binary random variables and develop the corrected control charts based on the EWMA statistic. In addition, the corrected control charts can detect out-of-control process accurately. The package contains a function to generate synthetic data and includes functions to determine the reasonable coefficient of control limit as well as estimate average run length. Moreover, for the visualization, we also provide the control charts to show the monitoring of in-control and out-of-control process. Finally, the functions in this package are clearly demonstrated, and numerical studies show the validity of the package. [ABSTRACT FROM AUTHOR]

Many papers have been recently devoted to the study of the radial behavior as z → 1 − of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like (1 + o (1)) C (z) / (1 − z) ρ for some ρ ∈ ℂ and C (z) is a real analytic function of z ∈ (0 , 1) such that C (z) = C (z r). They did not provide a formula for C (z) which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of C (z) as an exponential of a Fourier series in the variable log log (1 / z) / log (r) for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute C (z) associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in ℚ [ [ z ] ]), we provide an explicit expression of C (z) for every r-Mahler function of order 2 under mild assumptions on the coefficients in ℝ (z) of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1. [ABSTRACT FROM AUTHOR]

The causal shift-invariant convolution is studied from the point of view of inversion. Abel's algorithm, used in the tautochrone problem, is considered and Sonin's existence condition is deduced. To generate pairs of functions verifying Sonin's condition, the class of Mittag-Leffler type functions is used. In particular, functions that are impulse responses of ARMA(N,N) systems serve as a basis. The possible use of Abel's procedure as a support for introducing generalized fractional derivatives is evaluated. [ABSTRACT FROM AUTHOR]

AbstractInversion sequences of length n are positive integer sequences e1 e2 …en such that 1 ≤ ei ≤ i for all 1 ≤ i ≤ n. These sequences are in bijection with the permutations of [n]. This paper focuses on the polyomino or barpgraph representation of the inversion sequences. More precisely, we study the distribution of lattice points on these polyominoes. We find the generating functions respect to the length, the number of interior vertices, corners, and vertices of a given degree. We also give simple explicit formulas for the total values of these parameters over all polyominoes of inversion sequences of length n. Throughout this work, we use symbolic computer algebra to facilitate the computations. [ABSTRACT FROM AUTHOR]

We introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang–Mills equation. It turns out that the discovered recursion operators can be interpreted as infinite-dimensional matrices of differential functions which act on the generating vector functions of the nonlocal symmetries simply by matrix multiplication. To the best of our knowledge, there are no other examples of such recursion operators in the literature so far, so our approach is completely innovative. Further, we investigate the algebraic properties of the discovered operators and discuss the R -algebra structure on the set of all recursion operators for full-fledged nonlocal symmetries of the equation in question. Finally, we illustrate the action of the obtained recursion operators on particularly chosen full-fledged symmetries and emphasize their advantages compared to the action of traditionally used recursion operators for shadows. [ABSTRACT FROM AUTHOR]

We introduce and study the cumulative information generating function, which provides a unifying mathematical tool suitable to deal with classical and fractional entropies based on the cumulative distribution function and on the survival function. Specifically, after establishing its main properties and some bounds, we show that it is a variability measure itself that extends the Gini mean semi-difference. We also provide (i) an extension of such a measure, based on distortion functions, and (ii) a weighted version based on a mixture distribution. Furthermore, we explore some connections with the reliability of k-out-of-n systems and with stress–strength models for multi-component systems. Also, we address the problem of extending the cumulative information generating function to higher dimensions. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to analyze symmetric and asymmetric peaks in Dyck paths with air pockets (DAPs). These paths are formed by combining each maximal run of down-steps in ordinary Dyck paths into a larger, single down-step. To achieve this, we present a trivariate generating function that counts the number of DAPs based on their length and the number of symmetric and asymmetric peaks they contain. We determine the total numbers of symmetric and asymmetric peaks across all DAPs, providing an asymptotic for the ratio of these two quantities. Recursive relations and closed formulas are provided for the number of DAPs of length n, as well as for the total number of symmetric peaks, weight of symmetric peaks, and height of symmetric peaks. Furthermore, a recursive relation is established for the overall number of DAPs, similar to that for classic Dyck paths. A DAP is said to be non-decreasing if the sequence of ordinates of all local minima forms a non-decreasing sequence. In the last section, we focus on the sets of non-decreasing DAPs and examine their symmetric and asymmetric peaks. [ABSTRACT FROM AUTHOR]

