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This paper investigates a single-server queuing system with heterogeneous service, failure, and maintenance. The proposed model features a server acting as both the main and backup server. System failure can occur at any stage. When a failure happens, instead of stopping the service entirely, the main server functions as a backup, providing service at a reduced rate. Once all jobs in the system have been serviced, the backup server enters the maintenance state. Following the repair process during maintenance, the server transitions to an idle state, awaiting incoming jobs. Explicit expressions for both transient and steady-state behaviours of the system are derived. Additionally, key system performance metrics are discussed in this paper, accompanied by graphical illustrations to visualize system size probabilities and performance indices. [ABSTRACT FROM AUTHOR]

This paper investigates the thermodynamic properties of strongly interacting gravitational systems. By deriving the partition function for such systems, the authors obtain expressions for various thermodynamic quantities, including internal energy, specific heat, Helmholtz free energy, entropy, chemical potential, and pressure. The paper introduces an interaction parameter that quantifies the degree of non-ideality in the system, and explores its effects on the thermodynamic properties. The authors also calculate the moments generating function, which provides information about the distribution of particle positions. Additionally, the distribution function for the system is derived. The paper highlights the existence of an upper bound temperature beyond which the partition function becomes negative, indicating a limit on the validity of the model. [ABSTRACT FROM AUTHOR]

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order α and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program. [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]

In this paper, the transformed sine Dagum distribution is proposed using the Dagum distribution as a baseline distribution in the transformed sin-G family of distributions. The purpose of this paper is to propose and explore a modified version of the Dagum distribution, based on the transformed sin-G family of distributions. The density of the transformed sine Dagum distribution shows that it can adequately model datasets that are positively skewed, approximately symmetric and decreasing. The hazard rate plot shows that the proposed distribution can adequately model both monotonic and non-monotonic hazard rate datasets. Statistical properties such quantile function, moments, inverse moments, moments generating functions, mean and median deviation and order statistics are derived. The plots of the skewness and kurtosis show that the skewness is always positive and the kurtosis is increasing. Maximum likelihood estimation is used to estimate the parameters of the transformed sine Dagum distribution. Monte Carlo simulations are performed to ascertain the behavior of the estimators. The results show that the estimators are consistent. The proposed distribution is applied to two real life datasets. The results show that the proposed distribution provide a reasonable parametric fit to the datasets compared with the other competing distribution. [ABSTRACT FROM AUTHOR]

China, situated in the circum-Pacific seismic belt, experiences frequent seismic activity and faces diverse geological conditions, making structural stability of paramount importance, especially under seismic conditions. The majority of current earthquake generation methods do not consider the nonstationary nature of earthquakes. This paper introduces a spectral representation-random function model for generating nonstationary earthquakes, effectively simulating stochastic seismic ground motion. Furthermore, traditional slope stability analysis methods are deterministic and incapable of providing probabilistic assessments of slope instability. Therefore, this paper proposes a unified framework for static and dynamic structural reliability analysis based on the direct probability integration method, quantifying the impact of stochastic seismic ground motion on the dynamic reliability of slope stability. Finally, the proposed methods are applied to an excavation slope in Nanjing, using sliding displacement and safety factors as evaluation criteria to study the reliability of the slope under the influence of stochastic seismic events. [ABSTRACT FROM AUTHOR]

In a previous paper, we introduced the Collatz polynomials $ P_N\left (z \right) $ P N (z) , whose coefficients are the terms of the Collatz sequence of the positive integer N. Our work in this paper expands on our previous results, using the Eneström-Kakeya Theorem to tighten our old bounds of the roots of $ P_N\left (z \right) $ P N (z) and giving precise conditions under which these new bounds are sharp. In particular, we confirm an experimental result that zeros on the circle $ \ensuremath {\{z\in \ensuremath {\mathbb {C}}:\left | z \right | = 2\}} $ { z ∈ C : | z | = 2 } are rare: the set of N such that $ P_N\left (z \right) $ P N (z) has a root of modulus 2 is sparse in the natural numbers. We close with some questions for further study. [ABSTRACT FROM AUTHOR]

