In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-mode...

This is a pedagogical review on the integrability-based approach to the three-point function in N = 4 supersymmetric Yang-Mills theory. We first discuss the computation of the structure constant at weak coupling and show that the result can be recast as a sum over partitions of the rapidities of the magnons. We then introduce a non-perturbative framework, called the “hexagon approach”, and expl...

The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the BirkhoffGustavson(BG)-normalization: In solving an inverse problem of the BGnormalization on computer algebra, it is shown that if the perturbed harmonic oscillators with a homogeneous cubic-polynomial potential and with a homogeneous quartic-polynomial potenti...

We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan–Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obst...

The convergence in meaaa of a veighted sum k a.k(Xk EXk) of random elements in a separable Banach space is studied under a new hypothesis which relates the random elements with their respective weights in the sum: the {a.. }-compactly uniform integrability of {X. }. This condition, which is implied by the tightness of {X,,} and the {a,,k }-uniform integrability of {[IX,, II}, is weaker than the...

RIEMANN AND RIEMANN-STIELTJES INTEGRABILITY Looking back over the notes already posted on this and related subjects, I felt that their organization did not emphasize the things that are important on the subject of the caption. Hence these notes, which will initially reprise things that are already treated in the others, but with different emphasis. The material on almost-everywhere continuity a...

Correspondence should be addressed to Hongya Gao, [email protected] Received 30 March 2012; Revised 26 April 2012; Accepted 23 May 2012 Academic Editor: Martino Bardi Copyright q 2012 Hongya Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is...

Carl Gustav Jacob Jacobi (1804–1851), one of the nineteenth century Germany’s most accomplished scientists, developed the theory of determinants and transformations into a powerful tool for evaluating multiple integrals and solving differential equations. Since then, the Jacobian (determinant) has played a critical role in multidimensional analysis and related fields, including nonlinear elasti...

In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements o...

In this article we deal with the Riemann integral of functions from R into a real Banach space. The last theorem establishes the integrability of continuous functions on the closed interval of reals. To prove the integrability we defined uniform continuity for functions from R into a real normed space, and proved related theorems. We also stated some properties of finite sequences of elements o...