Point partition numbers: Decomposable and indecomposable critical graphs

Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number ${\chi }_t(G)$ is the least integer k for which G admits a coloring with k colors such that each color class induces a $(t?1)$-degenerate subgraph of G. So ${\chi }_1$ is the chromatic number and ${\chi }_2$ is the point arboricity. The point partition number ${\chi }_t$ with $t\ge 1$ was introduced by Lick and White. A graph G is called ${\chi }_t$-critical if every proper subgraph H of G satisfies ${\chi }_t(H)<{\chi }_t(G)$. In this paper we prove that if G is a ${\chi }_t$-critical graph whose order satisfies $|G|\le 2{\chi }_t(G)?2$, then G can be obtained from two non-empty disjoint subgraphs $G_1$ and $G_2$ by adding t edges between any pair $u,v$ of vertices with $u\in V(G_1)$ and $v\in V(G_2)$. Based on this result we establish the minimum number of edges possible in a ${\chi }_t$-critical graph G of order n and with ${\chi }_t(G)=k$, provided that $n\le 2k?1$ and t is even. For $t=1$ the corresponding two results were obtained in 1963 by Tibor Gallai.     

An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

Let ${\{{\phi }_i\}}_{i=0}^{\infty }$ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say ${\mathbb{E}}_n(\mu )$, of random polynomials

$P_n(z):=\sum _{i=0}^n{\eta }_i{\phi }_i(z),$

where ${\eta }_0,\dots ,{\eta }_n$ are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that ${\mathbb{E}}_n(|\mathrm{d}\xi |)$ admits an asymptotic expansion of the form

${\mathbb{E}}_n(|\mathrm{d}\xi |)～\frac{2}{\pi }\log ?(n+1)+\sum _{p=0}^{\infty }{A}_p{(n+1)}^{?p}$

(Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case ${\mathbb{E}}_n(\mu )$ admits an analogous expansion with the coefficients $A_p$ depending on the measure μ for $p\ge 1$ (the leading order term and $A_0$ remain the same).     

The secret life of μ-clubs

Let $\mu <\kappa <\lambda$ be three infinite cardinals, the first two being regular. We compare five versions for $P_{\kappa }(\lambda )$ of the ideal $N{S}_{\kappa }|E_{\mu }^{\kappa }$ (the restriction of the nonstationary ideal on κ to the set of all limit ordinals less than κ of cofinality μ): $N{S}_{\kappa ,\lambda }|E_{\mu }^{\kappa ,\lambda }$ (the restriction of the nonstationary ideal on $P_{\kappa }(\lambda )$ to the set of all a in $P_{\kappa }(\lambda )$ of uniform cofinality μ), $N{S}_{\mu ,\kappa ,\lambda }$ (the smallest $(\mu ,\kappa )$-normal ideal on $P_{\kappa }(\lambda )$), $J(\mu ,\kappa ,\lambda )$ (the smallest projection on $P_{\kappa }(\lambda )$ of a restriction of the nonstationary ideal on some $P_{\kappa }(\pi )$ to the set of all x in $P_{\kappa }(\pi )$ such that $x\cap \lambda$ can be reconstructed from a subset of x of size μ (and any of its subsets of size μ)), the ideal Nμ-$S_{\kappa ,\lambda }$ dual to the μ-club filter on $P_{\kappa }(\lambda )$ and the game ideal $N{G}_{\kappa ,\lambda }^{\mu }$. We show that if $\lambda <{\kappa }^{+\omega }$, then the first four ideals (and even all five ideals in case ${\rho }^{<\mu }<\kappa$ for any cardinal $\rho <\kappa$) coincide. Our main result asserts that if there are no large cardinals in an inner model, then Nμ-$S_{\kappa ,\lambda }=J(\mu ,\kappa ,\lambda )$. This throws some light on the so far rather mysterious μ-club filter.     

On the chromatic number of some P5-free graphs

Let G be a graph. We say that G is perfectly divisible if for each induced subgraph H of G, $V(H)$ can be partitioned into A and B such that $H[A]$ is perfect and $\omega (H[B])<\omega (H)$. We use $P_t$ and $C_t$ to denote a path and a cycle on t vertices, respectively. For two disjoint graphs $F_1$ and $F_2$, we use $F_1\cup F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)$, and use $F_1+F_2$ to denote the graph with vertex set $V(F_1)\cup V(F_2)$ and edge set $E(F_1)\cup E(F_2)\cup \{xy|x\in V(F_1)\text{ and }y\in V(F_2)\}$. In this paper, we prove that (i) $(P_5,C_5,K_{2,3})$-free graphs are perfectly divisible, (ii) $\chi (G)\le 2{\omega }^2(G)?\omega (G)?3$ if G is $(P_5,K_{2,3})$-free with $\omega (G)\ge 2$, (iii) $\chi (G)\le \frac{3}{2}({\omega }^2(G)?\omega (G))$ if G is $(P_5,K_1+2K_2)$-free, and (iv) $\chi (G)\le 3\omega (G)+11$ if G is $(P_5,K_1+(K_1\cup K_3))$-free.     