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

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)$$ U\left(n+1\right) $$‐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)$$ U\left(n+1\right) $$ [ABSTRACT FROM AUTHOR]

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

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

In many engineering optimization problems, the number of function evaluations is severely limited by the time or cost constraints. These limitations present a significant challenge in the field of global optimization, because existing metaheuristic methods typically require a substantial number of function evaluations to find optimal solutions. This paper presents a new metaheuristic optimization algorithm that considers the information obtained by a radial basis function neural network (RBFNN) in terms of the objective function for guiding the search process. Initially, the algorithm uses the maximum design approach to strategically distribute a set of solutions across the entire search space. It then enters a cycle in which the RBFNN models the objective function values from the current solutions. The algorithm identifies and uses key neurons in the hidden layer that correspond to the highest objective function values to generate new solutions. The centroids and standard deviations of these neurons guide the sampling process, which continues until the desired number of solutions is reached. By focusing on the areas of the search space that yield high objective function values, the algorithm avoids exhaustive solution evaluations and significantly reduces the number of function evaluations. The effectiveness of the method is demonstrated through a comparison with popular metaheuristic algorithms across several test functions, where it consistently outperforms existing techniques, delivers higher-quality solutions, and improves convergence rates. [ABSTRACT FROM AUTHOR]

This paper investigates the behavior of a fluid queue driven by a non-truncated Erlangian service queue, focusing on the symmetrical properties within the system. This study determines the formulations of the steady-state distribution of both the buffer content and stationary state probabilities of a background queueing system. The efficient generating function technique is employed, utilizing a new generalization of the modified Bessel function of the second kind. Performance metrics such as mean buffer content and throughput are calculated, and server utilization is examined. The results contribute to the understanding of fluid queueing systems and offer insights into their performance in various applications, including telecommunications, manufacturing systems, healthcare operations, and ecological models, where symmetry plays a critical role in optimizing performance. [ABSTRACT FROM AUTHOR]

It is well known that medication adherence is critical to patient outcomes and can decrease patient mortality. The Pharmacy Quality Alliance (PQA) has recognized and identified medication adherence as an important indicator of medication‐use quality. Hence, there is a need to use the right methods to assess medication adherence. The PQA has endorsed the proportion of days covered (PDC) as the primary method of measuring adherence. Although easy to calculate, the PDC has however several drawbacks as a method of measuring adherence. PDC is a deterministic approach that cannot capture the complexity of a dynamic phenomenon. Group‐based trajectory modeling (GBTM) is increasingly proposed as an alternative to capture heterogeneity in medication adherence. The main goal of this paper is to demonstrate, through a simulation study, the ability of GBTM to capture treatment adherence when compared to its deterministic PDC analogue and to the nonparametric longitudinal K‐means. A time‐varying treatment was generated as a quadratic function of time, baseline, and time‐varying covariates. Three trajectory models are considered combining a cat's cradle effect, and a rainbow effect. The performance of GBTM was compared to the PDC and longitudinal K‐means using the absolute bias, the variance, the c‐statistics, the relative bias, and the relative variance. For all explored scenarios, we find that GBTM performed better in capturing different patterns of medication adherence with lower relative bias and variance even under model misspecification than PDC and longitudinal K‐means. [ABSTRACT FROM AUTHOR]

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences Vn, first encountered as a special sequence (corresponding to n=2) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for Vn, and we derive an exponential generating function of Vn expressed by Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

