Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

We study approximation algorithms for several variants of the MaxCover problem, with the focus on algorithms that run in FPT time. In the MaxCover problem we are given a set N of elements, a family S of subsets of N, and an integer K. The goal is to find up to K sets from S that jointly cover (i.e., include) as many elements as possible. This problem is well-known to be NP-hard and, under standard complexity-theoretic assumptions, the best possible polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We first consider a variant of MaxCover with bounded element frequencies, i.e., a variant where there is a constant p such that each element belongs to at most p sets in S. For this case we show that there is an FPT approximation scheme (i.e., for each B there is a B-approximation algorithm running in FPT time) for the problem of maximizing the number of covered elements, and a randomized FPT approximation scheme for the problem of minimizing the number of elements left uncovered (we take K to be the parameter). Then, for the case where there is a constant p such that each element belongs to at least p sets from S, we show that the standard greedy approximation algorithm achieves approximation ratio exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted variant of MaxCover, and show approximation algorithms that run in exponential time and combine an exact algorithm with a greedy approximation. Some of our results improve currently known results for MaxVertexCover

On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

We study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t)$$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential

Pulsating flow and boundary layers in viscous electronic hydrodynamics

Motivated by experiments on a hydrodynamic regime in electron transport, we study the effect of an oscillating electric field in such a setting. We consider a long two-dimensional channel of width $L$, whose geometrical simplicity allows an analytical study as well as hopefully permitting experimental realisation. The response depends on viscosity $\nu$, driving frequency, $\omega$ and ohmic heating coefficient $\gamma$ via the dimensionless complex variable $\frac{L^2}{\nu}(i\omega +\gamma)=i\Omega +\Sigma$. While at small $\Omega$, we recover the static solution, a new regime appears at large $\Omega$ with the emergence of a boundary layer. This includes a splitting of the location of maximal flow velocity from the centre towards the edges of the boundary layer, an an increasingly reactive nature of the response, with the phase shift of the response varying across the channel. The scaling of the total optical conductance with $L$ differs between the two regimes, while its frequency dependence resembles a Drude form throughout, even in the complete absence of ohmic heating, against which, at the same time, our results are stable. Current estimates for transport coefficients in graphene and delafossites suggest that the boundary layer regime should be experimentally accessible.Comment: 5 pages, 3 figures, the title has been changed, the manuscript has been substantially modified and references update

Knots, BPS states, and algebraic curves

We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture stronger than the known M-theory integrality predictions. Furthermore we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.Comment: 43 pages, 6 figure