190,526 research outputs found
Generalized Orlicz spaces and Wasserstein distances for convex-concave scale functions
Given a strictly increasing, continuous function ,
based on the cost functional , we
define the -Wasserstein distance between
probability measures on some metric space . The function
will be assumed to admit a representation
as a composition of a convex and a concave function and , resp.
Besides convex functions and concave functions this includes all
functions.
For such functions we extend the concept of Orlicz spaces,
defining the metric space of measurable functions such that, for instance,
d_\vartheta(f,g)\le1\quad\Longleftrightarrow\quad
\int_X\vartheta(|f(x)-g(x)|)\,d\mu(x)\le1.$
Convex Integer Optimization by Constantly Many Linear Counterparts
In this article we study convex integer maximization problems with composite
objective functions of the form , where is a convex function on
and is a matrix with small or binary entries, over
finite sets of integer points presented by an oracle or by
linear inequalities.
Continuing the line of research advanced by Uri Rothblum and his colleagues
on edge-directions, we introduce here the notion of {\em edge complexity} of
, and use it to establish polynomial and constant upper bounds on the number
of vertices of the projection \conv(WS) and on the number of linear
optimization counterparts needed to solve the above convex problem.
Two typical consequences are the following. First, for any , there is a
constant such that the maximum number of vertices of the projection of
any matroid by any binary matrix is
regardless of and ; and the convex matroid problem reduces to
greedily solvable linear counterparts. In particular, . Second, for any
, there is a constant such that the maximum number of
vertices of the projection of any three-index
transportation polytope for any by any binary
matrix is ; and the convex three-index transportation problem
reduces to linear counterparts solvable in polynomial time
About t-norms on type-2 fuzzy sets.
Walker et al. defined two families of
binary operations on M (set of functions of [0,1]
in [0,1]), and they determined that, under certain
conditions, those operations are t-norms (triangular
norm) or t-conorms on L (all the normal and convex
functions of M). We define binary operations
on M, more general than those given by Walker et
al., and we study many properties of these general
operations that allow us to deduce new t-norms and
t-conorms on both L, and M
Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer
In a series of recent papers, W. M. Schmidt and L. Summerer developed a new
theory by which they recover all major generic inequalities relating exponents
of Diophantine approximation to a point in , and find new ones.
Given a point in , they first show how most of its exponents of
Diophantine approximation can be computed in terms of the successive minima of
a parametric family of convex bodies attached to that point. Then they prove
that these successive minima can in turn be approximated by a certain class of
functions which they call -systems. In this way, they bring the
whole problem to the study of these functions. To complete the theory, one
would like to know if, conversely, given an -system, there exists a
point in whose associated family of convex bodies has successive
minima which approximate that function. In the present paper, we show that this
is true for a class of functions which they call regular systems.Comment: 11 pages, 1 figure, to appear in Journal de th\'eorie des nombres de
Bordeau
On the minimization of Dirichlet eigenvalues
Results are obtained for two minimization problems: and where , is the 'th eigenvalue of the
Dirichlet Laplacian acting in , denotes the Lebesgue
measure of , denotes the perimeter of ,
and where is in a suitable class set functions. The latter
include for example the perimeter of , and the moment of inertia of
with respect to its center of mass.Comment: 15 page
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