Fermat-type configurations of lines in P3 and the containment problem

The purpose of this note is to show a new series of examples of homogeneous ideals I in $\mathbb{K}[x,y,z,w]$ for which the containment $I^{(3)}?I^2$ fails. These ideals are supported on certain arrangements of lines in ${\mathbb{P}}^3$, which resemble Fermat configurations of points in ${\mathbb{P}}^2$, see [14]. All examples exhibiting the failure of the containment $I^{(3)}?I^2$ constructed so far have been supported on points or cones over configurations of points. Apart from providing new counterexamples, these ideals seem quite interesting on their own.     

A new approach to the chromatic number of the square of Kneser graph K(2k+1,k)

The vertices of Kneser graph $K(n,k)$ are the subsets of $\{1,2,\dots ,n\}$ of cardinality $k$, two vertices are adjacent if and only if they are disjoint. The square $G^2$ of a graph $G$ is defined on the vertex set of $G$ with two vertices adjacent if their distance in $G$ is at most 2. Z. Füredi, in 2002, proposed the problem of determining the chromatic number of the square of the Kneser graph. The first non-trivial problem arises when $n=2k+1$. It is believed that $\chi \left(K^2(2k+1,k)\right)=2k+c$ where $c$ is a constant, and yet the problem remains open. The best known upper bounds are by Kim and Park: $8k∕3+20∕3$ for 1$k\ge 3$ (Kim and Park, 2014) and $32k∕15+32$ for $k\ge 7$ (Kim and Park, 2016). In this paper, we develop a new approach to this coloring problem by employing graph homomorphisms, cartesian products of graphs, and linear congruences integrated with combinatorial arguments. These lead to $\chi \left(K^2(2k+1,k)\right)\le 5k∕2+c$, where $c$ is a constant in $\{5∕2,9∕2,5,6\}$, depending on $k\ge 2$.     

A pairwise almost disjoint subspace of kN of maximal dimension

We show that if k is an infinite field, then there exists a subspace $W?k^{\mathbb{N}}$ of dimension $|k{|}^{{\mathrm{?}}_0}$, such that no nonzero member of W has infinitely many zeros. This generalizes a result from a paper by Bergman and Nahlus, and partly answers another question from the same paper.     

Planar graphs without 3-cycles adjacent to cycles of length 3 or 5 are (3,1)-colorable

Given a nonnegative integer $d$ and a positive integer $k$, a graph $G$ is said to be $(k,d)$-colorable if the vertices of $G$ can be colored with $k$ colors such that every vertex has at most $d$ neighbors receiving the same color as itself. Let $?$ be the family of planar graphs without $3$-cycles adjacent to cycles of length 3 or 5. This paper proves that everyone in $?$ is $(3,1)$-colorable. This is the best possible in the sense that there are members in $?$ which are not $(3,0)$-colorable.     

The list L(2,1)-labeling of planar graphs

Let $\mathbf{N}$ be the set of all positive integers. A list assignment of a graph $G$ is a function $L:V\left(G\right)?2^{\mathbf{N}}$ that assigns each vertex $v$ a list $L\left(v\right)$ for all $v\in V\left(G\right)$. We say that $G$ is $L$-$(2,1)$-choosable if there exists a function $?$ such that $?\left(v\right)\in L\left(v\right)$ for all $v\in V\left(G\right)$, $|?\left(u\right)??\left(v\right)|\ge 2$ if $u$ and $v$ are adjacent, and $|?\left(u\right)??\left(v\right)|\ge 1$ if $u$ and $v$ are at distance 2. The list-$L(2,1)$-labeling number ${\lambda }_l\left(G\right)$ of $G$ is the minimum $k$ such that for every list assignment $L=\{L\left(v\right):|L\left(v\right)|=k,v\in V\left(G\right)\}$, $G$ is $L$-$(2,1)$-choosable. We prove that if $G$ is a planar graph with girth $g\ge 8$ and its maximum degree $\Delta$ is large enough, then ${\lambda }_l\left(G\right)\le \Delta +3$. There are graphs with large enough $\Delta$ and $g\ge 8$ having ${\lambda }_l\left(G\right)=\Delta +3$.     

Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\mathfrak{g}=A?\mathfrak{k}$, where $\mathfrak{k}$ is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when $A={\mathrm{\Lambda }}_s(\mathbb{R})$ is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\mathfrak{k}$ is a simple compact Lie superalgebra with ${\mathfrak{k}}_1\ne \{0\}$, then each (projective) unitary representation of ${\mathrm{\Lambda }}_s(\mathbb{R})?\mathfrak{k}$ factors through a (projective) unitary representation of $\mathfrak{k}$ itself, and these are known by Jakobsen's classification. If ${\mathfrak{k}}_1=\{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford–Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.     

Moments about the mean of the size of a self-conjugate (s,t)-core partition

Johnson proved that if $s,t$ are coprime integers, then the $r$th moment of the size of an $(s,t)$-core is a polynomial of degree $2r$ in $t$ for fixed $s$. After that, by defining a statistic size on elements of affine Weyl group, which is preserved under the bijection between minimal coset representatives of ${\stackrel{?}{\mathfrak{S}}}_t∕{\mathfrak{S}}_t$ and $t$-cores, Thiel and Williams obtained the variance and the third moment about the mean of the size of an $(s,t)$-core. Later, Ekhad and Zeilberger stated the first six moments about the mean of the size of an $(s,t)$-core and the first nine moments about the mean of the size of an $(s,s+1)$-core using Maple. To get the moments about the mean of the size of a self-conjugate $(s,t)$-core, we proceed to follow the approach of Thiel and Williams, however, their approach does not seem to directly apply to the self-conjugate case. In this paper, following Johnson’s approach, by Ehrhart theory and Euler–Maclaurin theory, we prove that if $s,t$ are coprime integers, then the $r$th moment about the mean of the size of a self-conjugate $(s,t)$-core is a quasipolynomial of period 2 and degree $2r$ in $t$ for fixed odd $s$. Then, based on a bijection of Ford, Mai and Sze between self-conjugate $(s,t)$-cores and lattice paths in $?\frac{s}{2}?\times ?\frac{t}{2}?$ rectangle and a formula of Chen, Huang and Wang on the size of self-conjugate $(s,t)$-cores, we obtain the variance, the third moment and the fourth moment about the mean of the size of a self-conjugate $(s,t)$-core.     

On the spanning connectivity of the generalized Petersen graphs P(n,3)

In this paper, we employed lattice model to describe the three internally vertex-disjoint paths that span the vertex set of the generalized Petersen graph $P(n,3)$. We showed that the $P(n,3)$ is 3-spanning connected for odd $n$. Based on the lattice model, five amalgamated and one extension mechanisms are introduced to recursively establish the 3-spanning connectivity of the $P(n,3)$. In each amalgamated mechanism, a particular lattice trail was amalgamated with the lattice trails that was dismembered, transferred, or extended from parts of the lattice trails for $P(n?6,3)$, where a lattice tail is a trail in the lattice model that represents a path in $P(n,3)$.     

The Lie algebra of type G2 is rational over its quotient by the adjoint action

Let G be a split simple group of type $G_2$ over a field k, and let $\mathfrak{g}$ be its Lie algebra. Answering a question of J.-L. Colliot-Thélène, B. Kunyavski?, V.L. Popov, and Z. Reichstein, we show that the function field $k(\mathfrak{g})$ is generated by algebraically independent elements over the field of adjoint invariants $k{(\mathfrak{g})}^G$.     

Global,decaying solutions of a focusing energy-critical heat equation in R4

We study solutions of the focusing energy-critical nonlinear heat equation $u_t=\mathrm{\Delta }u?|u{|}^{2}u$ in ${\mathbb{R}}^4$. We show that solutions emanating from initial data with energy and ${\dot{H}}^1$-norm below those of the stationary solution W are global and decay to zero, via the “concentration-compactness plus rigidity” strategy of Kenig–Merle [33], [34]. First, global such solutions are shown to dissipate to zero, using a refinement of the small data theory and the $L^2$-dissipation relation. Finite-time blow-up is then ruled out using the backwards-uniqueness of Escauriaza–Seregin–Sverak [17], [18] in an argument similar to that of Kenig–Koch [32] for the Navier–Stokes equations.     

Uniformly bounded representations of SL(2,R)

We compute the “norm” of irreducible uniformly bounded representations of $\mathrm{SL}(2,\mathbb{R})$. We show that the Kunze–Stein version of the uniformly bounded representations has minimal norm in its similarity class of uniformly bounded representations.