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We introduce and study s-lecture hall P-partitions which is a generalization of s-lecture hall partitions to labeled (weighted) posets. We provide generating function identities for s-lecture hall P-partitions that generalize identities obtained by Savage and Schuster for s-lecture hall partitions, and by Stanley for P-partitions. We also prove that the corresponding (P,s)-Eulerian polynomials are real-rooted for certain pairs (P,s), and speculate on unimodality properties of these polynomials.

It is of importance to model and estimate the user influence in social networks, especially for advertisers who conduct viral marketing. In this paper, we are interested in the number of received messages incurred by a node generating a message, and introduce the concepts of individual influence and type influence, while type influence is got by averaging out individual influence over nodes of the same type. We propose a user behavior model and use generating function to analyze type influence (including the mean and variance) and diffusion threshold, and find these results are not accurate in finite-size networks. We then classify nodes into subtypes and redefine the network model, which achieves much more accurate results. We also propose a scalable approach to estimate individual influence, and find it can get good approximates for individual influence, subtype influence and type influence by only considering local neighbors and out-of-date information, which is useful in large-scale networks. All analysis results are verified by simulations in real-world networks. Models in this paper can be extended to consider more realistic situations, and we believe these results are of use in understanding the diffusion dynamics in social networks.

This paper addresses the fundamental difference between equilibria of thermodynamic systems and equilibria of autonomous state space systems. The notion of stability of thermodynamic equilibria is analyzed in terms of an entropy generating function that classifies as a Lyapunov function to prove asymptotic stability of thermal equilibria. The stability analysis is performed for both finite and infinite dimensional systems. It is shown how the proposed Lyapunov function naturally extends to assess stability of interconnected thermal systems. A number of examples is given to demonstrate the time evolution of the Lyapunov function.

The convolved (u, v)-Lucas first kind p-polynomials are defined using the generating function of the (u, v)-Lucas first kind p-polynomials. The determinantal and permanental representations of the convolved (u, v)-Lucas first kind p-polynomials are used to derive some identities of these polynomials.

Finite-dimensional integrable Hamiltonian system, obtained through the nonlinearization of the 3 × 3 spectral problem associated with the Boussinesq equation, is investigated. A generating function method starting from the Lax-Moser matrix is used to give an effective way to prove the involutivity of integrals. Finite-parameter solution of the Boussinesq equation is calculated based on the commutative system of ordinary differential equations with these integrals as Hamiltonians. The problem of the third order differential operator associated with the Boussinesq Neumann system put forward by H. Flaschka in 1983 is solved.

We discuss the OWA aggregation operation and the role the OWA weights play in determining the type of aggregation being performed. We introduce the idea of a weight generating function and describe its use in obtaining the OWA weights. We emphasize the importance of the weight generating function in prescribing the type of aggregation to be performed. We consider the problem of performing a prescribed OWA aggregation in the case when we have a probability distribution over the argument values. We show how we use the weight generating function to enable this type of aggregation. Next we consider the situation when we have a more general measure based uncertainty over the argument values. Here again we show how we can use the weight generating function to aid in performing the prescribed OWA aggregation in the face of this more general type of uncertainty. Finally we look at the task of obtaining a weight generating function from a given set of OWA weights.

We provide a contact analogue of the symplectic camel theorem that holds in $\mathbb{R}^{2n}\times S^1$, and indeed generalize the symplectic camel. Our proof is based on the generating function techniques introduced by Viterbo, extended to the contact case by Bhupal and Sandon, and builds on Viterbo's proof of the symplectic camel., Comment: 22 pages, 3 figures

In this paper we consider compositions of n as bargraphs. The depth of a cell inside this graphical representation is the minimum number of horizontal and/or vertical unit steps that are needed to exit to the outside. The depth of the composition is the maximum depth over all cells of the composition. We use finite automata to study the generating function for the number of compositions having a depth of at least r.

In this paper, we introduce a generating function of sums of two-parameter deformations of multiple polylogarithms, denoted by Φ2(a;p,q), and study a q-difference equation satisfied by it. We show that this q-difference equation can be solved by expanding Φ2(a;p,q) into power series of the parameter p and then using the method of variation of constants. By letting $p \rightarrow 0$ in the main theorem, we find that the generating function of sums of q-interpolated multiple zeta values can be written in terms of the q-hypergeometric function 3ϕ2, which is due to Li-Wakabayashi.

