A map on the space of rational functions

We describe dynamical properties of a map $\mathfrak{F}$ defined on the space of rational functions. The fixed points of $\mathfrak{F}$ are classified and the long time behavior of a subclass is described in terms of Eulerian polynomials

Lambda Polarization in Polarized Proton-Proton Collisions at RHIC

We discuss Lambda polarization in semi-inclusive proton-proton collisions, with one of the protons longitudinally polarized. The hyperfine interaction responsible for the $\Delta$-$N$ and $\Sigma$-$\Lambda$ mass splittings gives rise to flavor asymmetric fragmentation functions and to sizable polarized non-strange fragmentation functions. We predict large positive Lambda polarization in polarized proton-proton collisions at large rapidities of the produced Lambda, while other models, based on SU(3) flavor symmetric fragmentation functions, predict zero or negative Lambda polarization. The effect of $\Sigma^0$ and $\Sigma^*$ decays is also discussed. Forthcoming experiments at RHIC will be able to differentiate between these predictions.Comment: 18 pages, 5 figure

Game saturation of intersecting families

We consider the following combinatorial game: two players, Fast and Slow, claim $k$-element subsets of $[n]=\{1,2,...,n\}$ alternately, one at each turn, such that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed $k$-subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone the game's end as long as possible. The game saturation number is the score of the game when both players play according to an optimal strategy. To be precise we have to distinguish two cases depending on which player takes the first move. Let $gsat_F(\mathbb{I}_{n,k})$ and $gsat_S(\mathbb{I}_{n,k})$ denote the score of the saturation game when both players play according to an optimal strategy and the game starts with Fast's or Slow's move, respectively. We prove that $\Omega_k(n^{k/3-5}) \le gsat_F(\mathbb{I}_{n,k}),gsat_S(\mathbb{I}_{n,k}) \le O_k(n^{k-\sqrt{k}/2})$ holds

Upper and Lower Bounds for Weak Backdoor Set Detection

We obtain upper and lower bounds for running times of exponential time algorithms for the detection of weak backdoor sets of 3CNF formulas, considering various base classes. These results include (omitting polynomial factors), (i) a 4.54^k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Horn formulas; (ii) a 2.27^k algorithm to detect whether there is a weak backdoor set of at most k variables into the class of Krom formulas. These bounds improve an earlier known bound of 6^k. We also prove a 2^k lower bound for these problems, subject to the Strong Exponential Time Hypothesis.Comment: A short version will appear in the proceedings of the 16th International Conference on Theory and Applications of Satisfiability Testin

An Algorithm for Dualization in Products of Lattices and Its Applications

Let \cL=\cL_1×⋅s×\cL_n be the product of n lattices, each of which has a bounded width. Given a subset \cA\subseteq\cL, we show that the problem of extending a given partial list of maximal independent elements of \cA in \cL can be solved in quasi-polynomial time. This result implies, in particular, that the problem of generating all minimal infrequent elements for a database with semi-lattice attributes, and the problem of generating all maximal boxes that contain at most a specified number of points from a given n-dimensional point set, can both be solved in incremental quasi-polynomial time

Systems of Linear Equations over F2\mathbb{F}_2 and Problems Parameterized Above Average

In the problem Max Lin, we are given a system $Az=b$ of $m$ linear equations with $n$ variables over $\mathbb{F}_2$ in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC'06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least $k$, where $k$ is the parameter. It is not hard to see that we may assume that no two equations in $Az=b$ have the same left-hand side and $n={\rm rank A}$. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixed-parameter tractable for a wide special case: $m\le 2^{p(n)}$ for an arbitrary fixed function $p(n)=o(n)$. Max $r$-Lin AA is a special case of Max Lin AA, where each equation has at most $r$ variables. In Max Exact $r$-SAT AA we are given a multiset of $m$ clauses on $n$ variables such that each clause has $r$ variables and asked whether there is a truth assignment to the $n$ variables that satisfies at least $(1-2^{-r})m + k2^{-r}$ clauses. Using our maximum excess results, we prove that for each fixed $r\ge 2$, Max $r$-Lin AA and Max Exact $r$-SAT AA can be solved in time $2^{O(k \log k)}+m^{O(1)}.$ This improves $2^{O(k^2)}+m^{O(1)}$-time algorithms for the two problems obtained by Gutin et al. (IWPEC 2009) and Alon et al. (SODA 2010), respectively

Charge symmetry violation in the parton distributions of the nucleon

We point out that charge symmetry violation in both the valence and sea quark distributions of the nucleon has a non-perturbative source. We calculate this non-perturbative charge symmetry violation using the meson cloud model, which has earlier been successfully applied to both the study of SU(2) flavour asymmetry in the nucleon sea and quark-antiquark asymmetry in the nucleon. We find that the charge symmetry violation in the valence quark distribution is well below 1%, which is consistent with most low energy tests but significantly smaller than the quark model prediction about 5%-10%. Our prediction for the charge symmetry violation in the sea quark distribution is also much smaller than the quark model calculation.Comment: RevTex, 26 pages, 6 PostScript figure

Baryon Magnetic Moments and Proton Spin: A Model with Collective Quark Rotation

We analyse the baryon magnetic moments in a model that relates them to the parton spins $\Delta u$, $\Delta d$, $\Delta s$, and includes a contribution from orbital angular momentum. The specific assumption is the existence of a 3-quark correlation (such as a flux string) that rotates with angular momentum $\langle L_z \rangle$ around the proton spin axis. A fit to the baryon magnetic moments, constrained by the measured values of the axial vector coupling constants $a^{(3)}=F+D$, $a^{(8)}=3F-D$, yields $\langle S_z \rangle = 0.08 \pm 0.13$, $\langle L_z \rangle = 0.39 \pm 0.09$, where the error is a theoretical estimate. A second fit, under slightly different assumptions, gives $\langle L_z \rangle = 0.37 \pm 0.09$, with no constraint on $\langle S_z \rangle$. The model provides a consistent description of axial vector couplings, magnetic moments and the quark polarization $\langle S_z \rangle$ measured in deep inelastic scattering. The fits suggest that a significant part of the angular momentum of the proton may reside in a collective rotation of the constituent quarks.Comment: 16 pages, 3 ps-figures, uses RevTeX. Abstract, Sec. II, III and IV have been expande

Second moment of the Husimi distribution as a measure of complexity of quantum states

We propose the second moment of the Husimi distribution as a measure of complexity of quantum states. The inverse of this quantity represents the effective volume in phase space occupied by the Husimi distribution, and has a good correspondence with chaoticity of classical system. Its properties are similar to the classical entropy proposed by Wehrl, but it is much easier to calculate numerically. We calculate this quantity in the quartic oscillator model, and show that it works well as a measure of chaoticity of quantum states.Comment: 25 pages, 10 figures. to appear in PR