Proportional data arise frequently in a wide variety of fields of study. Such data often exhibit extra variation such as over/under dispersion, sparseness and zero inflation. For example, the hepatitis data present both sparseness and zero inflation with 19 contributing non-zero denominators of 5 or less and with 36 having zero seropositive out of 83 annual age groups. The whitefly data consists of 640 observations with 339 zeros (53%), which demonstrates extra zero inflation. The catheter management data involve excessive zeros with over 60% zeros averagely for outcomes of 193 urinary tract infections, 194 outcomes of catheter blockages and 193 outcomes of catheter displacements. However, the existing models cannot always address such features appropriately. In this paper, a new two-parameter probability distribution called Lindley–binomial (LB) distribution is proposed to analyze the proportional data with such features. The probabilistic properties of the distribution such as moment, moment generating function are derived. The Fisher scoring algorithm and EM algorithm are presented for the computation of estimates of parameters in the proposed LB regression model. The issues on goodness of fit for the LB model are discussed. A limited simulation study is also performed to evaluate the performance of derived EM algorithms for the estimation of parameters in the model with/without covariates. The proposed model is illustrated through three aforementioned proportional datasets. [ABSTRACT FROM AUTHOR]

In this paper, we propose quantum mechanical operator formalism for Touchard polynomials whose generating function is T n (x) = ∂ n ∂ λ n e (e λ − 1) x | λ = 0. That is replacing e λ x by e λ a † a , where a † a is the number operator, and using the method of integration within ordered product we find that (a † a) n is just the normal ordering form : T n (a † a) :. Then by virtue of the Weyl ordering form of quantum mechanical operator, we also introduce a new special polynomial whose generating function is G n (x) = ∂ n ∂ λ n 2 e λ + 1 exp [ 2 x e λ − 1 e λ + 1 ] | λ = 0. With use of the Weyl ordering form of e λ a † a , we prove (a † a) n = : T n (a † a) : = : : G n (a † a) : : , where : : : : denotes Weyl ordering. It seems that quantum mechancal operator formalism presents a new and simpler approach for generalizing Touchard polynomial theory. [ABSTRACT FROM AUTHOR]

In this paper, we present a simple algorithm for the explicit construction of n-particle harmonic oscillator states simultaneously belonging to irreducible representations of Sp(2, ℝ) (or SU(1, 1)) and O(n). For degenerate representations, the construction can be done using generating functions or hyperspherical harmonics. The cases with n = 2 and n = 3 are investigated at greater length. The analysis is extended to nondegenerate states, although the states are not so easily obtained as in the degenerate case. Finally, we briefly touch on the O(2) structure of Bell states. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study weighted estimates of the fractional type Marcinkiewicz integral and its commutator. By producing Guliyev type local pointwise estimates, we prove the boundedness of the fractional type Marcinkiewicz integral on the weighted Morrey–Guliyev spaces. Meanwhile, we also consider the corresponding weighted estimates of the commutators generated by a Lipschitz function and a BMO function. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of Balancing and BalancingLucas numbers. We describe some of their properties also we give the related matrix representation and divisibility properties. [ABSTRACT FROM AUTHOR]

Spinors are important objects in physics, which have found their place more and more after the discovery that particles have an intrinsic angular momentum shape and Cartan's mathematical expression of this situation. Recent studies using special number sequences have also revealed a new approach to the use of spinors in mathematics and have provided a different perspective for spinor research that can be used as a source for future physics studies. The purpose of this work is to expand the generalized Fibonacci quaternion polynomials to the generalized Fibonacci polynomial spinors by associating spinors with quaternions, and to introduce and investigate a new polynomial sequence that can be used to benefit from the potential advantages of spinors in physical applications, and thus, to provide mathematical arguments, such as new polynomials, for studies using spinors and quaternions in quantum mechanics. Starting from this point of view, in this paper we introduce and investigate a new family of sequences called generalized Fibospinomials (or generalized Fibonacci polynomial spinors or Horadam polynomial spinors). Being particular cases, we use (r , s) -Fibonacci and (r , s) -Lucas polynomial spinors. We present Binet's formulas, generating functions and the summation formulas for these polynomials. In addition, we obtain some special identities of these new sequences and matrices related to these polynomials. The importance of this study is that generalized Fibospinomials are currently the most generalized sequence in the literature when moving from Fibonacci quaternions to spinor structure, and that a wide variety of new spinor sequences can be obtained from this particular polynomial sequence. [ABSTRACT FROM AUTHOR]