Starting from three independent exponential random variables we have generated a bivariate random vector (U, V) having the marginal distributions as standard uniform. The joint distribution function and the survival function have been derived along with the moment generating function. We have also given an expression for the joint moment of order (r, s). The distributions of different functions of U and V have been derived. Different dependence measures between U and V have also been calculated. The reliability of the underlying stress-strength model has been obtained as an application of the distribution. A simulation exercise has been carried out to check for the goodness-of-fit of the model. We have also calculated the relative errors in different reliability measures under the assumption that the variables U and V are independent when actually they are not. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a combinatorial expression for the Perron root and eigenvectors of a non-negative integer matrix using techniques from symbolic dynamics. We associate such a matrix with a multigraph and consider the edge shift corresponding to it. This gives rise to a collection of forbidden words F which correspond to the non-existence of an edge between two vertices, and a collection of repeated words R with multiplicities which correspond to multiple edges between two vertices. In general, for given collections F of forbidden words and R of repeated words with pre-assigned multiplicities, we construct a generalized language as a multiset. A combinatorial expression that enumerates the number of words of fixed length in this generalized language gives the Perron root and eigenvectors of the adjacency matrix. We also obtain conditions under which such a generalized language is a language of an edge shift. [ABSTRACT FROM AUTHOR]

In this study, an improved extension of the Haq distribution was introduced. The new model is named "size-biased Haq distribution (SBHD)". Its two special cases length-biased and area-biased Haq distribution are discussed. The important mathematical properties such as moments, moment generating function, variance, skewness, kurtosis, Rényi entropy, survival function, hazard function, and mean residual life are derived. The model parameters are estimated using five different estimation methods. A comprehensive simulation study was utilized to illustrate the behavior of derived estimators. Two datasets from different fields are utilized to illustrate the flexibility of the new distributions. It is concluded that the new model is more flexible and efficiently analyzed when the datasets are compared to the considered competitive distributions. [ABSTRACT FROM AUTHOR]

In 2007, Andrews introduced Durfee symbols and k -marked Durfee symbols so as to give a combinatorial interpretation for the symmetrized moment function $\eta _{2k}(n)$ of ranks of partitions. He also considered the relations between odd Durfee symbols and the mock theta function $\omega (q)$ , and proved that the $2k$ th moment function $\eta _{2k}^0(n)$ of odd ranks of odd Durfee symbols counts $(k+1)$ -marked odd Durfee symbols of n. In this paper, we first introduce the definition of symmetrized positive odd rank moments $\eta _k^{0+}(n)$ and prove that for all $1\leq i\leq k+1$ , $\eta _{2k-1}^{0+}(n)$ is equal to the number of $(k+1)$ -marked odd Durfee symbols of n with the i th odd rank equal to zero and $\eta _{2k}^{0+}(n)$ is equal to the number of $(k+1)$ -marked Durfee symbols of n with the i th odd rank being positive. Then we calculate the generating functions of $\eta _{k}^{0+}(n)$ and study its asymptotic behavior. Finally, we use Wright's variant of the Hardy–Ramanujan circle method to obtain an asymptotic formula for $\eta _{k}^{0+}(n)$. [ABSTRACT FROM AUTHOR]

This article explores some properties of degenerate hypergeometric Bernoulli polynomials, which are defined through the following generating function t m e λ x (t) e λ x (t) - ∑ l = 0 m - 1 (1) l , λ t l l ! = ∑ n = 0 ∞ B n , λ [ m - 1 ] (x) t n n ! , | t | < min { 2 π , 1 | λ | } , λ ∈ ℝ \ { 0 }. We deduce their associated summation formulas and their corresponding determinant form. Also we focus our attention on the zero distribution of such polynomials and perform some numerical illustrative examples, which allow us to compare the behavior of the zeros of degenerate hypergeometric Bernoulli polynomials with the zeros of their hypergeometric counterpart. Finally, using a monomiality principle approach we present a differential equation satisfied by these polynomials. [ABSTRACT FROM AUTHOR]

