We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p-block for all primes p in the set. Our main focus is on the simple and quasi-simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

The paper is an addendum to D. Andrica and L. Funar, ‘On smooth maps with finitely many critical points’, J. London Math. Soc. (2) 69 (2004) 783–800.

The orientation preserving topological symmetry group of a graph embedded in the 3-sphere is the subgroup of the automorphism group of the graph consisting of those automorphisms which can be induced by an orientation preserving homeomorphism of the ambient space. We characterize all possible orientation preserving topological symmetry groups of embeddings of complete graphs in the 3-sphere.

Remez-type inequalities provide upper bounds for the uniform norms of polynomials $p$ on given compact sets $K$, provided that $|p(x)|\leq1$ for every $x\in K\setminus E$, where $E$ is a subset of $K$ of small measure. In this paper we prove sharp Remez-type inequalities for homogeneous polynomials on star-like surfaces in $\mathbb{R}^d$. In particular, this covers the case of spherical polynomials (when $d=2$ we deduce a result of Erdélyi for univariate trigonometric polynomials).

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any $x\in B$ either $T^nx=0$ for some positive integer n or the sequence $\{T^nx/\|T^nx\|\}$ weakly converges to zero in B. Antisupercyclicity of T means that the angle criterion of supercyclicity is not satisfied for T in the strongest possible way. Normal antisupercyclic operators and antisupercyclic bilateral weighted shifts are characterized.

As for the Volterra operator V, it is proved that if $1\leq p\leq\infty$ and any $f\in L_p[0,1]$ then the limit $\lim_{n\to\infty} (n!\|V^nf\|_p)^{1/n}$ does exist and equals $1-\inf\,\hbox{\rm supp}\,(f)$. Upon using this asymptotic formula it is proved that the operator V acting on the Banach space $L_p[0,1]$ is antisupercyclic for any $p\in(1,\infty)$. The same statement for $p=1$ or $p=\infty$ is false. The analogous results are proved for operators $V^zf(x)=(1/{\Gamma(z)})\int_0^x f(t) (x-t)^{z-1}\,dt$ when the real part of $z\in{\mathbb C}$ is positive.

This paper is devoted to the long-time behavior of solutions to the Cauchy problem of the porous medium equation $u_t=\Delta(u^m)-u^p$ in $\mathbb{R}^n\times (0,\infty)$ with $(1-2/n)_{+}<m<1$ and the critical exponent $p=m+2/n$. For the strictly positive initial data $u(x,0)=O(1+|x|)^{-k}$ with $n+mn(2-n+nm)/(2[2-m+mn(1-m)])\leq k<2/(1-m)$, we prove that the solution of the above Cauchy problem converges to a fundamental solution of $u_t=\Delta(u^m)$ with an additional logarithmic anomalous decay exponent in time as $t\to \infty$.

Given a sequence of integers aj, $j\ge 1$, a multiset is a combinatorial object composed of unordered components, such that there are exactly aj one-component multisets of size j. When $a_j\asymp j^{r-1} y^j$ for some r > 0, $y\geq 1$, then the multiset is called expansive. Let cn be the number of multisets of total size n. Using a probabilistic approach, we prove for expansive multisets that $c_n/c_{n+1}\to 1$ and that cn/cn+1 < 1 for large enough n. This allows us to prove monadic second-order limit laws for expansive multisets. The above results are extended to a class of expansive multisets with oscillation.

Moreover, under the condition $a_j=Kj^{r-1}y^j + O(y^{\nu j})$, where K > 0, r > 0, y > 1, $\nu\in (0,1)$, we find an explicit asymptotic formula for cn. In a similar way we study the asymptotic behavior of selections, which are defined as combinatorial objects composed of unordered components of distinct sizes.

The main objective of this paper is to establish a large deviation principle for heat kernel measures on loop spaces. It gives an extension of Fang and Zhang's results on loop groups. For the proof, we use the continuity theorem of Lyons' rough path theory.