We formulate the generic $\tau$-function of Painlev\'e II equation as a Fredholm determinant an integrable (Its-Izergin-Korepin-Slavnov) operator. The depends on isomonodromic time $t$ and two Stokes' parameters, vanishing locus $\tau$-function, called Malgrange divisor is determined by zeros determinant.

A hypoelliptic operator in the Heisenberg calculus on a compact contact manifold is Fredholm operator. Its symbol determines an element K-theory of noncommutative algebra symbols. We construct periodic cyclic cocycle which, when paired with Connes-Chern character principal symbol, calculates index. Our index formula local, i.e. given as local expression terms and connection TM its curvature. pr...

Theorem 1. Let M be a compact manifold, E and F be two (smooth) vector bundles over M , and P : C∞(M ;E) → C∞(M ;F ) be a differential operator (with smooth coefficients) of order n. Assume that P is elliptic, in the sense defined in the next section. Then: 1. The operator P (or rather its relevant extension) is Fredholm W n 2 (M ;E)→ L(M ;F ). Here W n 2 is a Sobolev space (introduced below). ...

Let G be a Jordan domain and K C G be relatively closed with Area(K) = 0. Let A2(G\K) and A2 (G) be the Bergman spaces on G\K, respectively G and define N = A2(G\K) e A2 (G). In this paper we show that with a mild restriction on K, every function in N has an analytic continuation across the analytic arcs of 8G that do not intersect K. This result will be used to discuss the Fredholm theory of t...

Abstract. The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is self– adjoint and compact, is considered here. The data function is assumed to be perturbed gently by an additive noise so that it still belongs to the range of the operator. First we estimate upper and lower bounds for the ε–capacity (and then for the...

Surjectivity results of Fredholm alternative type are obtained for nonlinear operator equations of the form λT (x)−S(x) = f , where T is invertible, and T, S satisfy various types of homogeneity conditions. We are able to answer some questions left open by Fuč́ık, Nečas, Souček, and Souček. We employ the concept of an a-stably-solvable operator, related to nonlinear spectral theory methodology. ...

We study the rapid stabilization of heat equation on 1-dimensional torus using backstepping method with a Fredholm transformation. This classical framework allows us to present transformations for Laplace operator in sharp functional setting, which is main objective this work. first prove that, under some assumptions control operator, two scalar controls are necessary and sufficient get control...