Search Results (257)

Several important functions, including the gamma function, as well as several infinite sums, admit integral representations involving the Hankel contour. In addition, the large t asymptotic analysis of several recently derived identities satisfied by the Riemann zeta function requires the computation of the asymptotic form of certain integrals which also involve the Hankel contour; these integrals depend on a real parameter, $\alpha$. A rigorous asymptotic technique is presented here for computing such integrals to all orders. For certain values of $\alpha$, the relevant formula, in addition to an asymptotic series of explicit terms, also contains a specific integral. It is shown that, remarkably, the leading behavior of this integral can be written in the form of the leading order of the Bleistein integral. The latter integral arises in the implementation of the classical steepest descent method in the case that the stationary point coincides with one of the boundary points of the integral under consideration. Full article

This paper investigates explicit coefficient-based estimates for a class of fractional Hermite functions defined through finite power series with Gamma-function coefficients. These functions may be viewed as a fractional Hermite-type family associated with the Caputo fractional derivative of order $\alpha \in (0,1]$. An explicit representation of the fractional derivative is obtained as a finite sum of monomials with computable Gamma coefficients. This representation is used to derive a preliminary uniform estimate on bounded intervals $[0,R]$ with an explicit constant depending on $\alpha$, n, and R. Consistency with the integer-order setting is established by showing that, when $\alpha =1$, the construction reduces to a Hermite-type polynomial family and the Caputo derivative coincides with the ordinary derivative. Explicit asymptotic formulas are obtained for the associated coefficient envelope as $R\to 0^{+}$ and $R\to \infty$. Numerical experiments up to degree $n=7$ show that the ratio between the coefficient envelope and the computed supremum norm remains below approximately $1.45$ for the tested parameter range. In addition, a weighted $L^2$ estimate is derived with respect to a fractional Gaussian-type weight, yielding an explicit coefficient-based bound. The estimates obtained in this work are preliminary in nature, being based on coefficient-wise majorization, and are not claimed to be optimal. Determining sharp constants and establishing genuine norm-comparison inequalities remain open problems. The results presented here provide a rigorous starting point for the study of explicit coefficient-based estimates for fractional Hermite functions and suggest several directions for future research in fractional approximation theory. Full article

This work focuses on a class of nonlinear second-order difference equations with non-autonomous periodic coefficients. We obtain explicit solution formulas that allow for a constructive characterization of their behavior. In particular, we derive necessary and sufficient conditions for the existence of periodic solutions, providing a complete description of when they arise. Several numerical examples are also included to illustrate the theoretical results. Full article

We investigate the asymptotic behavior of the product of Gaussian random variables, with a focus on tail probabilities. Although sums of normal variables are well understood through classical limit theorems, their products exhibit significantly more complex behavior and have no simple closed-form distributions. We analyze the extremes of such products by deriving precise asymptotic expressions for the tail probabilities as the threshold tends to infinity. The study covers both centered and shifted but independent Gaussian variables, as well as cases with heterogeneous variances and nonzero expectations. Using transformation techniques, geometric arguments in high-dimensional spaces, and asymptotic analysis based on Gaussian measures, we establish general results describing the decay rate of tail probabilities. The main theorems provide explicit asymptotic formulas that depend on the number of variables, their shifts, and the variance structure. Several corollaries present results for important particular cases, including the case of identically distributed random variables. Three individual examples are provided at the end of the paper to illustrate the resulting asymptotic formulas and compare them with Monte Carlo estimates. Full article

