220 research outputs found
Convex Hull of Planar H-Polyhedra
Suppose are planar (convex) H-polyhedra, that is, $A_i \in
\mathbb{R}^{n_i \times 2}$ and $\vec{c}_i \in \mathbb{R}^{n_i}$. Let $P_i =
\{\vec{x} \in \mathbb{R}^2 \mid A_i\vec{x} \leq \vec{c}_i \}$ and $n = n_1 +
n_2$. We present an $O(n \log n)$ algorithm for calculating an H-polyhedron
with the smallest such that
A Complete Characterization of the Gap between Convexity and SOS-Convexity
Our first contribution in this paper is to prove that three natural sum of
squares (sos) based sufficient conditions for convexity of polynomials, via the
definition of convexity, its first order characterization, and its second order
characterization, are equivalent. These three equivalent algebraic conditions,
henceforth referred to as sos-convexity, can be checked by semidefinite
programming whereas deciding convexity is NP-hard. If we denote the set of
convex and sos-convex polynomials in variables of degree with
and respectively, then our main
contribution is to prove that if and
only if or or . We also present a complete
characterization for forms (homogeneous polynomials) except for the case
which is joint work with G. Blekherman and is to be published
elsewhere. Our result states that the set of convex forms in
variables of degree equals the set of sos-convex forms if
and only if or or . To prove these results, we present
in particular explicit examples of polynomials in
and
and forms in
and , and a
general procedure for constructing forms in from nonnegative but not sos forms in variables and degree .
Although for disparate reasons, the remarkable outcome is that convex
polynomials (resp. forms) are sos-convex exactly in cases where nonnegative
polynomials (resp. forms) are sums of squares, as characterized by Hilbert.Comment: 25 pages; minor editorial revisions made; formal certificates for
computer assisted proofs of the paper added to arXi
A Bichromatic Incidence Bound and an Application
We prove a new, tight upper bound on the number of incidences between points
and hyperplanes in Euclidean d-space. Given n points, of which k are colored
red, there are O_d(m^{2/3}k^{2/3}n^{(d-2)/3} + kn^{d-2} + m) incidences between
the k red points and m hyperplanes spanned by all n points provided that m =
\Omega(n^{d-2}). For the monochromatic case k = n, this was proved by Agarwal
and Aronov.
We use this incidence bound to prove that a set of n points, no more than n-k
of which lie on any plane or two lines, spans \Omega(nk^2) planes. We also
provide an infinite family of counterexamples to a conjecture of Purdy's on the
number of hyperplanes spanned by a set of points in dimensions higher than 3,
and present new conjectures not subject to the counterexample.Comment: 12 page
A Transfer Matrix for the Backbone Exponent of Two-Dimensional Percolation
Rephrasing the backbone of two-dimensional percolation as a monochromatic
path crossing problem, we investigate the latter by a transfer matrix approach.
Conformal invariance links the backbone dimension D_b to the highest eigenvalue
of the transfer matrix T, and we obtain the result D_b=1.6431 \pm 0.0006. For a
strip of width L, T is roughly of size 2^{3^L}, but we manage to reduce it to
\sim L!. We find that the value of D_b is stable with respect to inclusion of
additional ``blobs'' tangent to the backbone in a finite number of points.Comment: 19 page
Computing the vertices of tropical polyhedra using directed hypergraphs
We establish a characterization of the vertices of a tropical polyhedron
defined as the intersection of finitely many half-spaces. We show that a point
is a vertex if, and only if, a directed hypergraph, constructed from the
subdifferentials of the active constraints at this point, admits a unique
strongly connected component that is maximal with respect to the reachability
relation (all the other strongly connected components have access to it). This
property can be checked in almost linear-time. This allows us to develop a
tropical analogue of the classical double description method, which computes a
minimal internal representation (in terms of vertices) of a polyhedron defined
externally (by half-spaces or hyperplanes). We provide theoretical worst case
complexity bounds and report extensive experimental tests performed using the
library TPLib, showing that this method outperforms the other existing
approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section
5 (using directed hypergraphs), detailed appendix; v3: major revision of the
article (adding tropical hyperplanes, alternative method by arrangements,
etc); v4: minor revisio
Lines, Circles, Planes and Spheres
Let be a set of points in , no three collinear and not
all coplanar. If at most are coplanar and is sufficiently large, the
total number of planes determined is at least . For similar conditions and
sufficiently large , (inspired by the work of P. D. T. A. Elliott in
\cite{Ell67}) we also show that the number of spheres determined by points
is at least , and this bound is best
possible under its hypothesis. (By , we are denoting the
maximum number of three-point lines attainable by a configuration of
points, no four collinear, in the plane, i.e., the classic Orchard Problem.)
New lower bounds are also given for both lines and circles.Comment: 37 page
Identically self-blocking clutters
A clutter is identically self-blocking if it is equal to its blocker. We prove that every identically self-blocking clutter different from is nonideal. Our proofs borrow tools from Gauge Duality and Quadratic Programming. Along the way we provide a new lower bound for the packing number of an arbitrary clutter
Localization via fractional moments for models on with single-site potentials of finite support
One of the fundamental results in the theory of localization for discrete
Schr\"odinger operators with random potentials is the exponential decay of
Green's function and the absence of continuous spectrum. In this paper we
provide a new variant of these results for one-dimensional alloy-type
potentials with finitely supported sign-changing single-site potentials using
the fractional moment method.Comment: LaTeX-file, 26 pages with 2 LaTeX figure
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure
From counting to construction of BPS states in N=4 SYM
We describe a universal element in the group algebra of symmetric groups,
whose characters provides the counting of quarter and eighth BPS states at weak
coupling in N=4 SYM, refined according to representations of the global
symmetry group. A related projector acting on the Hilbert space of the free
theory is used to construct the matrix of two-point functions of the states
annihilated by the one-loop dilatation operator, at finite N or in the large N
limit. The matrix is given simply in terms of Clebsch-Gordan coefficients of
symmetric groups and dimensions of U(N) representations. It is expected, by
non-renormalization theorems, to contain observables at strong coupling. Using
the stringy exclusion principle, we interpret a class of its eigenvalues and
eigenvectors in terms of giant gravitons. We also give a formula for the action
of the one-loop dilatation operator on the orthogonal basis of the free theory,
which is manifestly covariant under the global symmetry.Comment: 41 pages + Appendices, 4 figures; v2 - refs and acknowledgments adde
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