In this paper, we consider various Stirling numbers of both kinds, including the unsigned degenerate Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, the unsigned degenerate r-Stirling numbers of the first kind and the degenerate r-Stirling numbers of the second kind. The aim of this paper is by using generating functions to further study explicit expressions, some identities and equivalent relations for those Stirling numbers. [ABSTRACT FROM AUTHOR]

The motion of rigid spherical particles suspended in a low Reynolds number fluid can be related to the forces, torques, and stresslets acting upon them by 22 scalar resistance functions, commonly notated X 11 A , X 12 A , Y 11 A , etc. Near-field asymptotic forms of these resistance functions were derived in Jeffrey and Onishi ["Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow," J. Fluid Mech. 139, 261–290 (1984)] and Jeffrey ["The calculation of the low Reynolds number resistance functions for two unequal spheres," Phys. Fluids A 4, 16–29 (1992)]; these forms are now used in several numerical methods for suspension mechanics. However, the first of these important papers contains a number of small errors, which make it difficult for the reader to correctly evaluate the functions for parameters not explicitly tabulated. This short article comprehensively corrects these errors and adds formulas that were originally omitted from both papers, so that the reader can verify and implement the equations independently. The corrected expressions, rationalized and using contemporary nondimensionalization, are shown to match mid-field values of these scalars, which are calculated through an alternative method. A Python script to generate and evaluate these functions is provided. [ABSTRACT FROM AUTHOR]

With the sustainable and coordinated development of cities, the formulation of urban street policies requires multiangle analysis. In regard to the existing street research, a large number of studies have focused on specific landscapes or accessibility of streets, and there is a lack of research on the multiple functions of streets. Recent advances in sensor technology and digitization have produced a wealth of data and methods. Thus, we may comprehensively understand streets in a less labor-intensive way, not just single street functions. This paper defines an index system of the multiple functions of urban streets and proposes a framework for multifunctional street measurement. Via the application of deep learning to Baidu Street View (BSV) imagery, we generate three functions, namely, landscape, traffic, and economic functions. The results indicate that street facilities and features are suitably identified. According to the multifunctional perspective, this paper further classifies urban streets into multifunctional categories and provides targeted policy recommendations for urban street planning. There exist correlations among the various street functions, and the correlation between the street landscape and economic functions is highly significant. This framework can be widely applied in other countries and cities to better understand street differences in various cities. [ABSTRACT FROM AUTHOR]

We consider a system of two parallel discrete-time single-server queues, queue 1 and queue 2. The service time of any customer in either queue is equal to 1 time slot. Arrivals during consecutive slots occur independently from slot to slot. However, the arrival streams into both queues are possibly mutually interdependent, i.e., during any slot, the numbers of arrivals in queue 1 and queue 2 need not be statistically independent. Their joint probability generating function (pgf) A(x, y) fully characterizes the queueing model. As a consequence of the possible intra-slot correlation in the arrival process, the numbers of customers present ("system contents") in queues 1 and 2, at any given slot boundary, are not necessarily independent either. In a previous paper, we have already discussed the mathematical difficulty of computing their steady-state joint pgf U (z 1 , z 2) ; explicit closed-form results can only be obtained for specific choices of A(x, y). In this paper, we therefore look at the problem from an other angle. Specifically, we study the (asymptotic) conditional steady-state behavior of the system under the condition that the content of queue 1 is (temporarily) very high (goes to infinity). For ease of terminology, we refer to the system as the "asymptotic system" in these circumstances. We prove that the asymptotic system is nearly identical to the original (unconditional) system, but with a modified joint arrival pgf A ∗ (x , y) that can be computed explicitly from A(x, y). This fundamental result allows us to determine the stability condition of queue 2 in the asymptotic system, and explicitly compute the classical queueing performance metrics of queue 2, such as the pgf, the moments and the approximate tail distribution of its system content, when this condition is fulfilled. It also leads to accurate approximative closed-form expressions for the joint tail distribution of the system contents in both queues, in the original (unconditional) system. We extensively illustrate our methodology by means of various specific (popular) choices of A(x, y). In some examples, where an explicit solution for U (z 1 , z 2) or for the (approximative) joint tail distribution is known, we retrieve the known results easily. In other cases, new results are found for arrival pgfs A(x, y) for which no explicit results were known until now. [ABSTRACT FROM AUTHOR]

