Long-range order in the A-like phase of superfluid 3He in aerogel

A mutual action of the random anisotropy brought in the superfluid 3He by aerogel and of the global anisotropy caused by its deformation is considered. Strong global anisotropy tends to suppress fluctuations of orientation of the order parameter and stabilizes ABM order parameter. In a limit of vanishing anisotropy these fluctuations are getting critical. It is argued that still in a region of small fluctuations the average order parameter can acquire "robust" component. This component maintains a long-range order even in a limit of vanishing global anisotropy.Comment: A contribution to QFS 2007 in Kazan, revised for publication in the Proceeding

Robust superfluid phases of 3He in aerogel

Within a phenomenological approach possible forms of the order parameter of the superfluid phases of 3He in a vicinity of the transition temperature are discussed. Effect of aerogel is described by a random tensor field interacting with the orbital part of the order parameter. With respect to their interaction with the random tensor field a group of "robust" order parameters which can maintain long-range order in a presence of the random field is specified. Robust order parameters, corresponding to Equal Spin Pairing (ESP) states are found and proposed as candidates for the observed A-like superfluid phase of liquid 3He in aerogel.Comment: 5 pages, prepared for QFS 200

Space-time in light of Karolyhazy uncertainty relation

General relativity and quantum mechanics provide a natural explanation for the existence of dark energy with its observed value and predict its dynamics. Dark energy proves to be necessary for the existence of space-time itself and determines the rate of its stability.Comment: 5 pages, Two misprints are correcte

Polynomial kernelization for removing induced claws and diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fai

Cluster algebras in algebraic Lie theory

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.Comment: Invited survey; to appear in Transformation Group

Network and Seiberg Duality

We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a permutation, or equivalently a cell of the totally non-negative Grassmannian. The story is similar to a bipartite network on the torus classified by a Newton polygon. We then generalize the network to a general bordered Riemann surface and define IR SCFT from the geometric data of a Riemann surface. We also comment on IR R-charges and superconformal indices of our theories.Comment: 28 pages, 28 figures; v2: minor correction

Minimizing the Cost of Team Exploration

A group of mobile agents is given a task to explore an edge-weighted graph $G$, i.e., every vertex of $G$ has to be visited by at least one agent. There is no centralized unit to coordinate their actions, but they can freely communicate with each other. The goal is to construct a deterministic strategy which allows agents to complete their task optimally. In this paper we are interested in a cost-optimal strategy, where the cost is understood as the total distance traversed by agents coupled with the cost of invoking them. Two graph classes are analyzed, rings and trees, in the off-line and on-line setting, i.e., when a structure of a graph is known and not known to agents in advance. We present algorithms that compute the optimal solutions for a given ring and tree of order $n$, in $O(n)$ time units. For rings in the on-line setting, we give the $2$-competitive algorithm and prove the lower bound of $3/2$ for the competitive ratio for any on-line strategy. For every strategy for trees in the on-line setting, we prove the competitive ratio to be no less than $2$, which can be achieved by the $DFS$ algorithm.Comment: 25 pages, 4 figures, 5 pseudo-code

Superconductivity in a Mesoscopic Double Square Loop: Effect of Imperfections

We have generalized the network approach to include the effects of short-range imperfections in order to analyze recent experiments on mesoscopic superconducting double loops. The presence of weakly scattering imperfections causes gaps in the phase boundary $B(T)$ or $\Phi(T)$ for certain intervals of $T$, which depend on the magnetic flux penetrating each loop. This is accompanied by a critical temperature $T_c(\Phi)$, showing a smooth transition between symmetric and antisymmetric states. When the scattering strength of imperfections increases beyond a certain limit, gaps in the phase boundary $T_c(B)$ or $T_c(\Phi)$ appear for values of magnetic flux lying in intervals around half-integer $\Phi_0=hc/2e$. The critical temperature corresponding to these values of magnetic flux is determined mainly by imperfections in the central branch. The calculated phase boundary is in good agreement with experiment.Comment: 9 pages, 6 figure

Unit Interval Editing is Fixed-Parameter Tractable

Given a graph~$G$ and integers $k_1$, $k_2$, and~$k_3$, the unit interval editing problem asks whether $G$ can be transformed into a unit interval graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. We give an algorithm solving this problem in time $2^{O(k\log k)}\cdot (n+m)$, where $k := k_1 + k_2 + k_3$, and $n, m$ denote respectively the numbers of vertices and edges of $G$. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time $O(4^k \cdot (n + m))$. Another result is an $O(6^k \cdot (n + m))$-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time $O(6^k \cdot n^6)$.Comment: An extended abstract of this paper has appeared in the proceedings of ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an appendix is provided for a brief overview of related graph classe

Wigner crystallization in a polarizable medium

We present a variational study of the 2D and 3D Wigner crystal phase of large polarons. The method generalizes that introduced by S. Fratini,P.\ Qu{\'{e}}merais [Mod. Phys. Lett. B {\bf 12} 1003 (1998)]. We take into account the Wigner crystal normal modes rather than a single mean frequency in the minimization procedure of the variational free energy. We calculate the renormalized modes of the crystal as well as the charge polarization correlation function and polaron radius. The solid phase boundaries are determined via a Lindemann criterion, suitably generalized to take into account the classical-to-quantum cross-over. In the weak electron-phonon coupling limit, the Wigner crystal parameters are renormalized by the electron-phonon interaction leading to a stabilization of the solid phase for low polarizability of the medium. Conversely, at intermediate and strong coupling, the behavior of the system depends strongly on the polarizability of the medium. For weakly polarizable media, a density crossover occurs inside the solid phase when the renormalized plasma frequency approaches the phonon frequency. At low density, we have a renormalized polaron Wigner crystal, while at higher densities the electron-phonon interaction is weakened irrespective of the {\it bare} electron-phonon coupling. For strongly polarizable media, the system behaves as a Lorentz lattice of dipoles. The abrupt softening of the internal polaronic frequency predicted by Fratini and Quemerais is observed near the actual melting point only at very strong coupling, leading to a possible liquid polaronic phase for a wider range of parameters.Comment: 24 pages, 13 figures v1.