140,045 research outputs found
The Alexandrov-Fenchel type inequalities, revisited
Various Alexandrov-Fenchel type inequalities have appeared and played
important roles in convex geometry, matrix theory and complex algebraic
geometry. It has been noticed for some time that they share some striking
analogies and have intimate relationships. The purpose of this article is to
shed new light on this by comparatively investigating them in several aspects.
\emph{The principal result} in this article is a complete solution to the
equality characterization problem of various Alexandrov-Fenchel type
inequalities for intersection numbers of nef and big classes on compact
K\"{a}hler manifolds, extending earlier results of Boucksom-Favre-Jonsson,
Fu-Xiao and Xiao-Lehmann. Our proof combines a result of Dinh-Nguy\^{e}n on
K\"{a}hler geometry and an idea in convex geometry tracing back to Shephard. In
addition to this central result, we also give a geometric proof of the complex
version of the Alexandrov-Fenchel type inequality for mixed discriminants and a
determinantal type generalization of various Alexandrov-Fenchel type
inequalities.Comment: 18 pages, slightly revised version stressing our principal result,
comments welcom
Unbounded quasinormal operators revisited
Various characterizations of unbounded closed densely defined operators
commuting with the spectral measures of their moduli are established.In
particular, Kaufman's definition of an unbounded quasinormal operator is shown
to coincide with that given by the third-named author and Szafraniec. Examples
demonstrating the sharpness of results are constructed.Comment: 13 page
Hierarchic Superposition Revisited
Many applications of automated deduction require reasoning in first-order
logic modulo background theories, in particular some form of integer
arithmetic. A major unsolved research challenge is to design theorem provers
that are "reasonably complete" even in the presence of free function symbols
ranging into a background theory sort. The hierarchic superposition calculus of
Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we
demonstrate, not optimally. This paper aims to rectify the situation by
introducing a novel form of clause abstraction, a core component in the
hierarchic superposition calculus for transforming clauses into a form needed
for internal operation. We argue for the benefits of the resulting calculus and
provide two new completeness results: one for the fragment where all
background-sorted terms are ground and another one for a special case of linear
(integer or rational) arithmetic as a background theory
Hierarchic Superposition Revisited
Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory
The Steiner tree problem revisited through rectifiable G-currents
The Steiner tree problem can be stated in terms of finding a connected set of
minimal length containing a given set of finitely many points. We show how to
formulate it as a mass-minimization problem for -dimensional currents with
coefficients in a suitable normed group. The representation used for these
currents allows to state a calibration principle for this problem. We also
exhibit calibrations in some examples
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