Optimal Sparsification for Some Binary CSPs Using Low-degree Polynomials

This paper analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems, without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP is not contained in coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of CNF-SAT with d literals per clause, to equivalent instances with $O(n^{d-e})$ bits for any e > 0. For the Not-All-Equal SAT problem, a compression to size $\~O(n^{d-1})$ exists. We put these results in a common framework by analyzing the compressibility of binary CSPs. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with $O(n^{2-e})$ bits for any e > 0, unless NP is a subset of coNP/poly.Comment: Updated the cross-composition in lemma 18 (minor update), since the previous version did NOT satisfy requirement 4 of lemma 18 (the proof of Claim 20 was incorrect

Multicritical behavior in models with two competing order parameters

We employ the nonperturbative functional Renormalization Group to study models with an O(N_1)+O(N_2) symmetry. Here, different fixed points exist in three dimensions, corresponding to bicritical and tetracritical behavior induced by the competition of two order parameters. We discuss the critical behavior of the symmetry-enhanced isotropic, the decoupled and the biconical fixed point, and analyze their stability in the N_1, N_2 plane. We study the fate of non-trivial fixed points during the transition from three to four dimensions, finding evidence for a triviality problem for coupled two-scalar models in high-energy physics. We also point out the possibility of non-canonical critical exponents at semi-Gaussian fixed points and show the emergence of Goldstone modes from discrete symmetries.Comment: 16 pages, 7 figures, 5 tables, minor changes in updated version, identical to published one in Phys. Rev.

The structure of frontoparallel haptic space is task dependent

In three experiments, we investigated the structure of frontoparallel haptic space. In the first experiment, we asked blindfolded participants to rotate a matching bar so that it felt parallel to the reference bar, the bars could be at various positions in the frontoparallel plane. Large systematic errors were observed, in which orientations that were perceived to be parallel were not physically parallel. In two subsequent experiments, we investigated the origin of these errors. In Experiment 2, we asked participants to verbally report the orientation of haptically presented bars. In this task, participants made errors that were considerably smaller than those made in Experiment 1. In Experiment 3, we asked participants to set bars in a verbally instructed orientation, and they also made errors significantly smaller than those observed in Experiment 1. The data suggest that the errors in the matching task originate from the transfer of the reference orientation to the matching-bar position

Interaction of a Moreton/EIT wave and a coronal hole

We report high-cadence H-alpha observations of a distinct Moreton wave observed at Kanzelhoehe Solar Observatory associated with the 3B/X3.8 flare and CME event of 2005 January 17. The Moreton wave can be identified in about 40 H-alpha frames over a period of 7 min. The EIT wave is observed in only one frame but the derived propagation distance is close to that of the simultaneously measured Moreton wave fronts indicating that they are closely associated phenomena. The large angular extent of the Moreton wave allows us to study the wave kinematics in different propagation directions with respect to the location of a polar coronal hole (CH). In particular we find that the wave segment whose propagation direction is perpendicular to the CH boundary (``frontal encounter'') is stopped by the CH which is in accordance with observations reported from EIT waves (Thompson et al. 1998). However, we also find that at a tongue-shaped edge of the coronal hole, where the front orientation is perpendicular to the CH boundary (the wave ``slides along'' the boundary), the wave signatures can be found up to 100 Mm inside the CH. These findings are briefly discussed in the frame of recent modeling results.Comment: 14 pages, 6 figures, accepted for publication in the Ap