26,993 research outputs found
On the t-Term Rank of a Matrix
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to
be the largest number of 1s in A with at most one 1 in each column and at most
t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize
some basic results for the term rank to the t-term rank, including a formula
for the maximum term rank over a nonempty class of (0,1)-matrices with the the
same row sum and column sum vectors. We also show the surprising result that in
such a class there exists a matrix which realizes all of the maximum terms
ranks between 1 and t.Comment: 18 page
Level sets and non Gaussian integrals of positively homogeneous functions
We investigate various properties of the sublevel set
and the integration of on this sublevel set when and are positively
homogeneous functions. For instance, the latter integral reduces to integrating
on the whole space (a non Gaussian integral) and when is
a polynomial, then the volume of the sublevel set is a convex function of the
coefficients of . In fact, whenever is nonnegative, the functional is a convex function of for a large class of functions
. We also provide a numerical approximation scheme to compute
the volume or integrate (or, equivalently to approximate the associated non
Gaussian integral). We also show that finding the sublevel set of minimum volume that contains some given subset is a
(hard) convex optimization problem for which we also propose two convergent
numerical schemes. Finally, we provide a Gaussian-like property of non Gaussian
integrals for homogeneous polynomials that are sums of squares and critical
points of a specific function
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