For every natural number, let the sequence enumerate the number of subsets S of the set {1,2, . . . ,n } including no three consecutive odd integers. We give the generating function and the closed form formula of the sequence obtaining sixth order linear homogeneous recurrence relation with constant coefficients of the integer sequence. The sequence is associated with the Tribonacci sequence. The combinatorial representation of the sequence is obtained and limit of the ratios of consecutive terms of the sequence is found., Her doğal sayı için, {1,2, . . . ,n } kümesinin ardışık üç tek tam sayı içermeyen S alt kümelerinin sayısını veren tam sayı dizisini ele alalım . Dizinin altıncı dereceden sabit katsayılı lineer homojen rekürans bağıntısını elde ederek dizinin üreteç fonksiyonunu ve kapalı form formülünü verdik. Dizi Tribonacci sayı dizisi ile ilişkilendirildi. Dizinin kombinatoryal gösterimi elde edildi ve dizinin ardışık terimlerinin oranlarının limiti bulundu.

Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.

For a prototype quadratic Hamiltonian describing a driven, dissipative system, exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations of motion and operator calculus. The special and limiting cases are discussed., 15 pages, to be published in Scientific Reports

MIFARE CLASSIC CARD (hereinafter referred to as Mifare Classic) produced by NXP is the most common contactless smartcard on the market. Mifare Classics uses NXP's licensed patent stream cipher algorithm, CRYPTO-1, whose internal structure and algorithm parameters are commercial secrets and unknown to the outside world. At present, there is much research on the security of Mifare Classics. We analyze the security of the CRYPTO-1 algorithm in the Mifare Classics from the point of generating function of the linear feedback shift register (Linear feedback shift register, hereinafter referred to as LFSR) of the card and propose an optimization method for the vulnerability of generating function to improve the security of the Mifare Classics. Furthermore, the optimization method is generally applicable to the encryption algorithm based on the LFSR generating of pseudo-random sequences.

We construct (modified) scattering operators for the Vlasov–Poisson system in three dimensions, mapping small asymptotic dynamics as $$t\rightarrow -\infty$$ t → - ∞ to asymptotic dynamics as $$t\rightarrow +\infty$$ t → + ∞ . The main novelty is the construction of modified wave operators, but we also obtain a new simple proof of modified scattering. Our analysis is guided by the Hamiltonian structure of the Vlasov–Poisson system. Via a pseudo-conformal inversion, we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.

In this paper, we introduce a new family of distributions whose probability density function is defined as a weighted sum of two probability density functions; one is defined as a warped version of the other. We focus our attention on a special case based on the exponential distribution with three parameters, a dilation transformation and a weight with polynomial decay, leading to a new life-time distribution. The explicit expressions of the moments generating function, moments and quantile function of the proposed distribution are provided. For estimating the parameters, the method of maximum likelihood estimation is used. Two applications with practical data sets are given.

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

The trigintaduonions form a 32-dimensional Cayley–Dickson algebra. In this paper, we intend to make a new approach to introduce the concept of generalized Tribonacci trigintaduonions instead of and study some properties of this trigintaduonions like Binet’s formula, generating function, summation formula, norm value and matrix formulation.

Networks provide a mathematically rich framework to represent social contacts sufficient for the transmission of disease. Social networks are often highly clustered and fail to be locally tree-like. In this paper, we study the effects of clustering on the spread of sequential strains of a pathogen using the generating function formulation under a complete cross-immunity coupling, deriving conditions for the threshold of coexistence of the second strain. We show that clustering reduces the coexistence threshold of the second strain and its outbreak size in Poisson networks, whilst exhibiting the opposite effects on uniform-degree models. We conclude that clustering within a population must increase the ability of the second wave of an epidemic to spread over a network. We apply our model to the study of multilayer clustered networks and observe the fracturing of the residual graph at two distinct transmissibilities., 9 pages, 5 figures

State switching processes play an important role in the control of nanodevices and properties of many quasi-one-dimensional objects in physics, chemistry and biology. The paper considers the propagation of the state switching front from the sample boundary with the absorption of the emerging domains of the new phase at active centers in the bulk. The corresponding statistical-kinetic problem is solved with the calculation of the generating function of the distribution of domain wall positions, by means of which the average values of the “relay” runs are calculated.

