Explicit universal sampling sets in finite vector spaces

In this paper we construct explicit sampling sets and present reconstruction algorithms for Fourier signals on finite vector spaces $G$, with $|G|=p^r$ for a suitable prime $p$. The two sets have sizes of order $O(pt^2r^2)$ and $O(pt^2r^3\log(p))$ respectively, where $t$ is the number of large coefficients in the Fourier transform. The algorithms approximate the function up to a small constant of the best possible approximation with $t$ non-zero Fourier coefficients. The fastest of the algorithms has complexity $O(p^2t^2r^3\log(p))$

On the Generalised Colouring Numbers of Graphs that Exclude a Fixed Minor

The generalised colouring numbers $\mathrm{col}_r(G)$ and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the $r$-colouring number $\mathrm{col}_r$ and a polynomial bound for the weak $r$-colouring number $\mathrm{wcol}_r$. In particular, we show that if $G$ excludes $K_t$ as a minor, for some fixed $t\ge4$, then $\mathrm{col}_r(G)\le\binom{t-1}{2}\,(2r+1)$ and $\mathrm{wcol}_r(G)\le\binom{r+t-2}{t-2}\cdot(t-3)(2r+1)\in\mathcal{O}(r^{\,t-1})$. In the case of graphs $G$ of bounded genus $g$, we improve the bounds to $\mathrm{col}_r(G)\le(2g+3)(2r+1)$ (and even $\mathrm{col}_r(G)\le5r+1$ if $g=0$, i.e. if $G$ is planar) and $\mathrm{wcol}_r(G)\le\Bigl(2g+\binom{r+2}{2}\Bigr)\,(2r+1)$.Comment: 21 pages, to appear in European Journal of Combinatoric

Attractive Interaction between Vortex and Anti-vortex in Holographic Superfluid

Annihilation process of a pair of vortices in holographic superfluid is numerically simulated. The process is found to consist of two stages which are amazingly separated by vortex size $2r$. The separation distance $\delta(t)$ between vortex and anti-vortex as a function of time is well fitted by $\alpha (t_{0}-t)^{n}$, where the scaling exponent $n=1/2$ for $\delta (t)>2r$, and $n=2/5$ for $\delta(t)<2r$. Then the approaching velocity and acceleration as functions of time and as functions of separation distance are obtained. Thus the attractive force between vortex and anti-vortex is derived as $f(\delta)\propto 1/\delta^{3}$ for the first stage, and $f(\delta)\propto 1/\delta^{4}$ for the second stage. In the end, we explained why the annihilation rate of vortices in turbulent superfluid system obeys the two-body decay law when the vortex density is low.Comment: 14 pages, 5 figure

Differentials in the homological homotopy fixed point spectral sequence

We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E^2_{s,t} = H^{-s}_{gp}(T; H_t(R; F_p)), converging conditionally to the continuous homology H^c_{s+t}(R^{hT}; F_p) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations beta^epsilon Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the $^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E^infty-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C of T, and for the Tate- and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-27.abs.htm

A Cosmological Model with Dark Spinor Source

In this paper, we discuss the system of Friedman-Robertson-Walker metric coupling with massive nonlinear dark spinors in detail, where the thermodynamic movement of spinors is also taken into account. The results show that, the nonlinear potential of the spinor field can provide a tiny negative pressure, which resists the Universe to become singular. The solution is oscillating in time and closed in space, which approximately takes the following form g_{\mu\nu}=\bar R^2(1-\delta\cos t)^2\diag(1,-1,-\sin^2r ,-\sin^2r \sin^2\theta), with $\bar R= (1\sim 2)\times 10^{12}$ light year, and $\delta=0.96\sim 0.99$. The present time is about $t\sim 18^\circ$.Comment: 13 pages, no figure, to appear in IJMP

tt-Intersection sets in AG(r,q2)AG(r,q^2) and two-character multisets in PG(3,q2)PG(3,q^2)

In this article we construct new minimal intersection sets in $AG(r,q^2)$ with respect to hyperplanes, of size $q^2r-1$ and multiplicity $t$, where $t\in \ q^2r-3-q^(3r-5)/2, q^2r-3+q^(3r-5)/2-q^(3r-3)/2\$, for$r$odd or$t \in \ q^2r-3-q^(3r-4)/2, q^2r-3-q^r-2\$, for$r$even. As a byproduct, for any odd$q$we get a new family of two-character multisets in$PG(3,q^2)$. The essential idea is to investigate some point-sets in$AG(r,q^2)$ satisfying the opposite of the algebraic conditions required in [1] for quasi--Hermitian varieties