In this paper, we introduce and study a Szász-type family of positive linear operators generated by Euler polynomials of negative order on $[0,\infty )$. The construction is based on an explicit finite representation of these polynomials with non-negative terms, which ensures the positivity of the corresponding kernel. We prove the basic properties of the operators and show that they can be represented as finite convex combinations of shifted classical Szász operators. We also provide a probabilistic representation of the kernel as a finite mixture of Poisson distributions, which clarifies the role of the parameter k and the resulting moment structure. The corresponding algebraic and central moment identities are derived and used to establish convergence on compact intervals and to obtain quantitative estimates in terms of the modulus of continuity, Lipschitz-type classes, and Peetre’s K-functional. Furthermore, Voronovskaya-type asymptotic results are obtained, including a quantitative form and a second-order asymptotic formula. Numerical tables and a graphical illustration are presented for selected test functions and parameter values, and the results are consistent with the theoretical convergence behaviour. The paper shows that Euler polynomials of negative order provide a positive and structurally tractable framework for constructing Szász-type approximation operators on the positive real axis. Full article

This paper designs a novel aperiodic intermittent control (AIC) strategy using discrete-time observation information. It can stabilize unstable hybrid McKean–Vlasov stochastic differential equations and reduce control consumption effectively. Key contributions include the following: (1) Lévy noise is introduced into the hybrid McKean–Vlasov framework to describe discontinuous disturbances. We further derive the existence, uniqueness and generalized Itô formula for the above system. (2) A new distribution-dependent Lyapunov functional to prove moment finiteness, mean square, and asymptotic exponential stability is constructed. (3) We derive explicit ranges for the AIC time rate and observation intervals. By tightening the state error bound via an innovative technique, the control design constraints are effectively relaxed. (4) We prove the equivalence of exponential stability between the controlled system and its particle approximation. This approach avoids the computational intractability of the exact probability distribution. Finally, the efficacy of our method is demonstrated through a numerical example. Full article

This paper addresses the stabilization problem of hybrid stochastic McKean–Vlasov differential equations via a discrete-time state observation feedback control strategy. Utilizing the coupling method and particle system approximation, Itô’s formula for Markovian switching stochastic McKean–Vlasov differential equations is established. Based on the derived formula, we construct two novel Lyapunov functionals that incorporate state processes, probability distributions, and Markovian switching signals. Using the proposed Lyapunov functionals, we further analyze three stability properties of the closed-loop system, including H_∞ stability, asymptotic stability, and mean-square exponential stability. Due to the time-varying characteristics of system distributions, numerical simulation lacks fixed reference benchmarks and faces considerable difficulties. To overcome this challenge, this paper introduces a particle system approximation scheme. We further prove the exponential stability equivalence between the controlled McKean–Vlasov system and its corresponding particle system. This equivalence relation provides an effective new approach for the stability analysis of such controlled hybrid stochastic systems. Finally, an illustrative example is given to verify our theory results. Full article

We study the degree-d Jensen polynomials $J_{d,n}\left(X\right)$ built from the moment sequence $M_n={\int }_0^{\infty }{\mathsf{\Phi }}_1\left(u\right)u^{2n}du$ of the Riemann $\mathsf{\Xi }$-function, which coincides with the classical Pólya–Jensen family. Using bridge coordinates, the staircase law, and Plancherel–Rotach asymptotics, we prove that $J_{d,n}^{\gamma }$ is hyperbolic for all $n\ge C_0^{\infty }{d}^4$ ($C_0^{\infty }\approx 0.020$; the analytic formula for $C_0^{\infty }$ is rigorous but agrees with the numerically observed value to within $2.6\%$); combined with the GORZ theorem for $d\le 8$, this covers the entire asymptotic regime. We identify a phase-transition law $n^{*}\left(d\right)=C_0^{\infty }{d}^4+\alpha d^3+\beta {(-1)}^d{d}^2+O\left(d\right)$: the leading constant $C_0^{\infty }\approx 0.0195$ is computed analytically and verified to within $2.6\%$ of the empirical large-d limit; the formula for $\alpha$ is derived; its numerical value of ≈$-0.2$ to $-0.3$ is numerical evidence; the parity structure $\beta {(-1)}^d{d}^2$ is proved. For the finite strip $0\le n<C_0^{\infty }{d}^4$ with $d\ge 9$, the sole remaining gap, whose closure is equivalent to the Riemann Hypothesis under standard transversality, we establish four structural obstructions: ratio-barrier saturation (no usable margin, certified and numerical); frozen zero count (parity blocks any ladder, certified for $d\le 21$); interlacing-lift vacuity (proved); and a discriminant equivalence (proved under transversality), showing that all known local and inductive mechanisms fail simultaneously in this region. The problem reduces to $Disc\left(J_{d,n}^{\gamma }\right)>0$ for all $d\ge 9$ and $0\le n<C_0^{\infty }{d}^4$; this requires moment data $M_k$ for $k\ge 130$, which are currently inaccessible. Full article

