Daily Papers

Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables z_i, we derive a system of partial differential equations w.r.t.\ new variables x_j, which parameterize the differentiable constraints z_i=y_i(x_j). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

We use a simple method that leads to the integrals involved in obtaining the traveling wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in terms of hypergeometric functions are obtained while when that term is nonzero all the basic traveling wave solutions of Liouville, Tzitzeica and their variants, as well as sine/sinh-Gordon equations with important applications in the phenomenology of nonlinear physics and dynamical systems are found through a detailed study of the corresponding elliptic equations

We present an analysis of the Rayleigh-Plesset equation for a three dimensional vacuous bubble in water. In the simplest case when the effects of surface tension are neglected, the known parametric solutions for the radius and time evolution of the bubble in terms of a hypergeometric function are briefly reviewed. By including the surface tension, we show the connection between the Rayleigh-Plesset equation and Abel's equation, and obtain the parametric rational Weierstrass periodic solutions following the Abel route. In the same Abel approach, we also provide a discussion of the nonintegrable case of nonzero viscosity for which we perform a numerical integration

It was demonstrated recently that the W_{1+infty} algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized widetilde W algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the widetilde W algebras studied earlier. We suggest a definition of the generalized widetilde W algebra as differential operators in variables p_k basing on the matrix realization of the W_{1+infty} algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the W_{1+infty} algebra than is contained in its commutative subalgebras. The positive integer rays are associated with widetilde W algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric tau-functions corresponding to the completed cycles.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric mathcal D-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic mathcal D-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. Over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric D-module defines an F-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.