961 research outputs found

    Removing Degeneracy in LP-Type Problems Revisited

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    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

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    We elucidate aspects of the one-loop anomalous dimension of so(6)so(6)-singlet multi-trace operators in N=4 SU(Nc)\mathcal{N}=4\ SU(N_c) SYM at finite NcN_c. First, we study how 1/Nc1/N_c corrections lift the large NcN_c degeneracy of the spectrum, which we call the operator submixing problem. We observe that all large NcN_c zero modes acquire non-positive anomalous dimension starting at order 1/Nc21/N_c^2, 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 2Nc2N_c. The dimension of such operators becomes more negative as NcN_c increases, which will eventually diverge in a double scaling limit. Third, we examine the structure of level-crossing at finite NcN_c in view of unitarity. Finally we find out a correspondence between the large NcN_c 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

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    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

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    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

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    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
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