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## The Cauchy problem in Sobolev spaces for Dirac operators

### Russian Mathematics (2009-07-01) 53: 43-54 , July 01, 2009

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator *A* in Sobolev spaces in a bounded domain *D* ⊂ ℝ^{n} with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one.

Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function *u* in a Sobolev space *H*^{s}(*D*), *s* ∈ ℕ, from its values on Γ and values *Au* in *D*, where Γ is an open connected subset of the boundary *∂D*. It is worth pointing out that we impose no assumptions about geometric properties of the domain *D*, except for its connectedness.

## The pointwise estimates on walsh overconvergence

### Approximation Theory and its Applications (1990-03-01) 6: 46-64 , March 01, 1990

Let f∈A_{p}. For any positive integer l, the quantity Δ_{1,n−1}(*f*:*z*) has been studied extensively. Here we give some quantitative estimates for
$$\mathop {\overline {\lim } \max }\limits_{n \to \infty |x| = R} |\vartriangle _{l,n - 1}^{(r)} (f;z)|^{1/n} $$
and investigate some pointwise estimates of Δ
_{l,n−1}^{(r)}
(*f*;*z*).

## Stability Theory of General Contractions for Delay Equations

### Integral Equations and Operator Theory (2010-11-01) 68: 413-426 , November 01, 2010

We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations *v*′ = *L*(*t*)*v*_{t} with a *nonuniform* exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation *v*′ = (*L*(*t*) + *M*(*t*))*v*_{t}, thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation *v*′ = *L*(*t*)*v*_{t} + *f*(*t*, *v*_{t}). In addition, we consider general contractions *e*^{−λρ(t)} determined by an increasing function *ρ* that includes the usual exponential behavior with *ρ*(*t*) = *t* as a very special case. We also obtain corresponding results in the case of discrete time.

## Periods of principal homogeneous spaces of algebraic tori

### Vestnik St. Petersburg University: Mathematics (2010-03-01) 43: 39-43 , March 01, 2010

A generic torsor of an algebraic torus *S* over a field *F* is the generic fiber of a *S*-torsor *P* → *T*, where *P* is a quasi-trivial torus containing *S* as a subgroup and *T* = *P/S*. The period of a generic *S*-torsor over a field extension *K/F*, i.e., the order of the class of the torsor in the group *H*^{p1}(*K, S*) does not depend on the choice of a generic torsor. In the paper we compute the period of a generic torsor of *S* in terms of the character lattice of the torus *S*.

## Back Matter - The Resolution Calculus

### The Resolution Calculus (1997-01-01) , January 01, 1997

## Notes on global existence for the nonlinear Schrödinger equation involves derivative

### Acta Mathematica Sinica, English Series (2014-10-01) 30: 1735-1747 , October 01, 2014

In this paper, we consider the scattering for the nonlinear Schrödinger equation with small, smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schrödinger equation with nonlinear term |*u*|^{2} involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with *p*-order nonlinearity with derivatives, the similar results hold for
$p \geqslant \tfrac{{2d + 3}}
{{2d - 1}}
$
and *d* ≥ 2, which is lower than the Strauss exponents.

## When is the multiplicity of a weight equal to 1?

### Functional Analysis and Its Applications (1990-10-01) 24: 259-269 , October 01, 1990

## Exkurs: Aspekte der Risikoanalyse — das Duration — Konzept

### Einführung in die Finanzmathematik (2003-01-01): 321-349 , January 01, 2003

### Zusammenfassung

Festverzinsliche Wertpapiere *(Bonds*, *Anleihen*,*... siehe Kapitel 6)* gehören zu den besonders wichtigen Finanzinstrumenten, und zwar insbesondere dann, wenn der Schuldner/Emittent hohe Bonität genießt. In diesem Fall dürften somit die zukünftigen Rückflüsse aus einem Bond nach Höhe und Zeitpunkt sicher sein *(d.h. das*,,*Bonitätsrisiko “darf dann vernachlässigt werden)*.