For simplicity, we use the following convention: i, n denote natural numbers, a, r, x, y denote real numbers, A denotes a closed-interval subset of R, C denotes a non empty set, and X denotes a set. We now state several propositions: (1) For every element D of divsA such that vol(A) = 0 holds lenD = 1. (2) χA,A is integrable on A and integralχA,A = vol(A). (3) For every partial function f from ...

Using the Laplace derivative a Perron type integral, is defined. Moreover, it shown that this integral includes and to show inclusion proper, an example of function constructed, which integrable but not integrable. Properties integrals such as fundamental theorem calculus, Hake's theorem, integration by parts, convergence theorems, mean value remainder form Taylor's with estimation remainder, a...

In this paper, we define fractal bases and frames of L2(I×J), where I J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification L2(I×J) with tensor product space L2(I)⨂L2(J). First, recall procedure constructing perturbation continuous or integrable function. Then, composed such functions. We also obtain we...

Abstract There are different methods of discretizing integrable systems. We consider semi-discrete analog two-dimensional Toda lattices associated to the Cartan matrices simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on notion Darboux integrability. Generalized known be continuous case (that is, they admit complete families characteristic integrals bot...

Abstract We study the SYK 2 model of N Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling to two- four-point sources. In particular, we define generalized form factor (SFF) level spacing distribution function generalizing from partition generating function. For = 2, obtain an exact solution SFF. It exhibits qualitatively similar behavior hi...

In this paper we explore the general conditions in order that a 2-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that the invariant is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary t...

The constructive definition usually begins with a function f, then by the process of using Riemann sums and limits, we arrive the definition of the integral of f, ∫ b a f. On the other hand, a descriptive definition starts with a primitive F satisfying certain condition(s) such as F ′ = f and F is absolutely continuous if f is Lebesgue integrable, and F is generalized absolutely continuous if f...

We analyze the one-loop correction to the three-point function coefficient of scalar primary operators in N = 4 SYM theory. By applying constraints from the superconformal symmetry, we demonstrate that the type of Feynman diagrams that contribute depends on the choice of renormalization scheme. In the planar limit, explicit expressions for the correction are interpreted in terms of the hamilton...

Renormalization of twist-three operators and integrable lattice models. Abstract We address the problem of solution of the QCD three-particle evolution equations which govern the Q 2-dependence of the chiral-even quark-gluon-quark and three-gluon correlators contributing to a number of asymmetries at leading order and the transversely polarized structure function g 2 (x Bj). The quark-gluon-qua...

is integrable in the first argument on [a + ε, b − ε] for arbitrary ρ ∈ [0,+∞[ and ε ∈ ]0, (b− a)/2[ . Of special interest for us is the case where the function f (and therefore the function f) is non-integrable in the first argument on [a, b], having singularities at the ends of this segment. In this sense the problems (1), (2) and (1), (3) are singular ones. Singular boundary value problems f...