On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Higher level Zhu algebras and modules for vertex operator algebras

Motivated by the study of indecomposable, nonsimple modules for a vertex operator algebra V, we study the relationship between various types of V-modules and modules for the higher level Zhu algebras for V, denoted $A_n(V)$, for $n\in \mathbb{N}$, first introduced by Dong, Li, and Mason in 1998. We resolve some issues that arise in a few theorems previously presented when these algebras were first introduced, and give examples illustrating the need for certain modifications of the statements of those theorems. We establish that whether or not $A_{n?1}(V)$ is isomorphic to a direct summand of $A_n(V)$ affects the types of indecomposable V-modules which can be constructed by inducing from an $A_n(V)$-module, and in particular whether there are V-modules induced from $A_n(V)$-modules that were not already induced by $A_0(V)$. We give some characterizations of the V-modules that can be constructed from such inducings, in particular as regards their singular vectors. To illustrate these results, we discuss two examples of $A_1(V)$: when V is the vertex operator algebra associated to either the Heisenberg algebra or the Virasoro algebra. For these two examples, we show how the structure of $A_1(V)$ in relationship to $A_0(V)$ determines what types of indecomposable V-modules can be induced from a module for the level zero versus level one Zhu algebras. We construct a family of indecomposable modules for the Virasoro vertex operator algebra that are logarithmic modules and are not highest weight modules.     

Higher Laplacians on pseudo-Hermitian symmetric spaces

One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians $L_m$ are defined for each positive integer m, where $L_1$ coincides with the Laplace–Beltrami operator. Despite their natural definition, these higher Laplacians have not yet been studied in detail. In this paper, we consider the setting of simple pseudo-Hermitian symmetric spaces, i.e., let $X=G/H$ be a symmetric space for a real simple Lie group G, equipped with a G-invariant complex structure. We show that the higher Laplacians $L_1,L_3,\dots ,L_{2r?1}$ form a set of algebraically independent generators for the algebra ${\mathbb{D}}_G(X)$ of G-invariant differential operators on X, where r denotes the rank of X. For higher rank, this is the first instance of a set of generators for ${\mathbb{D}}_G(X)$ defined explicitly in purely geometric terms, and confirms a conjecture of Engli? and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces.     

Symmetrizers for Schur superalgebras

For the Schur superalgebra $S=S(m|n,r)$ over a ground field K of characteristic zero, we define the symmetrizer $T^{\lambda }[i:j]$ of the ordered pairs of tableaux $(T_i,T_j)$ of the shape λ. We show that the K-span $A_{\lambda ,K}$ of all symmetrizers $T^{\lambda }[i:j]$ has a basis consisting of $T^{\lambda }[i:j]$ for $T_i$ and $T_j$ semistandard. In particular, $A_{\lambda ,K}\ne 0$ if and only if λ is an $(m|n)$-hook partition. In this case, the S-superbimodule $A_{\lambda ,K}$ is identified as $D_{\lambda }{?}_K{D}_{\lambda }^o$, where $D_{\lambda }$ and $D_{\lambda }^o$ are left and right irreducible S-supermodules of the highest weight λ.We define modified symmetrizers $T^{\lambda }\{i:j\}$ and show that their $\mathbb{Z}$-span forms a $\mathbb{Z}$-form $A_{\lambda ,\mathbb{Z}}$ of $A_{\lambda ,\mathbb{Q}}$. We show that every modified symmetrizer $T^{\lambda }\{i:j\}$ is a $\mathbb{Z}$-linear combination of modified symmetrizers $T^{\lambda }\{i:j\}$ for $T_i,T_j$ semistandard. Using modular reduction to a field K of characteristic $p>2$, we obtain that $A_{\lambda ,K}$ has a basis consisting of modified symmetrizers $T^{\lambda }\{i:j\}$ for $T_i$ and $T_j$ semistandard.     

Extremal general affine surface areas

For a convex body K in ${\mathbb{R}}^n$, we introduce and study the extremal general affine surface areas, defined by${\mathrm{\text{IS}}}_{\phi }(K):=\underset{K^{\prime }?K}{\sup }?{\mathrm{as}}_{\phi }(K),{\mathrm{\text{os}}}_{\psi }(K):=\underset{K^{\prime }?K}{\inf }?{\mathrm{as}}_{\psi }(K)$ where ${\mathrm{as}}_{\phi }(K)$ and ${\mathrm{as}}_{\psi }(K)$ are the $L_{\phi }$ and $L_{\psi }$ affine surface area of K, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ${\mathrm{\text{IS}}}_{\phi }$ and ${\mathrm{\text{os}}}_{\psi }$ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Schütt and Werner (2020), who introduced and studied the extremal $L_p$ affine surface areas.