5,275 research outputs found
Hahn decomposition and Radon-Nikodym theorem with a parameter
The paper contains a simple proof of the classical Hahn decomposition theorem
for charges and, as a corollary, an explicit measurable in parameter
construction of a Radon-Nikodym derivative of one measure by another
Developments in General Relativity: Black Hole Singularity and Beyond
At the 20-th Texas Symposium on Relativistic Astrophysics there was a plenary
talk devoted to the recent developments in classical Relativity. In that talk
the problems of gravitational collapse, collisions of black holes, and of black
holes as celestial bodies were discussed. But probably the problems of the
internal structure of black holes are a real great challenge. In my talk I want
to outline the recent achievements in our understanding of the nature of the
singularity (and beyond!) inside a realistic rotating black hole. This
presentation also addresses the following questions: Can we see what happens
inside a black hole? Can a falling observer cross the singularity without being
crushed? An answer to these questions is probably "yes".Comment: 13 page
Stars creating a gravitational repulsion
In the framework of the Theory of General Relativity, models of stars with an
unusual equation of state where is the mass density
and is the pressure, are constructed. These objects create outside
themselves the forces of gravitational repulsion. The equilibrium of such stars
is ensured by a non-standard balance of forces. Negative mass density, acting
gravitationally on itself, creates an acceleration of the negative mass,
directed from the center. Therefore in the absence of pressure such an object
tends to expand. At the same time, the positive pressure, which falls just like
in ordinary stars from the center to the surface, creates a force directed from
the center. This force acts on the negative mass density, which causes
acceleration directed the opposite of the acting force, that is to the center
of the star. This acceleration balances the gravitational repulsion produced by
the negative mass. Thus, in our models gravity and pressure change roles: the
negative mass tends to create a gravitational repulsion, while the gradient of
the pressure acting on the negative mass tends to compress the star. In this
paper, we construct several models of such a star with various equations of
state.Comment: 6 pages, 4 figure
Oscillation of Fourier Integrals with a spectral gap
Suppose that Fourier transform of a function f is zero on the interval
[-a,a]. We prove that the lower density of sign changes of f is at least a/pi,
provided that f is a locally integrable temperate distribution in the sense of
Beurling, with non-quasianalytic weight. We construct an example showing that
the last condition cannot be omitted.Comment: 1 Figur
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