The problem of linearly predicting a scalar response Y from a functional (random) explanatory variable X = X (t) , t ∈ I is considered. It is argued that the term "linearly" can be interpreted in several meaningful ways. Thus, one could interpret that (up to a random noise) Y could be expressed as a linear combination of a finite family of marginals X (t i) of the process X, or a limit of a sequence of such linear combinations. This simple point of view (which has some precedents in the literature) leads to a formulation of the linear model in terms of the RKHS space generated by the covariance function of the process X(t). It turns out that such RKHS-based formulation includes the standard functional linear model, based on the inner product in the space L 2 [ 0 , 1 ] , as a particular case. It includes as well all models in which Y is assumed to be (up to an additive noise) a linear combination of a finite number of linear projections of X. Some consistency results are proved which, in particular, lead to an asymptotic approximation of the predictions derived from the general (functional) linear model in terms of finite-dimensional models based on a finite family of marginals X (t i) , for an increasing grid of points t j in I. We also include a discussion on the crucial notion of coefficient of determination (aimed at assessing the fit of the model) in this setting. A few experimental results are given. [ABSTRACT FROM AUTHOR]

This paper focuses on integrals and Borel transforms concerning λ$$ \lambda $$‐Bessel and λ$$ \lambda $$‐Tricomi functions, employing an umbral perspective as a central approach. Differintegral techniques, presently utilized in calculus, offer a rich reserve of tools that can be utilized across a broad spectrum of problems encompassing the realm of special functions and beyond. Employing integral transforms akin to the Borel type and the related formalism proves to be a potent approach, facilitating a connection between umbral and operational methodologies. By integrating these perspectives, we establish a novel and effective method for deriving integrals of hybrid special functions and achieving the summation of their corresponding generating functions. [ABSTRACT FROM AUTHOR]

This paper proposes an improved two‐degree‐of‐freedom active disturbance rejection controller for the coupling problem of asynchronous motor vector system. To simplify the analysis process and accommodate observers of different types, a unified expression based on different controllers for the system output is developed. The closed‐loop transfer function generated by the reference input and load disturbance is given. For the coupling problem of motor output speed and immunity, the structure of a higher‐order extended state observer is reconstructed. The extended state observer estimates both the output speed and the total system disturbance, which serve as feedback and feed‐forward compensation quantities. Compared to the PI controller and traditional active disturbance rejection controller, the proposed controller achieves decoupling of output response speed and immunity, simplifies the process of parameter tuning. Finally, simulation and experiment results verify the feasibility and effectiveness of the algorithm in this paper. [ABSTRACT FROM AUTHOR]

In this paper, using the symmetrizing operator δ e 1 e 2 2 − l , we derive new generating functions of the products of p , q -modified Pell numbers with various bivariate polynomials, including Mersenne and Mersenne Lucas polynomials, Fibonacci and Lucas polynomials, bivariate Pell and bivariate Pell Lucas polynomials, bivariate Jacobsthal and bivariate Jacobsthal Lucas polynomials, bivariate Vieta–Fibonacci and bivariate Vieta–Lucas polynomials, and bivariate complex Fibonacci and bivariate complex Lucas polynomials. [ABSTRACT FROM AUTHOR]

The generating function of the Segre integrals on Hilbert schemes of points on a surface |$X$| can be determined by five universal series |$A_{0}(z)$|⁠ , |$A_{1}(z)$|⁠ , |$A_{2}(z)$|⁠ , |$A_{3}(z)$|⁠ , |$A_{4}(z)$|⁠ , due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface |$X$| and depend on the element |$\alpha \in K(X)$|⁠ , to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined |$A_{0}(z),A_{1}(z),A_{2}(z)$| for all ranks. For rank 0, it is easy to see |$A_{4}(z)=1$|⁠. Marian–Oprea–Pandharipande also conjectured that |$A_{3}(z)=A_{0}(z)A_{1}(z)$| for rank 0. We prove this conjecture by showing that when |$X$| is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined. [ABSTRACT FROM AUTHOR]

