We give new proofs of a global and local property the integrals which compute closed string theory amplitudes at genus zero. Both kinds properties are related to newborn single-valued periods, our provide an intuitive understanding this relation. The property, known in physics as KLT formula, is factorisation into products pairs open integrals. deduce it by identifying with special values corre...

Abstract We relate the low-energy expansions of world-sheet integrals in genus-one amplitudes open- and closed-string states. The respective expansion coefficients are elliptic multiple zeta values (eMZVs) open-string case non-holomorphic modular forms dubbed ‘modular graph (MGFs)’ for closed strings. By inspecting differential equations degeneration limits suitable generating series integrals,...

In this paper an algorithm is presented for the regularization of singular integrals with any degrees of singularity, which may be employed in all three-dimensional problems analyzed by Boundary Elements. The integrals in Boundary Integrals Equations are inherently singular. For example, one can mention the integrals confronted in potential problems to evaluate the flow or the gradient of the f...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/notes 2012-13/06c cpt ab gps.pdf] 1. Approximate identities on topological groups 2. Uniqueness of invariant measure 3. Simultaneous eigenfunctions for integral operators 4. Simultaneous eigenfunctions are characters The spectral theory for normal compact operators on Hil...

This paper presents Stieltjes-type integration for operator-valued functions with respect to spectral families. The relation between RiemannStieltjes integrals associated with some classes of spectral families including, in particular, those that arise in the context of the Stone representation theorem for Banach spaces is established. The developed approach is applied to the structural analysi...

Probability measures (quasi probability mass), given in the form of integrals of Wigner function over areas of the underlying phase space, give rise to operator valued probability measures (OVM). General construction methods of OVMs, are investigated in terms of geometric positive trace increasing maps (PTI), for general 1D domains, as well as 2D shapes e.g. circles, disks. Spectral properties ...

The paper pursues the definition of a maxitive integral on all real-valued functions (i.e., the integral of the pointwise maximum of two functions must be the maximum of their integrals). This definition is not determined by maxitivity alone: additional requirements on the integral are necessary. The paper studies the consequences of additional requirements of invariance with respect to affine ...