Random integer lattices are fundamental to lattice-based cryptography and algorithmic number theory. A new random integer lattice model, free of any restrictions on the Hermite Normal Form (HNF), was introduced by in 2016. It was also observed that the probability of such a lattice being in a simple HNF form is approximately 44%. In this paper, the gap between general random integer lattices and those in a simple HNF is bridged by introducing the concept of the $(n,k)$-simple random integer lattice, where the first k diagonal entries of the HNF are fixed to 1. We derive the asymptotic counting formula for such lattices and compute their density among all integer lattices. Furthermore, a generation algorithm for the $(n,k)$-simple random integer lattice based on rejection sampling and inverse sampling methods are proposed, with the analysis showing that it achieves $O\left(n^2\right)$ expected running time. This work provides a theoretical foundation and practical toolkit for constructing structured random lattices with controlled HNF forms. Full article

Static wetting is a widely adopted method for determining surface and interface tensions of materials. In this study, the effects of elasticity and thickness are analytically investigated using a spherical cap model, for an incompressible elastic thin layer bonded to a rigid substrate. The asymptotic solutions are derived based on the principle of stationary total potential energy. Explicit formulas are provided to link the droplet shape to the surface/interface energies, the thickness and the Young’s modulus of the layer. Furthermore, the model is refined with the aid of numerical results of spherical indentation. The refined model shows reasonable agreement with existing experimental data of contact angles for various layer thicknesses with a maximum deviation of 5°, whereas the deviation yielded by the classical Young’s equation exceeds 15°. Furthermore, the predicted Young’s modulus based on the explicit formulas is 3.2 kPa, close to 3 kPa obtained from macro bulk rheometry experiments. These agreements demonstrate that the present model has the potential to capture the effects of thickness and elasticity, thereby improving the prediction accuracy of surface energies, and provide an alternative method to characterize the Young’s modulus of the layer based on measurements of contact angles across various contact radii. Full article

This article introduces and develops a comprehensive theory of the Mixed Degenerate Gould–Hopper–Appell Type Polynomials $MDGHA$-$TPs$, constructed by embedding an Appell factor into the framework of degenerate Gould–Hopper generating functions. Beginning with the generating function formulation, we derive explicit series representations, monomial-type operational identities, recurrence relations, and a determinantal form that encodes the algebraic structure of the family. Summation identities expressed via Stirling numbers of the first kind and addition-type formulas are established. A detailed numerical investigation of the zero distributions of these polynomials is then carried out, with graphical illustrations revealing symmetry patterns and geometric arrangements in the complex plane. Connections with classical sequences of Appell, Hermite, and Gould–Hopper are explored throughout. The article concludes with remarks on open problems including the orthogonality of the $MDGHA$-$TPs$ with respect to suitable weight functions, the asymptotic behaviour of their zeros as the degree tends to infinity, and potential applications to boundary-value problems in heat diffusion, perturbation expansions in quantum mechanics, and signal processing in non-homogeneous media. Full article

