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Newly discovered, incomplete forms of special functions are increasing the interest of both pure and applied mathematicians. The main purpose of this work is to derive four theorems on partial derivatives with incomplete Aleph functions of two variables and generalize them up to r-variables. In addition to these theorems, we also established some novel formulae on the partial derivatives that play a key role in deriving the main results in terms of finite sum. Further, we generalize the result and obtain the finite sum for the incomplete Aleph functions with r-variables. Here, we also established some particular cases that are in most general character and including the results given earlier by Buschman and Deshpande and may prove significant in numerous interesting situations appearing in the literature on mathematical analysis, applied mathematics and mathematical physics. [ABSTRACT FROM AUTHOR]

Starting from the pseudo‐differential decomposition D=(−Δ)12H of the Dirac operator D=∑j=1nej∂xj in terms of the fractional operator (−Δ)12 of order 1 and of the Riesz–Hilbert type operator H we will investigate the fundamental solutions of the space‐fractional Dirac equation of Lévy–Feller type ∂tΦα(x,t;θ)=−(−Δ)α2expiπθ2HΦα(x,t;θ)involving the fractional Laplacian −(−Δ)α2 of order α, with 2m≤α < 2m + 2 (m∈ℕ), and the exponentiation operator expiπθ2H as the hypercomplex counterpart of the fractional Riesz–Hilbert transform carrying the skewness parameter θ, with values in the range |θ|≤min{α−2m,2m+2−α}. Such model problem permits us to obtain hypercomplex counterparts for the fundamental solutions of higher‐order heat‐type equations ∂tFM(x,t)=κM(∂x)MFM(x,t)(M=2,3,...) in case where the even powers resp. odd powers D2m=(−Δ)m (M=α=2m) resp. D2m+1=(−Δ)m+12H (M=α=2m+1) of D are being considered. [ABSTRACT FROM AUTHOR]

The new generalized hypergeometric function (Yang 핐-function) described by the contour-type Mellin–Barnes integral representation serves as the inspiration for this study. The incomplete Yang 핐-functions γ핐 r,sp,q(z) and Γ핐 r,sp,q(z) that we shall introduce here are the appropriate extension of a class of 핐-functions by merit of the gamma functions of incomplete type, γ(σ,x) and Γ(σ,x). In this work, our objective is to study several remarkable characteristics of these newly introduced incomplete 핐-functions viz; reduction formula, derivative, various integral transformations. Additionally, using these newly established incomplete functions, we formulate and solve a generalized fractional kinetic equation in terms of these incomplete 핐-functions. Further, we also propose the incomplete 핎-functions γ핎 p,qr,s and Γ핎 p,qr,s corresponding to a new generalization of the 핐-function. [ABSTRACT FROM AUTHOR]

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDE s) associated with the quantum cohomology of P 1 -bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the P 1 -bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17] , called Borel (α , β) -multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen P 1 -bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDE s. As an example, we show how to integrate the qDE of blow-up of P 2 at one point via Borel multitransforms of solutions of the qDE of P 1. [ABSTRACT FROM AUTHOR]

We introduce the quantum currents for quantum systems with an SU(1,1) dynamic symmetry group whose evolution is governed by a non-linear Hamiltonian possessing a continuous spectrum and apply them to the analysis of the tunneling dynamics on the hyperbolic phase space. [ABSTRACT FROM AUTHOR]

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose new formulas for representing the solution of the first (Dirichlet), second (Neumann) and third (Robin) initial-boundary value problems for one-dimensional fractional diffusion equations with the time-fractional Caputo derivative. These formulas are obtained by the continuation method used in the theory of differential equations with integer derivatives. The Green's functions of the problems are constructed in terms of the Fox -function. As an example of the use of the constructed formulas, the inverse problem of determining the zero-coefficient of the fractional diffusion equation with the Neumann boundary condition and the overdetermination condition at the boundary of the spatial domain is studied. The existence and uniqueness of the solution of the considered inverse problem is obtained by a method based on the integral equations using Green's function. It is shown that this solution depends continuously on the input data. [ABSTRACT FROM AUTHOR]