In this paper, I will answer a question that Albrecht asked me when preparing a paper on spectral instabilities: Is the ε -pseudospectrum of a Toeplitz operator T (a) , say with a continuous generating function a , always contained in the ε -pseudospectrum of the compactly perturbed operator T (a) + K ? [ABSTRACT FROM AUTHOR]

The generalization of probability distributions has a rich history, with various techniques employed to introduce new families of distributions. One notable approach is the transformed‐transformer (T‐X) method, which has been used by several authors to introduce many of novel probability distributions. In this study, we present the new Lindley‐X class, a novel family of continuous probability distributions, and we introduce some submodels within this framework. Our research involves deriving mathematical expressions for various statistical measures, including moments, the generating function, the quantile function, and Renyi entropy. To estimate the parameters of the new family of distributions, we utilize a range of methods, including maximum likelihood, maximum product spacing, least squares, Cramer‐von Mises, and Anderson–Darling estimations. Furthermore, we illustrate the practical applicability of this new family of distributions through the analysis of real‐life datasets. [ABSTRACT FROM AUTHOR]

AbstractThe 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]

This study presents a modified one-parameter Ailamujia distribution called the Entropy Transformed Ailamujia distribution (ETAD) is introduced to handle both symmetric and asymmetric lifetime data sets. The ETAD properties like order and reliability statistics, entropy, moment and moment generating function, quantile function, and its variability measures were derived. The maximum likelihood estimation (MLE) method was used in estimating the parameter of ETAD and through simulation at different sample sizes, the MLE was found to be consistent, efficient, and unbiased for estimating the ETAD parameter. The flexibility of ETAD was shown by fitting it to six different real lifetime data sets and compared it alongside seven competing oneparameter distributions. The goodness of fit (GOF) results from Akaike information criteria, Bayesian information criteria, corrected Akaike information criteria, and Hannan-Quinn information criteria show that the ETAD was the best fit amongst all the seven competing distributions across all the six data sets. [ABSTRACT FROM AUTHOR]

This paper introduces a new statistical distribution called length biased quasi suja distribution (LBQS). It explores its properties, including moments, moment generating function(MGF), characteristic function(CF), harmonic mean, reliability, hazard rate and reverse hazard rate. Order statistics of the above distribution is obtained. Furthermore, the paper also examines various entropy which measures the randomness of system, like Renyi entropy and Tsalli's entropy. It also evaluates Bonferroni and Lorenz curves which are useful in measuring the inequality. It also discusses parameter estimation techniques specifically maximum likelihood estimation and likelihood ratio testing. Moreover, a simulation study has been conducted to demonstrate how well the distribution would perform in real-life situation. The validity of the distribution is also demonstrated with real-world data example of failure data, highlighting its potential for practical applications in data analysis. [ABSTRACT FROM AUTHOR]

In this paper, a 4-parameter Exponentiated Skew Laplace distribution is defined and studied. Various statistical properties including its moment generating function, characteristics function, hazard function, and reliability function of the proposed ESLD were derived. The estimation of its parameters was carried out using the maximum likelihood method of estimation. The performance of the proposed ESLD compared with other similar distributions was demonstrated empirically with daily returns of S & P 500 between 2/02/24 and 28/03/2024 and daily returns of Bitcoin between 2/02/24 and 1/04/24 as obtained from Yahoo Finance. The fitness performance of the proposed distribution was evaluated based on log-likelihood, AIC, and BIC. Results obtained show that the proposed ESLD reported the highest log likelihood as well as the lowest AIC and BIC in the two data sets. This study therefore underscores the superiority of the proposed distribution over the some of the similar existing distributions. [ABSTRACT FROM AUTHOR]

The suggested queueing model describes a single-server feedback retrial queueing system with starting failure, Bernoulli working vacation and vacation interruptions. The server departs on a working vacation as soon as orbit is empty. During the working vacation period, the server provides a slower level of service. The supplementary variable method was utilized to determine the steady-state probability-generating functions for the system and its orbit. If there are consumers in the system at the end of each vacation, the server becomes idle and ready to serve new customers. The average busy time and the average busy cycle are presented as important system performance indicators. Additionally, the adaptive neuro-fuzzy interface system has compared the numerical results with the neuro-fuzzy results. Finally, particle swarm optimization (PSO) were utilized to obtain the best (optimal) cost for the system in this study. We have examined the convergence of these optimization strategies. [ABSTRACT FROM AUTHOR]

