581 research outputs found
Bad semidefinite programs: they all look the same
Conic linear programs, among them semidefinite programs, often behave
pathologically: the optimal values of the primal and dual programs may differ,
and may not be attained. We present a novel analysis of these pathological
behaviors. We call a conic linear system {\em badly behaved} if the
value of is finite but the dual program has no
solution with the same value for {\em some} We describe simple and
intuitive geometric characterizations of badly behaved conic linear systems.
Our main motivation is the striking similarity of badly behaved semidefinite
systems in the literature; we characterize such systems by certain {\em
excluded matrices}, which are easy to spot in all published examples.
We show how to transform semidefinite systems into a canonical form, which
allows us to easily verify whether they are badly behaved. We prove several
other structural results about badly behaved semidefinite systems; for example,
we show that they are in in the real number model of computing.
As a byproduct, we prove that all linear maps that act on symmetric matrices
can be brought into a canonical form; this canonical form allows us to easily
check whether the image of the semidefinite cone under the given linear map is
closed.Comment: For some reason, the intended changes between versions 4 and 5 did
not take effect, so versions 4 and 5 are the same. So version 6 is the final
version. The only difference between version 4 and version 6 is that 2 typos
were fixed: in the last displayed formula on page 6, "7" was replaced by "1";
and in the 4th displayed formula on page 12 "A_1 - A_2 - A_3" was replaced by
"A_3 - A_2 - A_1
On monochromatic triangles
AbstractLet A and B be two disjoint finite sets in R2. Simple conditions that guarantee the existence of a triangle with vertices in one of the sets and with no points from the other set in its interior are given. The analogous problem for d-simplices in Rd is treated. Conditions are derived that guarantee the existence of a triangle with vertices in one of the sets and with no points from either set on its boundary
N=4 Characters in Gepner Models, Orbits and Elliptic Genera
We review the properties of characters of the N=4 SCA in the context of a
non-linear sigma model on , how they are used to span the orbits, and how
the orbits produce topological invariants like the elliptic genus. We derive
the same expression for the elliptic genus using three different Gepner
models (, and theories), detailing the orbits and verifying
that their coefficients are given by elementary modular functions. We
also reveal the orbits for the , and theories. We
derive relations for cubes of theta functions and study the function for .Comment: 39 pages; errors corrected in section 6, section 7 added (mixed
Gepner models), ref adde
Phase transition in a log-normal Markov functional model
We derive the exact solution of a one-dimensional Markov functional model
with log-normally distributed interest rates in discrete time. The model is
shown to have two distinct limiting states, corresponding to small and
asymptotically large volatilities, respectively. These volatility regimes are
separated by a phase transition at some critical value of the volatility. We
investigate the conditions under which this phase transition occurs, and show
that it is related to the position of the zeros of an appropriately defined
generating function in the complex plane, in analogy with the Lee-Yang theory
of the phase transitions in condensed matter physics.Comment: 9 pages, 5 figures. v2: Added asymptotic expressions for the
convexity-adjusted Libors in the small and large volatility limits. v3: Added
one reference. Final version to appear in Journal of Mathematical Physic
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