961 research outputs found
Removing Degeneracy in LP-Type Problems Revisited
LP-type problems is a successful axiomatic framework for optimization problems capturing, e.g., linear programming and the smallest enclosing ball of a point set. In Matoušek and Škovroň (Theory Comput. 3:159-177, 2007), it is proved that in order to remove degeneracies of an LP-type problem, we sometimes have to increase its combinatorial dimension by a multiplicative factor of at least 1+ε with a certain small positive constant ε. The proof goes by checking the unsolvability of a system of linear inequalities, with several pages of calculations. Here by a short topological argument we prove that the dimension sometimes has to increase at least twice. We also construct 2-dimensional LP-type problems with −∞ for which removing degeneracies forces arbitrarily large dimension increas
Negative anomalous dimensions in N=4 SYM
We elucidate aspects of the one-loop anomalous dimension of -singlet
multi-trace operators in SYM at finite . First,
we study how corrections lift the large degeneracy of the
spectrum, which we call the operator submixing problem. We observe that all
large zero modes acquire non-positive anomalous dimension starting at
order , and they mix only among the operators with the same number of
traces at leading order. Second, we study the lowest one-loop dimension of
operators of length equal to . The dimension of such operators becomes
more negative as increases, which will eventually diverge in a double
scaling limit. Third, we examine the structure of level-crossing at finite
in view of unitarity. Finally we find out a correspondence between the
large zero modes and completely symmetric polynomials of Mandelstam
variables.Comment: 34+31 pages, many figures, a Mathematica file attached, v2: typos
corrected, references added, section 5 revised, v3: revised Section 4 on
correlators, and small detail
Kakeya sets of curves
We investigate analogues for curves of the Kakeya problem for straight lines.
These arise from H"ormander-type oscillatory integrals in the same way as the
straight line case comes from the restriction and Bochner-Riesz problems. Using
some of the geometric and arithmetic techniques developed for the straight line
case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results
for families of parabolas whose coefficients satisfy certain algebraic
conditions.Comment: 32 pages. To appear in GAF
Wavelet analysis on symbolic sequences and two-fold de Bruijn sequences
The concept of symbolic sequences play important role in study of complex
systems. In the work we are interested in ultrametric structure of the set of
cyclic sequences naturally arising in theory of dynamical systems. Aimed at
construction of analytic and numerical methods for investigation of clusters we
introduce operator language on the space of symbolic sequences and propose an
approach based on wavelet analysis for study of the cluster hierarchy. The
analytic power of the approach is demonstrated by derivation of a formula for
counting of {\it two-fold de Bruijn sequences}, the extension of the notion of
de Bruijn sequences. Possible advantages of the developed description is also
discussed in context of applied
Random Sampling with Removal
Random sampling is a classical tool in constrained optimization. Under favorable conditions, the optimal solution subject to a small subset of randomly chosen constraints violates only a small subset of the remaining constraints. Here we study the following variant that we call random sampling with removal: suppose that after sampling the subset, we remove a fixed number of constraints from the sample, according to an arbitrary rule. Is it still true that the optimal solution of the reduced sample violates only a small subset of the constraints?
The question naturally comes up in situations where the solution subject to the sampled constraints is used as an approximate solution to the original problem. In this case, it makes sense to improve cost and volatility of the sample solution by removing some of the constraints that appear most restricting. At the same time, the approximation quality (measured in terms of violated constraints) should remain high.
We study random sampling with removal in a generalized, completely abstract setting where we assign to each subset R of the constraints an arbitrary set V(R) of constraints disjoint from R; in applications, V(R) corresponds to the constraints violated by the optimal solution subject to only the constraints in R. Furthermore, our results are parametrized by the dimension d, i.e., we assume that every set R has a subset B of size at most d with the same set of violated constraints. This is the first time this generalized setting is studied.
In this setting, we prove matching upper and lower bounds for the expected number of constraints violated by a random sample, after the removal of k elements. For a large range of values of k, the new upper bounds improve the previously best bounds for LP-type problems, which moreover had only been known in special cases. We show that this bound on special LP-type problems, can be derived in the much more general setting of violator spaces, and with very elementary proofs
- …