In this paper, we give some new generating functions of the products of (p, q)-Fibonacci numbers, (p.q)-Lucas numbers, (p. q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p, q)-Jacobsthal Lucas numbers with k-Pell numbers and Chebyshev polynomials of the second, third and fourth kinds. [ABSTRACT FROM AUTHOR]

In a recent article [Kawamoto, J. Phys. Complexity 4, 035005 (2023)], Kawamoto evoked statistical physics methods for the problem of counting graphs with a prescribed degree sequence. This treatment involved truncating a particular Taylor expansion at the first two terms, which resulted in the Bender-Canfield estimate for the graph counts. This is surprisingly successful since the Bender-Canfield formula is asymptotically accurate for large graphs, while the series truncation does not a priori suggest a similar level of accuracy. We upgrade this treatment in three directions. First, we derive an exact formula for counting d-regular graphs in terms of a d-dimensional Gaussian integral. Second, we show how to convert this formula into an integral representation for the generating function of d-regular graph counts. Third, we perform explicit saddle point analysis for large graph sizes and identify the saddle point configurations responsible for graph count estimates. In these saddle point configurations, only two of the integration variables condense to significant values, while the remaining ones approach zero for large graphs. This provides an underlying picture that justifies Kawamoto's earlier findings. [ABSTRACT FROM AUTHOR]

The Weil–Petersson volumes of moduli spaces of hyperbolic surfaces with geodesic boundaries are known to be given by polynomials in the boundary lengths. These polynomials satisfy Mirzakhani's recursion formula, which fits into the general framework of topological recursion. We generalize the recursion to hyperbolic surfaces with any number of special geodesic boundaries that are required to be tight. A special boundary is tight if it has minimal length among all curves that separate it from the other special boundaries. The Weil–Petersson volume of this restricted family of hyperbolic surfaces is shown again to be polynomial in the boundary lengths. This remains true when we allow conical defects in the surface with cone angles in (0, π) in addition to geodesic boundaries. Moreover, the generating function of Weil–Petersson volumes with fixed genus and a fixed number of special boundaries is polynomial as well, and satisfies a topological recursion that generalizes Mirzakhani's formula. This work is largely inspired by recent works by Bouttier, Guitter, and Miermont [Ann. Henri Lebesgue 5, 1035–1110 (2022)] on the enumeration of planar maps with tight boundaries. Our proof relies on the equivalence of Mirzakhani's recursion formula to a sequence of partial differential equations (known as the Virasoro constraints) on the generating function of intersection numbers. Finally, we discuss a connection with Jackiw–Teitelboim (JT) gravity. We show that the multi-boundary correlators of JT gravity with defects are expressible in the tight Weil–Petersson volume generating functions, using a tight generalization of the JT trumpet partition function. [ABSTRACT FROM AUTHOR]

In this current paper, we propose to study a three-dimensional Moran model (X n (1) , X n (2) , X n (3)) , where each random walk (X n (i)) ∈ { 1 , 2 , 3 } increases by one unit or is reset to zero at each unit of time. We analyze the joint law of its final altitude X n = max (X n (1) , X n (2) , X n (3)) via the moment generating tools. Furthermore, we show that the limit distribution of each random walk follows a shifted geometric distribution with parameter 1 − q i , and we analyze the maximum of these three walks, also giving explicit expressions for the mean and variance. [ABSTRACT FROM AUTHOR]

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we also derive some identities including a symmetry property for the Bernstein type polynomials. We give some relations among the Bernstein type polynomials, Bernoulli numbers, Stirling numbers, Dahee numbers, the Legendre polynomials, and the coefficients of the classical superoscillatory function associated with the weak measurements. We introduce some integral formulae for these polynomials. By using these integral formulae, we derive some new combinatorial sums involving the Bernoulli numbers and the combinatorial numbers. Moreover, we define Bezier type curves in terms of these polynomials. [ABSTRACT FROM AUTHOR]