We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$-function on ${\rm PGSp}_6$. We deduce that for a wide class of cuspidal automorphic representations ...$\pi$, the partial Spin $L$-function is holomorphic except for a possible simple pole at $s=1$, and that the presence of such a pole indicates that $\pi$ is an exceptional theta lift from ${\rm G}_2$. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these $\pi$ upon which the aforementioned model becomes unique. The other integral can be regarded as a higher rank analogue of the integral of Kohnen-Skoruppa on ${\rm GSp}_4$.
We propose to define F1-algebras as integral bi-rings with the co-ring structure being the descent data from Z to F1. The coordinate bi-ring of Spec(Z)DIVISION SLASHF1 is then the co-ring of integral ...linear recursive sequences equipped with the Hadamard product. We associate a noncommutative moduli space to this setting and show that it is defined over F1 and has motive Formula omitted. .
The work focuses on the solution of a problem of approximation theory. The task is to investigate approximative properties of the Weierstrass integrals on the classes WβrHα. We obtain asymptotic ...equalities for the upper borders of defluxion of functions from the classes WβrHα from the Weierstrass integrals.
We consider the problem of additional analytic first integrals within the known problem of the perturbed motion of a satellite in a field determined by the Barrard gravitational potential. The paper ...demonstrates the absence of additional (different from known) first integrals of the problem.
We compute the four-loop contributions to the photon quark and Higgs quark form factors involving two closed fermion loops. We present analytical results for all nonplanar master integrals of the two ...nonplanar integral families which enter our calculation.
In this paper, a new computational method based on the generalized hat basis functions is proposed for solving stochastic Itô–Volterra integral equations. In this way, a new stochastic operational ...matrix for generalized hat functions on the finite interval 0,T is obtained. By using these basis functions and their stochastic operational matrix, such problems can be transformed into linear lower triangular systems of algebraic equations which can be directly solved by forward substitution. Also, the rate of convergence of the proposed method is considered and it has been shown that it is O(1n2). Further, in order to show the accuracy and reliability of the proposed method, the new approach is compared with the block pulse functions method by some examples. The obtained results reveal that the proposed method is more accurate and efficient in comparison with the block pule functions method.
This advanced monograph is concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of ...singularities for the wave equation and their counterparts in classical analysis. In particular, the author uses microlocal analysis to study problems involving maximal functions and Riesz means using the so-called half-wave operator. To keep the treatment self-contained, the author begins with a rapid review of Fourier analysis and also develops the necessary tools from microlocal analysis. This second edition includes two new chapters. The first presents Hörmander's propagation of singularities theorem and uses this to prove the Duistermaat–Guillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff.
In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and the level of its applications. In essence, the fractional calculus ...theory is a mathematical analysis tool applied to studying integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until a few years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing, and entropy theory. Since fractional order's differential and integral operators are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations; this is why applying fractional calculus theory has become a focus of international academic research. This Special Issue, “Applied Mathematics and Fractional Calculus II,” has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide, such as the USA, Ireland, Romania, Bulgaria, Türkiye, China, Pakistan, Iran, Egypt, India, Iraq, and Saudi Arabia.
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, ...while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE.
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