New optimal subsystem codes

Subsystem codes (also known as operator quantum error-correcting codes) are a generalization of noiseless subsystems, decoherence-free subspaces, and quantum error-correcting codes. In this note, we present a construction for new subsystem codes with parameters _{[[q2+1,(q−1)2,4,q−1]]q}${\left[[q^2+1,{(q-1)}^2,4,q-1]\right]}_q$, where q=2^m$q=2^m$, and m≥1$m\ge 1$ is a positive integer, whose parameters are not covered by the codes available in the literature. Moreover, the constructed subsystem codes are optimal.     

On the general excess bound for binary codes with covering radius one

Let K(n,1)$K(n,1)$ denote the minimal cardinality of a binary code of length n$n$ and covering radius one. Fundamental for the theory of lower bounds for K(n,1)$K(n,1)$ is the covering excess method introduced by Johnson and van Wee. Let _δi${\delta }_i$ denote the covering excess on a sphere of radius i$i$, 0≤i≤n$0\le i\le n$. Generalizing an earlier result of van Wee, Habsieger and Honkala showed _δp−1≥p−1${\delta }_{p-1}\ge p-1$ whenever n≡−1$n\equiv -1$ (mod p$p$) for an odd prime p$p$ and _δ0=_δ1=?=_δp−2=0${\delta }_0={\delta }_1=?={\delta }_{p-2}=0$ holds. In the present paper we give the new estimation _δp−1≥(p−2)p−1${\delta }_{p-1}\ge (p-2)p-1$ instead. This answers a question of Habsieger and yields a “general improvement of the general excess bound” for binary codes with covering radius one. The proof uses a classification theorem for certain subset systems as well as new congruence properties for the δ$\delta$-function, which were conjectured by Habsieger.     

Constructing cubature formulae of degree 5 with few points

This paper is devoted to construct a family of fifth degree cubature formulae for n$n$-cube with symmetric measure and n$n$-dimensional spherically symmetrical region. The formula forn$n$-cube contains at most n²+5n+3$n^2+5n+3$ points and for n$n$-dimensional spherically symmetrical region contains only n²+3n+3$n^2+3n+3$ points. Moreover, the numbers can be reduced to n²+3n+1$n^2+3n+1$ and n²+n+1$n^2+n+1$ if n=7$n=7$ respectively, the latter of which is minimal.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.     

Stable categories of higher preprojective algebras

We introduce (n+1)$(n+1)$-preprojective algebras of algebras of global dimension n$n$. We show that if an algebra is n$n$-representation-finite then its (n+1)$(n+1)$-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)$(n+1)$-preprojective algebra is (n+1)$(n+1)$-Calabi–Yau, and, more precisely, it is the (n+1)$(n+1)$-Amiot cluster category of the stable n$n$-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)$(n+1)$-cluster tilting object. We show that even if the (n+1)$(n+1)$-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}$n\in \{1,2\}$) the results above still hold for the stable category of Cohen–Macaulay modules.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.     

On the Lazarev–Lieb extension of the Hobby–Rice theorem

O. Lazarev and E.H. Lieb proved that, given _f1,…,_fn∈L¹([0,1];C)$f_1,\dots ,f_n\in L^1([0,1];\mathbb{C})$, there exists a smooth function Φ$\Phi$ that takes values on the unit circle and annihilates span{_f1,…,_fn}$\text{span}\{f_1,\dots ,f_n\}$. We give an alternative proof of that fact that also shows the W^1,1$W^{1,1}$ norm of Φ$\Phi$ can be bounded by 5πn+1$5\pi n+1$. Answering a question raised by Lazarev and Lieb, we show that if p>1$p>1$ then there is no bound for the W^1,p$W^{1,p}$ norm of any such multiplier in terms of the norms of _f1,…,_fn$f_1,\dots ,f_n$.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.