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems. [ABSTRACT FROM AUTHOR]

We review the hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment. Upcoming measurements will reduce the experimental uncertainty of this observable by a factor of four; therefore, the theoretical precision must improve accordingly to fully harness such an experimental breakthrough. With regards to the HLbL contribution, this implies a study of the high-energy intermediate states that are neglected in dispersive estimates. We focus on the maximally symmetric high-energy regime and in-quark loop approximation of perturbation theory, following the method of the OPE with background fields proposed by Bijnens et al. in 2019 and 2020. We confirm their results regarding the contributions to the muon g − 2 . For this, we use an alternative computational method based on a reduction in the full quark loop amplitude, instead of projecting on a supposedly complete system of tensor structures motivated by first principles. Concerning scalar coefficients, mass corrections have been obtained by hypergeometric representations of Mellin–Barnes integrals. By our technique, the completeness of such kinematic singularity/zero-free tensor decomposition of the HLbL amplitude is explicitly checked. [ABSTRACT FROM AUTHOR]

It has been shown in recent works that JT gravity with matter with two boundaries has a type II∞ algebra on each side. As the bulk spacetime between the two boundaries fluctuates in quantum nature, we can only define the entanglement wedge for each side in a pure algebraic sense. As we take the semiclassical limit, we will have a fixed long wormhole spacetime for a generic partially entangled thermal state (PETS), which is prepared by inserting heavy operators on the Euclidean path integral. Under this limit, with appropriate assumptions of the matter theory, geometric notions of the causal wedge and entanglement wedge emerge in this background. In particular, the causal wedge is manifestly nested in the entanglement wedge. Different PETS are orthogonal to each other, and thus the Hilbert space has a direct sum structure over sub-Hilbert spaces labeled by different Euclidean geometries. The full algebra for both sides is decomposed accordingly. From the algebra viewpoint, the causal wedge is dual to an emergent type III1 subalgebra, which is generated by boundary light operators. To reconstruct the entanglement wedge, we consider the modular flow in a generic PETS for each boundary. We show that the modular flow acts locally and is the boost transformation around the global RT surface in the semiclassical limit. It follows that we can extend the causal wedge algebra to a larger type III1 algebra corresponding to the entanglement wedge. Within each sub-Hilbert space, the original type II∞ reduces to type III1. [ABSTRACT FROM AUTHOR]

In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: \begin{equation*} \left (\partial ^{\beta }_t+\dfrac {\nu }{2}\left (-\Delta \right)^{\alpha / 2}\right) u(t, x) = \: I_{t}^{\gamma }\left [\lambda u(t, x) \dot {W}(t, x)\right ] \quad t>0,\: x\in \mathbb {R}^d, \end{equation*} with constants \lambda \ne 0 and \nu >0, where \partial ^{\beta }_t is the Caputo fractional derivative of order \beta \in (0,2], I_{t}^{\gamma } refers to the Riemann-Liouville integral of order \gamma \ge 0, and \left (-\Delta \right)^{\alpha /2} is the standard fractional/power of Laplacian with \alpha >0. We concentrate on the scenario where the noise \dot {W} is the space-time white noise. The existence and uniqueness of solution in the Itô-Skorohod sense is obtained under Dalang's condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the p-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the p-th moment Lyapunov exponents. In particular, by letting \beta = 2, \alpha = 2, \gamma = 0, and d = 1, we confirm the following standing conjecture for the stochastic wave equation: \begin{align*} \frac {1}{t}\log \mathbb {E}[|u(t,x)|^p ] \asymp p^{3/2}, \quad \text {for p\ge 2 as t\to \infty.} \end{align*} The method for the lower bounds is inspired by a recent work of Hu and Wang, where the authors focus on the space-time colored Gaussian noise case. [ABSTRACT FROM AUTHOR]

