The algebra of singular operators with terminal symbol on a piecewise smooth curve: I. Convolution type operators on a semiaxis

2000 ◽  
Vol 36 (9) ◽  
pp. 1337-1347
Author(s):  
A. P. Soldatov
Keyword(s):  
Smooth Curve ◽  
Piecewise Smooth ◽  
Convolution Type ◽  
Terminal Symbol ◽  
Singular Operators

The Morse Formula for Curves Which are not Locally Simple

2000 ◽  
Vol 7 (4) ◽  
pp. 599-608
Author(s):  
R. Abdulaev
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Critical Points ◽  
Smooth Curve ◽  
Piecewise Smooth ◽  
Number Of Solutions

Abstract Let be an interior mapping of the unit disk, continuous in D2 and such that the restriction of f to the unit circle S 1 is a locally simple curve γ. Suppose that f(a) ≠ a on S 1 and denote by n(a) the number of solutions of the equation f(z) = a in D2 , by μ(f) the sum of multiplicities of the critical points of f in , by q(a) the angular order of γ with respect to a, and by τ(γ) the angular order of γ. It is proved that the Morse formula 2n(a) – μ(f) – 2q(a) + τ(γ) – 1 = 0 remains correct for a piecewise smooth curve which is not locally simple.

The Riemann–Hilbert Problem in a Domain with Piecewise Smooth Boundaries in Weight Classes of Cauchy Type Integrals with a Density from Variable Exponent Lebesgue Spaces

2009 ◽  
Vol 16 (4) ◽  
pp. 737-755 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Vakhtang Paatashvili
Keyword(s):  
Variable Exponent ◽  
Hilbert Problem ◽  
Domain Boundary ◽  
Smooth Curve ◽  
Piecewise Smooth ◽  
Cauchy Type ◽  
Lebesgue Spaces ◽  
Variable Exponent Lebesgue Spaces ◽  
Cauchy Type Integrals ◽  
Riemann Hilbert Problem

Abstract The Riemann–Hilbert problem for an analytic function is solved in weighted classes of Cauchy type integrals in a simply connected domain not containing 𝑧 = ∞ and having a density from variable exponent Lebesgue spaces. It is assumed that the domain boundary is a piecewise smooth curve. The solvability conditions are established and solutions are constructed. The solution is found to essentially depend on the coefficients from the boundary condition, the weight, space exponent values at the angular points of the boundary curve and also on the angle values. The non-Fredholmian case is investigated. An application of the obtained results to the Neumann problem is given.

On the geometric properties of Vandermonde's mapping and on the problem of moments

1989 ◽  
Vol 112 (3-4) ◽  
pp. 203-211 ◽  
Author(s):  
V. P. Kostov
Keyword(s):  
Half Space ◽  
Smooth Curve ◽  
Random Variable ◽  
Piecewise Smooth ◽  
Geometric Properties ◽  
Problem Of Moments ◽  
Euclidian Distance

SynopsisIn this paper we prove that the domain of hyperbolicity of the polynomial xn + λ2nn−2+λ3xn−3+ … + λn,λiϵR intersected by the half-space λ2 ≧ – 1, has the property of Whitney, i.e., every two points of this set can be connected by a piecewise-smooth curve belonging to it, whose length is ≦C times greater than the euclidian distance between the points, where the constant C does not depend on the choice of the points. Parallel with this, we show that the values x1≦x2≦…≦xn of a random variable are uniquely determined by the corresponding probabilities and by thefirst n moments.

On the Riemann–Hilbert Problem in the Domain with a Nonsmooth Boundary

1997 ◽  
Vol 4 (3) ◽  
pp. 279-302
Author(s):  
V. Kokilashvili ◽  
V. Paatashvili
Keyword(s):  
Hilbert Problem ◽  
Smooth Curve ◽  
Piecewise Smooth ◽  
Analytical Function ◽  
Boundary Values ◽  
Nonsmooth Boundary ◽  
Smirnov Class ◽  
Angular Boundary ◽  
Riemann Hilbert Problem

Abstract The following Riemann–Hilbert problem is solved: find an analytical function Φ from the Smirnov class Ep (D), whose angular boundary values satisfy the condition Re[(a(t) + ib(t))Φ+ (t)] = ƒ(t). The boundary Γ of the domain D is assumed to be a piecewise smooth curve whose nonintersecting Lyapunov arcs form, with respect to D, the inner angles with values νkπ, 0 < νk ≤ 2.

Smooth curve drawing using a Bessel function

2015 ◽  
Author(s):  
Y. Q. Du ◽  
M. J. Pan ◽  
Q. Li ◽  
L. Li
Keyword(s):  
Bessel Function ◽  
Smooth Curve
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