In this paper, we present the physics and design-space exploration of a novel pulse compression photoconductive switch (PCPS) using semi-insulating gallium arsenide (GaAs) operating in the negative differential mobility (NDM) regime of electron transport. We systematically quantify the relationship between the PCPS performance and various design options, including contact separation, laser energy and placement, and trap dynamics. Specifically, we report the full-width at half-maximum and the peak output current generated by the PCPS as a function of applied electrical and optical bias. We discuss the optimal spacing between the electrodes and the distance of the laser spot to the anode to achieve higher electron confinement and superior radio-frequency (RF) metrics. Reducing the laser energy is important to prevent the appearance of secondary peaks due to diffusive transport, but there exists a trade-off between the bandwidth and the maximum current of the PCPS. We also compare the PCPS response with and without trap dynamics and find that the electrostatic screening from the trap-induced space charge is time-independent when the trapping time constant is set larger than the recombination lifetime. Overall, trap dynamics are detrimental to performance, unless the compensation doping scheme to achieve semi-insulating GaAs is carefully selected. Results presented in this paper can be used by experimentalists to fine-tune the PCPS design parameters to meet the specifications of various RF applications. Moreover, our results will provide a strong theoretical basis to the measurements of PCPS devices using GaAs and other NDM materials under investigation. [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]

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 paper, using operational rules, we derive some explicit expansion formulas associated with Brenke type polynomials. In particular, the corresponding connection and linearization coefficients are expressed either in terms of double sums or integral representations or sometimes in closed hypergeometric terms. The derived results are applied to Generalized Gould–Hopper and q-Laguerre polynomials. Some well-known duplication and convolution formulas are deduced as particular cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of the generalized k-Leonardo numbers and study their properties. We investigate the Gaussian Leonardo numbers and associated new families of these Gaussian forms. We obtain combinatorial identities like Binet formula, Cassini’s identity, partial sum, etc. in the closed form. Moreover, we give various generating and exponential generating functions. [ABSTRACT FROM AUTHOR]

Trees are acyclic connected graphs. Plane trees, d-ary trees, binary trees, noncrossing trees and their generalizations, which are families of trees, have been enumerated by many authors using various statistics. These trees are known to be enumerated by Catalan or Catalan-like formulas (Fuss-Catalan numbers). One of the most common approaches to the enumeration of these trees is by means of generating functions. Another method that can be used to enumerate them is by constructing bijections between sets of the same cardinality. The bijective method is preferred to other methods by many combinatorialists. So, in this paper, we construct bijections relating k-plane trees, k-noncrossing increasing trees, k-noncrossing trees, k-binary trees and weakly labelled k-trees. [ABSTRACT FROM AUTHOR]

In this paper, we study the generating functions of multiple t-star values with an arbitrary number of blocks of twos, which are based on the results of the corresponding generating functions of multiple t-harmonic star sums. These generating functions can deduce an explicit expression of multiple t-star values. As applications, we obtain some evaluations of multiple t-star values that represent the relations between multiple t-star values and alternating multiple t-half values. [ABSTRACT FROM AUTHOR]

The paper studies quasilinear interpolation by minimal splines constructed on nonuniform grids with multiple nodes. Asymptotic representations for normalized splines are obtained. The sharpness of biorthogonal approximation and the order of accuracy of quasilinear interpolation with respect to the grid stepsize are established. Results of numerical experiments on approximating some test functions, which demonstrate the effect of choosing a generating vector function in constructing the corresponding minimal spline, are presented. [ABSTRACT FROM AUTHOR]