The aim of this paper is to not only provide a definition of a new family of special numbers and polynomials of higher-order with their generating functions, but also to investigate their fundamental properties in the spirit of probabilistic distributions. By applying generating functions methods, we derive miscellaneous novel identities and formulas involving the Chu--Vandermonde-type convolution formulas, combinatorial sums, Bernstein basis functions, and the other well-known special numbers and polynomials. Moreover, we provide a computational algorithm which returns special values of these numbers and polynomials. In addition, we show that our new identities and formulas are connected with the interpolation functions of the Apostol-type numbers and polynomials. Finally, we present some theoretical and applied details on probabilistic distributions arising from the aforementioned Chu-Vandermonde-type convolution formulas.

We introduce a new continuous model with strong physical motivations and wide applications upon compounding the diecreate zero truncated Poisson model and a new continuous model called the Burr X Pareto type II distribution. Some of its mathematical and statistical properties are derived as well as four applications to real data sets are provided with detailes to illustrate the wide importance of the new model. We conclude that the new model is better than other nine competitive models via the four applications. Method of maximum likelihood is used to estimate the unknown parameters of the new model. The new model provide adequate Öts as compared to other related models in the four applications.

We analyze controlled mass transportation plans with free end-time that minimize the transport cost induced by the generating function of a Lagrangian within a bounded domain, in addition to costs incurred as export and import tariffs at entry and exit points on the boundary. We exhibit a dual variational principle \`a la Kantorovich, that takes into consideration the additional tariffs. We then show that the primal optimal transport problem has an equivalent Eulerian formulation whose dual involves the resolution of a Hamilton-Jacobi-Bellman quasi-variational inequality with non-homogeneous boundary conditions. This allows us to prove existence and to describe the solutions for both the primal optimization problem and its Eulerian counterpart., Comment: 25 pages. Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/

The Eulerian distribution on the involutions of the symmetric group is unimodal, as shown by Guo and Zeng. In this paper we prove that the Eulerian distribution on the involutions of the hyperoctahedral group, when viewed as a colored permutation group, is unimodal in a similar way and we compute its generating function, using signed quasisymmetric functions.

In this paper asymptotic behavior of the service process for the multi-channel network with input flow controlled by semi-Markov process is studied. Conditions in which a stationary regime exists are pointed out. Integral representation for generating function of the stationary distribution is constructed. In case of exponentially distributed service times limit Gaussian and diffusion Ornstein–Uhlenbeck processes for normalized service process of network processing under heavy traffic conditions are obtained.

Andrews et al. (Acta Arith. 105:51–66, 2002) introduced and studied the partition function PD(n), the number of partitions of n with designated summands. Recently, congruences involving the number of $$\ell $$ -regular partitions with designated summands, denoted by $$PD_{\ell }(n)$$ , have been explored for specific fixed values of $$\ell $$ . In this paper, we provide several families containing infinitely many congruences for $$PD_k(n)$$ for various values of k.

The present article is concerned with proving a well-known and very useful equivalent formula for the Pitman asymptotic relative efficiency A.R.E between rank tests. Furthermore, we also weaken the assumptions needed on the score generating function by using techniques arising from functional analysis.

The main objective of this article is to introduce the 2D q-Bessel functions by means of the generating function and to frame these q -special functions into the context of the representation of the 2-dimensional Euclidian quantum algebra eq(2). The quantum algebra eq(2) and its representation present an algebraic derivation for an explicit expression of the 2D q-Bessel functions in terms of the product of the q -Bessel functions of the first and second kinds. The series expansion, differential recurrence relations and functional identities for these functions are derived. Further, the q -Hermite—Bessel functions are introduced and their series expansions are derived. These expansions are used to derive the hypergeometric representation and generating equation for the q-Hermite—Bessel functions.

An example of the evolution of a random graph is used to discuss the approach to stochastic dynamics of complex systems based on evolutionary equations. For the case of a graph, these equations describe temporal changes in the structure of the graph associated with the process of randomly adding new bonds to it. Such a process is closely related to the coalescence of individual irreducible components of the graph and leads to the appearance of singularities in the spectra and their moments during finite time intervals. These singularities arise due to the appearance of a giant connected component whose order is comparable with the total order of the entire graph. The paper demonstrates a method for analyzing the dynamics of the process of evolution of a random graph based on the exact solution of an evolutionary equation that describes the time dependence of the generating functional for the probability of finding in the system a given distribution of connected components of the graph. A derivation of the nonlinear integral equation for the generating function distribution on the number of connected components is given and outlined the methods of its analysis. In the concluding part, the possibilities of applying this approach to solving a number of evolutionary problems of statistical geodynamics are discussed.

