Convolution equations in a bounded region in spaces with weighted norms

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

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Fields Connected with Particles in Terms of Complex-Riemannian Geometry

Zeitschrift für Naturforschung A ◽

10.1515/zna-1990-0201 ◽

1990 ◽

Vol 45 (2) ◽

pp. 81-94

Author(s):

Julian Ławrynowicz ◽

Katarzyna Kędzia ◽

Leszek Wojtczak

Keyword(s):

Field Potential ◽

Riemannian Metrics ◽

Interaction Matrix ◽

Explicit Calculation ◽

Maxwell System ◽

Curved Space ◽

Starting Point ◽

Autocorrelation Time ◽

The Fourier Transform ◽

Convolution Equations

AbstractA complex analytical method of solving the generalised Dirac-Maxwell system has recently been proposed by two of us for a certain class of complex Riemannian metrics. The Dirac equation without the field potential in such a metric appeared to be equivalent to the Dirac-Maxwell system including the field potentials produced by the currents of a particle in question. The method proposed is connected with applying the Fourier transform with respect to the electric charge treated as a variable, with the consideration of the mass as an eigenvalue, and with solving suitable convolution equations. In the present research an explicit calculation based on linearization of the spinor connections is given. The conditions for the motion are interpreted as a starting point to seek selection rules for curved space-times corresponding to actually existing particles. Then the same method is applied to solids. Namely, by a suitable transformation of the configuration space in terms of elements of the interaction matrix corresponding to the Coulomb, exchange, and dipole integrals, the interaction term in the hamiltonian becomes zero, thus leading to experimentally verificable formulae for the autocorrelation time

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An elastic singularity in a bounded region: volume change and related effects

Journal of the Mechanics and Physics of Solids ◽

10.1016/s0022-5096(00)00044-2 ◽

2001 ◽

Vol 49 (3) ◽

pp. 551-569 ◽

Cited By ~ 2

Author(s):

L.J. Walpole

Keyword(s):

Volume Change ◽

Bounded Region

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Almost vanishing polynomials and an application to the Hough transform

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500571 ◽

2014 ◽

Vol 13 (08) ◽

pp. 1450057 ◽

Cited By ~ 7

Author(s):

Maria-Laura Torrente ◽

Mauro C. Beltrametti

Keyword(s):

Hough Transform ◽

Affine Space ◽

Bounded Region ◽

Pattern Recognition Technique ◽

Local Study ◽

Affine Hypersurface ◽

Automated Recognition ◽

Real Affine ◽

Necessary And Sufficient ◽

Recognition Technique

We consider the problem of deciding whether or not an affine hypersurface of equation f = 0, where f = f(x1, …, xn) is a polynomial in ℝ[x1, …, xn], crosses a bounded region 𝒯 of the real affine space 𝔸n. We perform a local study of the problem, and provide both necessary and sufficient numerical conditions to answer the question. Our conditions are based on the evaluation of f at a point p ∈ 𝒯, and derive from the analysis of the differential geometric properties of the hypersurface z = f(x1, …, xn) at p. We discuss an application of our results in the context of the Hough transform, a pattern recognition technique for the automated recognition of curves in images.

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