Type II fuzzy sets consider the uncertainties involved in the membership function of classical fuzzy set theory. The membership function of a Type II fuzzy set is obtained by blurring the boundaries of the original fuzzy set membership function. The interval-based modified Type II fuzzy set method is presented in this paper to measure the fuzziness present in medical images. Using Hamacher T-conorm as the aggregation operator, the membership functions of the upper and lower intervals have been combined to obtain the contrast-enhanced image. For experimental analysis, quantitative and qualitative metrics have been evaluated for different kinds of medical data sets. To test the efficiency of the proposed technique, the computed results are compared with state-of-the-art techniques. The qualitative and quantitative results clearly demonstrate that the performance of the proposed techniques is much better than the existing techniques for almost all the image data sets. The results evaluated for average values with standard deviation for all the datasets bear witness to the performance of the proposed technique. The mean opinion score and the processing time also support the efficacy of the proposed technique, which is much better than most state-of-the-art techniques except at some of the cases. [ABSTRACT FROM AUTHOR]

As a structural topological index, the number of subtrees has great significance for the analysis and design of hybrid locally reliable networks. In this paper, with generating function and introducing a novel two-forest dual transformation technique, we solve the subtree enumerating problems of two representatives of the self-similar networks, such as the hierarchical lattice and |$(u,v)$| -flower networks. Moreover, by means of the circle weight transfer technique, two linear time algorithms of computing the subtree generation functions of these two families of networks are also proposed. The subtree density of two special cases for these self-similar networks is briefly discussed as an application. [ABSTRACT FROM AUTHOR]

In this paper, we look at the distributions of runs and scans in multistate Markov exchangeable sequences. The joint distributions of runs of several lengths under four types of enumeration schemes are analyzed. We evaluate the upper tail probabilities for ratchet scan statistics exactly. By utilizing the expansion of the generating functions, we propose effective computational tools for the derivation of probability functions. The results presented here provide approaches for the evaluation of the exact distributions of runs and scans in a wide class of practical problems. Finally, we discuss several applications and numerical examples to show how our theoretical results are applied to the investigation of runs and scans, as well as a parameter estimation problem. [ABSTRACT FROM AUTHOR]

This paper studies an M/M/1 retrial queue with negative customers, passive breakdown, and delayed repairs. Assume that the breakdown behavior of the server during idle periods is different from that during busy periods. Passive breakdowns may occur when the server is idle, due to the lack of monitoring of the server during idle periods. When the passive breakdown occurs, the server does not get repaired immediately and enters a delayed repair phase. Negative customers arrive during the busy period, which will cause the server to break down and remove the serving customers. Under steady-state conditions, we obtain explicit expressions of the probability generating functions for the steady-state distribution, together with some important performance measures for the system. In addition, we present some numerical examples to illustrate the effects of some system parameters on important performance measures and the cost function. Finally, based on the reward-cost structure, we discuss Nash equilibrium and socially optimal strategy and numerically analyze the influence of system parameters on optimal strategies and optimal social benefits. [ABSTRACT FROM AUTHOR]

Active contour model (ACM) is considered as one of the most frequently employed models in image segmentation due to its effectiveness and efficiency. However, the segmentation results of images with intensity non-uniformity processed by the majority of existing ACMs are possibly inaccurate or even wrong in the forms of edge leakage, long convergence time and poor robustness. In addition, they usually become unstable with the existence of different initial contours and unevenly distributed intensity. To better solve these problems and improve segmentation results, this paper puts forward an ACM approach using adaptive local pre-fitting energy (ALPF) for image segmentation with intensity non-uniformity. Firstly, the pre-fitting functions generate fitted images inside and outside contour line ahead of iteration, which significantly reduces convergence time of level set function. Next, an adaptive regularization function is designed to normalize the energy range of data-driven term, which improves robustness and stability to different initial contours and intensity non-uniformity. Lastly, an improved length constraint term is utilized to continuously smooth and shorten zero level set, which reduces the chance of edge leakage and filters out irrelevant background noise. In contrast with newly constructed ACMs, ALPF model not only improves segmentation accuracy (Intersection over union(IOU)), but also significantly reduces computation cost (CPU operating time T), while handling three types of images. Experiments also indicate that it is not only more robust to different initial contours as well as different noise, but also more competent to process images with intensity non-uniformity. [ABSTRACT FROM AUTHOR]