Recently, George Andrews gave a detailed study of partitions with even parts below odd parts in which only the largest even part appears an odd number of times. In this note, we provide a combinatorial proof of a generating function identity related to such partitions. This answers a problem of Andrews.

It was known that one-point functions in the ABJM matrix model (obtained by applying the localization technique to one-point functions of the half-BPS Wilson loop operator in the ABJM theory) satisfy the Jacobi-Trudi formula, which strongly indicates the integrable structure of the system. In this paper, we identify the integrable structure of two-point functions in the ABJM matrix model as the two-dimensional Toda lattice hierarchy. The identification implies infinitely many non-linear differential equations for the generating function of the two-point functions., 35 pages, 7 eps figures; v2: references added

Nonlocal equations of the coagulation theory are studied by the group analysis methods. In addition to the integro-differential Smoluchowski equation, equivalent models are also considered, including the equation for the Laplace transform of the original equation, an infinite system of equations for the power moments of its solution, and the equation for the generating function of the power moments. Admitted Lie groups for the considered equations are found, their relationships are studied, and the corresponding invariant solutions are analyzed.

In a very recent work (Murad, 2018) one of the present authors represents an algorithm for constructing all the static spherically symmetric exact anisotropic solutions of Einstein’s field equations of embedding class 1 based on a single generating function. In this work we apply the same algorithm to construct stellar models by generalizing some previous and offering some new forms of the generating function. The resulting equation of state and energy density, radial and tangential pressures, pressure anisotropy for a compact star having mass and radius similar to PSR J1614-2230 have been analyzed and the numerical values are found quite in agreement with the authors’ previous works.

Discretization is fundamental for digital realization of the fractional order system. The fractional order systems are digitally implemented by using either indirect or direct discretization methods. All these methods are typically developed in the conventional z domain in the form of s to z transforms. In this paper, a new direct discretization method is presented as a substitute of conventional z domain direct discretization method. The proposed approach capitalizes the feature of delta operator to unify both the discretized system and its continuous-time counterpart at the lower sampling time limit. The discretization is accomplished in two stages. At first, the delta operator is reframed by using the trapezoidal quadrature rule to obtain the suitable generating function. Then, the generating function is further expanded using the continued fraction expansion method to obtain the discrete-time approximation of the fractional order differentiator. Illustrative examples are presented to exhibit the efficacy of the proposed method with respect to two prevalent direct discretization methods of z domain on the basis of simulation results obtained using MATLAB.

We show that for any n and q, the number of real conjugacy classes in $ \rm{PGL}(\it{n},\mathbb{F}_q) $ is equal to the number of real conjugacy classes of $ \rm{GL}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SL}(\it{n},\mathbb{F}_q) $, refining a result of Lehrer [J. Algebra36(2) (1975), 278–286] and extending the result of Gill and Singh [J. Group Theory14(3) (2011), 461–489] that this holds when n is odd or q is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $ \rm{PGU}(\it{n},\mathbb{F}_q) $, and equal to the number of real conjugacy classes of $ \rm{U}(\it{n},\mathbb{F}_q) $ which are contained in $ \rm{SU}(\it{n},\mathbb{F}_q) $, refining results of Gow [Linear Algebra Appl.41 (1981), 175–181] and Macdonald [Bull. Austral. Math. Soc.23(1) (1981), 23–48]. We also give a generating function for this common quantity.

This paper discusses a discrete-time queueing system in which an arriving customer may adopt four different strategies; two of them correspond to a LCFS discipline where displacements or expulsions occur, and in the other two, the arriving customer decides to follow a FCFS discipline or to become a negative customer eliminating the customer in the server, if any. The different choices of the involved parameters make this model to enjoy a great versatility, having several special cases of interest. We carry out a thorough analysis of the system, and using a generating function approach, we derive analytical results for the stationary distributions obtaining performance measures for the number of customers in the queue and in the system. Also, recursive formulae for calculating the steady-state distributions of the queue and system size has been developed. Making use of the busy period of an auxiliary system, the sojourn times of a customer in the queue and in the system have also been obtained. Finally, some numerical examples are given.