We investigate the radiative QED corrections to the lepton (L = e, μ and τ) anomalous magnetic moment due to the contributions of diagrams with insertions of the photon vacuum polarisation operator consisting solely of four closed lepton (l = e, μ and τ) loops. Moreover, we focus on specific operators with two loops formed by leptons L of the same type as the external one, the other two formed by leptons ℓ different from L. The approach is essentially based on the employment of the Mellin–Barnes representation of the x -parametrization of the corresponding Feynman diagrams. This allows one to obtain, for the first time, exact analytical expressions for the radiative corrections of the tenth order w.r.t. the electromagnetic coupling constant e. Analytically, the radiative corrections are expressed in terms of the ratio r = m ℓ / m L of the internal to external lepton masses. The dependence on r is investigated numerically in the whole interval of r, 0 < r < ∞. To make comparisons with earlier published results possible, our exact analytical expressions are expanded as r → 0 and r → ∞ and compared with the corresponding asymptotic expansions known in the literature. [ABSTRACT FROM AUTHOR]

The Matérn and the Generalized Cauchy families of covariance functions have a prominent role in spatial statistics as well as in a wealth of statistical applications. The Matérn family is crucial to index mean-square differentiability of the associated Gaussian random field; the Cauchy family is a decoupler of the fractal dimension and Hurst effect for Gaussian random fields that are not self-similar. Our effort is devoted to prove that a scale-dependent family of covariance functions, obtained as a reparameterization of the Generalized Cauchy family, converges to a particular case of the Matérn family, providing a somewhat surprising bridge between covariance models with light tails and covariance models that allow for long memory effect. [ABSTRACT FROM AUTHOR]

Multiple-input–multiple-output (MIMO) systems have effectively addressed today's high demand for 5G communications and beyond. Further, MIMO-assisted space–time block codes (STBCs) have been shown to enhance the system's performance and provide a complete variety of coherent flat-fading channels. Additionally, the closed-form expression of the probability density function of the summation of correlated Weibull random variables remains unknown. In this work, we investigate the performance analysis of MIMO-STBC-enabled systems subject to the Weibull fading channel. New tight approximate expressions for numerous system performance metrics, e.g., outage probability, average capacity under various rate adaption methods, and average symbol/bit error rate, have been obtained. The Monte Carlo simulation method has corroborated all the presented results. [ABSTRACT FROM AUTHOR]

The extended conformable k-hypergeometric function finds various applications in physics due to its ability to describe complex mathematical relationships arising in different physical scenarios. Here are a few instances of its uses in physics, including nuclear physics, fluid dynamics, quantum mechanics, and astronomy. The main objectives of this paper are to introduce the extended conformable k-hypergeometric and confluent hypergeometric functions by utilizing the new definition of the α , k -beta function and studying its important properties, like integral representation, summation formula, derivative formula, transform formula, and generating function. Also, introduce the extension of the Riemann–Liouville fractional derivative and establish some results related to the newly defined fractional operator, such as the Mellin transform and relations to extended α , k -hypergeometric functions. [ABSTRACT FROM AUTHOR]

We combine perturbation theory with analytic and numerical bootstrap techniques to study the critical point of the long-range Ising (LRI) model in two and three dimensions. This model interpolates between short-range Ising (SRI) and mean-field behaviour. We use the Lorentzian inversion formula to compute infinitely many three-loop corrections in the two-dimensional LRI near the mean-field end. We further exploit the exact OPE relations that follow from bulk locality of the LRI to compute infinitely many two-loop corrections near the mean-field end, as well as some one-loop corrections near SRI. By including such exact OPE relations in the crossing equations for LRI we set up a very constrained bootstrap problem, which we solve numerically using SDPB. We find a family of sharp kinks for two- and three-dimensional theories which compare favourably to perturbative predictions, as well as some Monte Carlo simulations for the two-dimensional LRI. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the recent article "Multi-parametric Le Roy function", Fract. Calc. Appl. Anal. 26(5), 54–69 (2023), https://doi.org/10.1007/s13540-022-00119-y, by S. Rogosin and M. Dubatovskaya. Here we present further analytic properties of the Le Roy function depending on several parameters. Using the Mellin-Barnes representations we determine relations of the multi-parametric Le Roy function F (α , β) m (γ) m to the Fox H-function in the case of integer parameters γ j . For non-integer γ j the function F (α , β) m (γ) m becomes a particular case of a new special HG-function generalizing the Fox H-function and related to other generalizations of the Fox H-function, namely H ¯ -function and I-function (I integral). Extension to other values of the parameters α j , γ j of the multi-parametric Le Roy function are discussed too. Novel formulas for double-, triple- and multiple-Laplace transform of F (α , β) m (γ) m are established. Riemann-Liouville fractional operators of the multi-parametric Le Roy function are calculated. [ABSTRACT FROM AUTHOR]

