66 research outputs found
Dvoretzky type theorems for multivariate polynomials and sections of convex bodies
In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type
theorem) for homogeneous polynomials on , and improve bounds on
the number in the analogous conjecture for odd degrees (this case
is known as the Birch theorem) and complex polynomials. We also consider a
stronger conjecture on the homogeneous polynomial fields in the canonical
bundle over real and complex Grassmannians. This conjecture is much stronger
and false in general, but it is proved in the cases of (for 's of
certain type), odd , and the complex Grassmannian (for odd and even and
any ). Corollaries for the John ellipsoid of projections or sections of a
convex body are deduced from the case of the polynomial field conjecture
Ten Proofs of the Generalized Second Law
Ten attempts to prove the Generalized Second Law of Thermodyanmics (GSL) are
described and critiqued. Each proof provides valuable insights which should be
useful for constructing future, more complete proofs. Rather than merely
summarizing previous research, this review offers new perspectives, and
strategies for overcoming limitations of the existing proofs. A long
introductory section addresses some choices that must be made in any
formulation the GSL: Should one use the Gibbs or the Boltzmann entropy? Should
one use the global or the apparent horizon? Is it necessary to assume any
entropy bounds? If the area has quantum fluctuations, should the GSL apply to
the average area? The definition and implications of the classical,
hydrodynamic, semiclassical and full quantum gravity regimes are also
discussed. A lack of agreement regarding how to define the "quasi-stationary"
regime is addressed by distinguishing it from the "quasi-steady" regime.Comment: 60 pages, 2 figures, 1 table. v2: corrected typos and added a
footnote to match the published versio
The generalized Rademacher functions
In [A & G], the authors introuced the so-called generalized Rademacher functions and used the to prove that every continuous multilinear form A : x ... x
Behavior of entire functions on balls in a Banach space
In this paper we prove that given any two disjoint balls in an infinite dimensional complex Banach space, there exists an entire function which is bounded on one and unbounded on the other
Estimates by polynomials
Consider the following possible properties which a Banach space X may have: (P): If (x(j)) and (y(j)) are are bounded sequences in X such that for all n greater than or equal to 1 and for every continuous n-homogeneous polynomial P on X, P(x(j)) - P(y(j)) --> 0, then, Q(x(j) - y(j)) --> 0 for all m greater than or equal to 1 and for every continuous us m-homogeneous polynomial Q on X.
(RP): If (x(j)) and (y(j)) are bounded sequences in X such that for all n greater than or equal to 1 and for every continuous n-homogeneous polynomial P on X, P(x(j) - y(j)) --> 0, then Q(x(j)) - Q(y(j)) --> 0 for all m greater than or equal to 1 and for every continuous m-homogeneous polynomial Q on X. We study properties (P) and (RP) and their relation with the Schur property, Dunford-Pettis property, Lambda, and others. Several. applications of these properties are given
Dirichlet approximation and universal dirichlet series
We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and to the chordal one as well, analogous to classical results of Runge, Mergelyan and Vituškin. We also strengthen the notion of universal Dirichlet series. © 2017 American Mathematical Society
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