In this paper, the valuation of barrier options is studied when the underlying asset is driven by a hybrid model of stochastic volatility and constant elasticity of variance. Using an asymptotic expansion approach and the Fourier transform method, a semi-analytical approximate pricing formula for up-and-out call options are derived under the proposed hybrid model. We validate the approximate pricing formula by comparing its outputs with those produced by Monte Carlo simulation and the binomial tree method. In addition, we perform a sensitivity analysis numerically on the key model parameters and investigate limiting regimes of the hybrid model. It is verified that the approximation is properly anchored to simpler benchmark models when one or both perturbative effects vanish. Full article

The objective of this work is to investigate the $\mathbb{T}=(s,t)$-deformed free convolution ${⊞}_{\mathbb{T}}$ for $s>0$ and $t\in \mathbb{R}$ and to clarify its structural and asymptotic properties within the framework of Cauchy–Stieltjes kernel (CSK) families. The methodology is based on the analysis of the associated variance functions (VFs), which provide an effective analytic tool for describing deformation mechanisms, invariance properties, and convolution structures. In particular, we derive an explicit formula for the VF of convolution powers and exploit this representation to develop approximation procedures for distributions in CSK families generated by the $(s,t)$-deformed free Gaussian and free Poisson laws. We also establish several limit theorems describing the asymptotic behavior of the deformation. These findings highlight intrinsic symmetry and scaling properties and reveal connections with free additive, Boolean additive, and free multiplicative convolutions, thereby placing the $(s,t)$-deformation within a unified probabilistic framework governed by transformation, invariance, and structural regularity. Full article

We construct and analyze a family of support-defined Brauer-type configurations canonically associated with the Boolean geometry underlying the Grassmann algebra. The construction is governed by an x-support map on monomial labels, which identifies the vertex set with the Boolean lattice $\mathcal{P}\left(\right[n\left]\right)$. This identification yields a Boolean support quiver isomorphic to the directed Hasse diagram of $\mathcal{P}\left(\right[n\left]\right)$, equivalently, to an oriented hypercube. We then equip the family with a canonical cyclic ordering at each vertex and obtain a genuine connected reduced Brauer configuration in the standard sense, together with its associated Brauer configuration algebra and its standard Brauer quiver. A ghost-variable mechanism is introduced to obtain a connected realization without altering any support-controlled invariants. We prove that polygon membership, valencies, multiplicities, Boolean stratification, and the support quiver are invariant under support-preserving ghost relabelings. We also give an explicit description of the standard Brauer quiver and show that it is different from the Boolean support quiver. On the algebraic side, we derive closed formulas for the center dimension, the algebra dimension, and the normalization constant of the induced weighted distribution. On the probabilistic side, we distinguish the vertex entropy from the layer entropy, establish an exact decomposition of the former by Hamming layers, and show that the layer distribution is asymptotically concentrated on the middle layers, while extremal vertices and any fixed maximal path contribute a negligible fraction of the total weight. As a consequence, the layer entropy satisfies a logarithmic asymptotic law. We also investigate geometric consequences of the Boolean model transported through the support identification. Coordinate projections produce a rigidity phenomenon for antipodal pairs, providing a combinatorial analogue of Greenberger–Horne–Zeilinger (GHZ)-type fragility, whereas the first Boolean layer exhibits a persistence property analogous to W-type robustness. Together, these results exhibit a concrete bridge between Grassmann combinatorics, Brauer configuration theory, hypercube geometry, and entropy asymptotics. Full article

In this paper, we study the weighted approximation properties of a family of Szász–Mirakyan-type operators preserving two exponential functions on the unbounded interval $[0,\infty )$. The operators act on exponential weighted spaces and are analyzed within the framework of positive linear operator theory. We first establish their well-definedness and boundedness between suitable weighted spaces. By applying a weighted Korovkin-type theorem, we prove convergence in the corresponding weighted norm. Furthermore, we obtain quantitative estimates in terms of a weighted modulus of continuity and derive an order of convergence result. A Voronovskaya-type asymptotic formula is also established, describing the precise asymptotic behavior of the operators. Numerical examples are included to support the theoretical results. Full article