We compute scattering amplitudes involving one massive scalar and two, three, or four gravitons. We show that when the conformal dimension of the massive scalar is set to zero, the resulting celestial correlators depend only on the coordinates of the gravitons. Such correlators of gravitons are well-defined and do not suffer from divergences associated with the Mellin transform of usual graviton amplitudes. Moreover, they are non-distributional and take the form of standard CFT correlators. We show that they are consistent with the usual OPEs but the statement of the soft theorem is modified. [ABSTRACT FROM AUTHOR]

We study the self-energies of weakly interacting scalar fields in de Sitter space with one field much lighter than the Hubble scale. We argue that self-energies drastically simplify in this light limit. We illustrate this in theories with two scalar fields, one heavy and one light, interacting with one another through either cubic or quartic interactions. To regulate infrared divergences, we compute these self-energies in Euclidean de Sitter space and then carefully analytically continue to Lorentzian signature. In particular, we do this for the most general renormalizable theory of two scalar fields with even interactions to leading order in the coupling and the mass of the light field. These self-energies are determined by de Sitter sunset diagrams, whose analytic structure and UV divergences we derive. Even at very weak couplings, the light field can substantially change how the heavy field propagates over long distances. The light field’s existence may then be inferred from how it modifies the heavy field’s oscillatory contribution to the primordial bispectrum in the squeezed limit, i.e. its cosmological collider signal. [ABSTRACT FROM AUTHOR]

We perform an analysis of the one- and two-loop massive quark form factor in QED in a region expansion, up to next-to-leading power in the quark mass. This yields an extensive set of regional integrals, categorized into three topologies, against which factorization theorems at next-to-leading power could be tested. Our analysis reveals a number of subtle aspects involving rapidity regulators, as well as additional regions that manifest themselves only beyond one loop, at the level of single diagrams, but which cancel in the form factor. [ABSTRACT FROM AUTHOR]

We study the product P m of m real Ginibre matrices with Gaussian elements of size N, which has received renewed interest recently. Its eigenvalues, which are either real or come in complex conjugate pairs, become all real with probability one when m → ∞ at fixed N. In this regime the statistics becomes deterministic and the Lyapunov spectrum has been derived long ago. On the other hand, when N → ∞ and m is fixed, it can be expected that away from the origin the same local statistics as for a single real Ginibre ensemble at m = 1 prevails. Inspired by analogous findings for products of complex Ginibre matrices, we introduce a critical scaling regime when the two parameters are proportional, m = α N . We derive the expected number, variance and rescaled density of real eigenvalues in this critical regime. This allows us to interpolate between previous recent results in the above mentioned limits when α → ∞ and α → 0 , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we delve into the challenge of identifying an unknown source in a space-time fractional diffusion-wave equation. Through an analysis of the exact solution, it becomes evident that the problem is ill-posed. To address this, we employ both the Tikhonov regularization method and the Quasi-boundary regularization method, aiming to restore the stability of the solution. By adhering to both a priori and a posteriori regularization parameter choice rules, we derive error estimates that quantify the discrepancies between the regularization solutions and the exact solution. Finally, we present numerical examples to illustrate the effectiveness and stability of the proposed methods. [ABSTRACT FROM AUTHOR]

This study investigates an extension of the extended Hurwitz-Lerch zeta function. along with related integral images and derivatives. by extending the extended beta function. Also established is a link between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. It 5.P f is also demonstrated how to use the enlarged Hurwitz-Lerch zeta function Cv.Xcx; c. a.p.q) to probability distributions. Some (old and new) observations are offered here as specific illustrations of our theories. [ABSTRACT FROM AUTHOR]