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$. In this article, we discuss congruences satisfied by $p_{k}(n)$ when $k$ is a rational number.

A passivity based control method is presented for the diffusion process by the use of the concept of available storage and irreversible thermodynamics. A convex extension using the internal energy as generating function will be employed to define a Lyapunov functional for the distributed irreversible port Hamiltonian system (the diffusion process). The latter functional, will be used to suggest a Lyapunov stability condition by applying the so called La Salle’s Invariance principle for infinite dimensional systems on the heat and mass diffusion process.

Partitions play an important role in numerous combinatorial optimization problems. Here we introduce the number of ordered 3-partitions of a multiset M having equal sums denoted by S(m1,…, mn; α 1 ,…, α n ), for which we find the generating function and give a useful integral formula. Some recurrence formulae are then established and new integer sequences are added to OEIS, which are related to the number of solutions for the 3-signum equation.

Recently, parametric $$(q,\omega )$$ -exponential functions were defined in [4]. In the present paper, we obtain non-parametric Hahn exponential functions by using the characteristic properties of the usual discrete and q-exponential functions. We introduce Hahn-Appell polynomials and prove that their generating function is written by means of this new non-parametric exponential function. We further obtain a recurrence relation, difference equation and determinantal representation of Hahn-Appell polynomials. Finally, we determine the generating function of the Hahn-Appell polynomials which are also d-orthogonal and obtain the form of d-dimensional functional vector which ensures the d- orthogonality.

Starting from a total Lagrangian describing an oscillator-bath system, a novel derivation of exact quantum propagator is presented. Having the quantum propagator, the exact density matrix, reduced density matrix of the main oscillator and thermal equilibrium fixed point are obtained. The problem is generalised to the cases where the main oscillator is under the influence of a classical external force. By introducing generalised auxiliary classical fields, the generalised quantum propagator or generating functional of position correlation functions is obtained., Comment: 21 pages, to be published in Physics Letters A

In 1944 Lars Onsager published the exact partition function of the ferromagnetic Ising model on the infinite square lattice in terms of a definite integral. Only in the literature of the last decade, however, has it been recast in terms of special functions. Until now all known formulas for the partition function in terms of special functions have been restricted to the important special case of the isotropic Ising model with symmetric couplings. Indeed, the anisotropic model is more challenging because there are two couplings and hence two reduced temperatures, one for each of the two axes of the square lattice. Hence, standard special functions of one variable are inadequate to the task. Here, we reformulate the partition function of the anisotropic Ising model in terms of the Kamp\'e de F\'eriet function, which is a double hypergeometric function in two variables that is more general than the Appell hypergeometric functions. Finally, we present hypergeometric formulas for the generating function of multipolygons of given length on the infinite square lattice, for isotropic as well as anisotropic edge weights. For the isotropic case, the results allow easy calculation, to arbitrary order, of the celebrated series found by Cyril Domb., Comment: 13 pages, submitted for publication

This paper proposes a phase angle-modulated bat algorithm (P-AMBA) for high-dimensional binary optimization. The idea was to reduce the optimization time by introducing angle modulation technology to reduce the optimization dimensions. Different from the original angle-modulated bat algorithm (AMBA), the control of the trigonometric generating function cosine wave is by introducing new parameters, thereby improving the perturbation ability of the function curve near the x-axis. P-AMBA can explore more 0/1 solutions, and it has advantages in optimizing convergence speed and global search capabilities. The numerical results of the 0–1 knapsack problem tests show that P-AMBA is superior to the contrast algorithms on optimization ability and optimization time. Finally, the experimental result of a compact dual-band planar monopole antenna design showed the effectiveness of P-AMBA in engineering applications.

A mean-field type model with random growth and reset terms is considered. The stationary distributions resulting from the corresponding master equation are relatively easy to obtain, however, for practical applications one also needs to know the convergence to stationarity. The present work contributes to this direction, studying the transient dynamics in the discrete version of the model by two different approaches. The first method is based on mathematical induction by the recursive integration of the coupled differential equations for the discrete states. The second method transforms the coupled ordinary differential equation system into a partial differential equation for the generating function. We derive analytical results for some important, practically interesting cases and discuss the obtained results for the transient dynamics.