This paper deals with a discrete-time Geo/G/1 queue with repeated attempts and starting failure. If the server fails to start, it is sent for repair. During a repair process, alterations in the repair times is permitted based on current requirements. Customers are served on priority by the pre-emptive resume queue discipline. The distributions of the various system states when the system is in stable are analysed using the generating function technique. Analytical expressions are supported by numerical illustrations to exhibit the influence of the various parameters of the system on the performancemeasures. [ABSTRACT FROM AUTHOR]

This paper introduces a novel hybrid numerical method, SAUSM, designed for accurate and robust simulation of compressible flows governed by the Euler equations. While the AUSM + scheme provides proper resolution of smooth flow features, it is susceptible to anomalies, particularly the carbuncle phenomenon near strong shock discontinuities. Conversely, the AUFS scheme offers inherent stability in capturing shocks; however, it lacks the accuracy of AUSM + in smooth regions. The proposed SAUSM method combines AUSM + and AUFS through an adaptive weighting function, facilitating a seamless transition between the schemes. This approach preserves the accuracy of AUSM + in smooth regions while ensuring robust shock-capturing capabilities near discontinuities. The effectiveness of the SAUSM method is rigorously demonstrated through a comprehensive suite of progressively complex test cases. Numerical experiments demonstrate SAUSM's proficiency in resolving intense shock patterns and discontinuities without introducing anomalies. In the selected test cases, SAUSM agrees with reference solutions and effectively mitigates anomalies observed in AUSM + , including kinked Mach stems. In the challenging test case involving hypersonic blunt body flow over a cylinder, SAUSM adapts dissipation effectively by utilizing its adaptive weighting function to generate smooth pressure distributions, thereby eliminating the carbuncle instability linked to AUSM + when applied to a high aspect ratio grid. The consistent formulation of flux splitting and the adaptive weighting in SAUSM prevent excessive dissipation away from discontinuities, thus preserving accuracy comparable to that of exact Riemann solvers. Consequently, SAUSM emerges as a promising and innovative approach to accurately and robustly simulate a wide range of compressible Euler flows. The comprehensive results obtained from the validation tests firmly establish SAUSM as a highly effective general-purpose technique for computational fluid dynamics in academic research. [ABSTRACT FROM AUTHOR]

In this paper, we find an explicit formula for the generating function for the number of smooth squared (triangular, hexagonal) bargraphs according to the perimeter and number of columns. In particular, we show that the number of smooth squared, triangular, and hexagonal bargraphs with perimeter ... is the smallest positive root of the polynomial p16-2p14+p12-2p11-2p10+2p9+4p8-5p6-2p5+p4-2p³-2p²+1 and cs, ct, ch are three constants, as n 7→ ∞. [ABSTRACT FROM AUTHOR]

In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R [ t ] / (t 2) . Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q , R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra F q [ t ] / (t r) . We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation. [ABSTRACT FROM AUTHOR]

This paper derives the bivariate version of the extended standard U-quadratic (eSU) distribution using the compounding method or pseudo-family of distributions. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape distributions with the following properties: (i) X and Y have bathtub shapes; (ii) X has a constant distribution and Y has a bathtub shape; and (iii) X has an inverted bathtub and Y has a bathtub shape. Moreover, some properties of this proposed distribution are derived such as the marginal distribution, conditional distribution, conditional moments, product and ratio moments, Pearson correlation coefficient, joint moment generating function, and the stress - strength parameter. Further, the maximum likelihood estimation is performed to estimate the parameters of the derived distribution. A simulation study is carried out to evaluate the behavior of the estimates of the parameters. A new bivariate Kumaraswamy distribution is derived and used to simulate bivariate data with X and Y having bathtub shapes. A new bivariate version of the Cubic Transmuted Uniform (CTU) distribution is also derived. Finally, the proposed Bivariate eSU distribution is applied to simulated data and compared with the Bivarite Cubic Transmuted Uniform distribution. The results show that the proposed Bivariate eSU distribution provides a better fit on the simulated dataset compared with the Bivariate CTU distribution. [ABSTRACT FROM AUTHOR]