1,648 research outputs found
Hyperbolic groups acting improperly
In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The
first main result (Theorem A) is a structural result about the Sageev
construction, in which we relate quasi-convexity of hyperplane stabilizers with
quasi-convexity of cell stabilizers. The second main result (Theorem D)
generalizes both Agol's theorem on cubulated hyperbolic groups and Wise's
Quasi-convex Hierarchy Theorem.Comment: 52pp. In v3, some unnecessary assumptions are dropped from some
technical results, especially in Section 5 and Corollary 6.5. The main
results are unchanged, but the improved technical results are expected to be
useful in future work. Several other small improvements to the exposition
have been mad
Quasi-convexity in mixtures for generalized rank-dependent functions
Quasi-convexity in probabilistic mixtures is a common and useful property in
decision analysis. We study a general class of non-monotone mappings, called
the generalized rank-dependent functions, which include the preference models
of expected utilities, dual utilities, and rank-dependent utilities as special
cases, as well as signed Choquet integrals used in risk management. As one of
our main results, quasi-convex (in mixtures) signed Choquet integrals precisely
include two parts: those that are convex (in mixtures) and the class of scaled
quantile-spread mixtures, and this result leads to a full characterization of
quasi-convexity for generalized rank-dependent functions. Seven equivalent
conditions for quasi-convexity in mixtures are obtained for dual utilities and
signed Choquet integrals. We also illustrate a conflict between convexity in
mixtures and convexity in risk pooling among constant-additive mappings
Which finitely generated Abelian groups admit isomorphic Cayley graphs?
We show that Cayley graphs of finitely generated Abelian groups are rather
rigid. As a consequence we obtain that two finitely generated Abelian groups
admit isomorphic Cayley graphs if and only if they have the same rank and their
torsion parts have the same cardinality. The proof uses only elementary
arguments and is formulated in a geometric language.Comment: 16 pages; v2: added reference, reformulated quasi-convexity, v3:
small corrections; to appear in Geometriae Dedicat
Some Numerical Simulations Based on Dacorogna Example Functions in Favor of Morrey Conjecture
Morrey Conjecture deals with two properties of functions which are known as
quasi-convexity and rank-one convexity. It is well established that every
function satisfying the quasi-convexity property also satisfies rank-one
convexity. Morrey (1952) conjectured that the reversed implication will not
always hold. In 1992, Vladimir Sverak found a counterexample to prove that
Morrey Conjecture is true in three dimensional case. The planar case remains,
however, open and interesting because of its connections to complex analysis,
harmonic analysis, geometric function theory, probability, martingales,
differential inclusions and planar non-linear elasticity. Checking analytically
these notions is a very difficult task as the quasi-convexity criterion is of
non-local type, especially for vector-valued functions. That's why we perform
some numerical simulations based on a gradient descent algorithm using
Dacorogna and Marcellini example functions. Our numerical results indicate that
Morrey Conjecture holds true
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