We prove that a Küchle fourfold X$X$ of type d3 has a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of X$X$.

Let F$F$ be an algebraically closed field of characteristic 0$0$ and let \operatorname{sp}(2l,F)$\operatorname{sp}(2l,F)$ be the rank l$l$ symplectic algebra of all 2l\times 2l$2l\times 2l$ matrices x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over F$F$, where A^{t}$A^{t}$ is the transpose of A$A$ and B,C$B,C$ are symmetric matrices of order l$l$. The commuting graph \unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of \operatorname{sp}(2l,F)$\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in \operatorname{sp}(2l,F)$\operatorname{sp}(2l,F)$ and two distinct vertices x$x$ and y$y$ are adjacent if and only if xy=yx$xy=yx$. We prove that the diameter of \unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is 4$4$ when l>2$l>2$.

Let G$G$ be a finite group and let p$p$ be a prime factor of |G|$|G|$. Suppose that G$G$ is solvable and P$P$ is a Sylow p$p$-subgroup of G$G$. In this note, we prove that P{\vartriangleleft}G$P{\vartriangleleft}G$ and G/P$G/P$ is nilpotent if and only if \unicode[STIX]{x1D711}(1)^{2}$\unicode[STIX]{x1D711}(1)^{2}$ divides |G:\ker \unicode[STIX]{x1D711}|$|G:\ker \unicode[STIX]{x1D711}|$ for all irreducible monomial p$p$-Brauer characters \unicode[STIX]{x1D711}$\unicode[STIX]{x1D711}$ of G$G$.

We prove a lower bound for the large sieve with square moduli.

A strongly concave composition of n$n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

Let q$q$ be a prime and let A$A$ be an elementary abelian group of order at least q^{3}$q^{3}$ acting by automorphisms on a finite q^{\prime }$q^{\prime }$-group G$G$. We prove that if |\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$|\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$ for any a\in A^{\#}$a\in A^{\#}$, then the order of \unicode[STIX]{x1D6FE}_{\infty }(G)$\unicode[STIX]{x1D6FE}_{\infty }(G)$ is m$m$-bounded. If F(C_{G}(a))$F(C_{G}(a))$ has index at most m$m$ in C_{G}(a)$C_{G}(a)$ for any a\in A^{\#}$a\in A^{\#}$, then the index of F_{2}(G)$F_{2}(G)$ is m$m$-bounded.

Let G$G$ be a finite group and let \text{Irr}(G)$\text{Irr}(G)$ be the set of all irreducible complex characters of G$G$. Let \unicode[STIX]{x1D70C}(G)$\unicode[STIX]{x1D70C}(G)$ be the set of all prime divisors of character degrees of G$G$. The character degree graph \unicode[STIX]{x1D6E5}(G)$\unicode[STIX]{x1D6E5}(G)$ associated to G$G$ is a graph whose vertex set is \unicode[STIX]{x1D70C}(G)$\unicode[STIX]{x1D70C}(G)$, and there is an edge between two distinct primes p$p$ and q$q$ if and only if pq$pq$ divides \unicode[STIX]{x1D712}(1)$\unicode[STIX]{x1D712}(1)$ for some \unicode[STIX]{x1D712}\in \text{Irr}(G)$\unicode[STIX]{x1D712}\in \text{Irr}(G)$. We prove that \unicode[STIX]{x1D6E5}(G)$\unicode[STIX]{x1D6E5}(G)$ is k$k$-regular for some natural number k$k$ if and only if \overline{\unicode[STIX]{x1D6E5}}(G)$\overline{\unicode[STIX]{x1D6E5}}(G)$ is a regular bipartite graph.

Let G$G$ be a group, p$p$ be a prime and P\in \text{Syl}_{p}(G)$P\in \text{Syl}_{p}(G)$. We say that a p$p$-Brauer character \unicode[STIX]{x1D711}$\unicode[STIX]{x1D711}$ is monolithic if G/\ker \unicode[STIX]{x1D711}$G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that P$P$ is normal in G$G$ if and only if p\nmid \unicode[STIX]{x1D711}(1)$p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character \unicode[STIX]{x1D711}\in \text{IBr}(G)$\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When G$G$ is p$p$-solvable, we also prove that P$P$ is normal in G$G$ and G/P$G/P$ is nilpotent if and only if \unicode[STIX]{x1D711}(1)^{2}$\unicode[STIX]{x1D711}(1)^{2}$ divides |G:\ker \unicode[STIX]{x1D711}|$|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible p$p$-Brauer characters \unicode[STIX]{x1D711}$\unicode[STIX]{x1D711}$ of G$G$.

Suppose that f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$f(x)=x^{n}+A(Bx+C)^{m}\in \mathbb{Z}[x]$, with n\geq 3$n\geq 3$ and 1\leq m<n$1\leq m<n$, is irreducible over \mathbb{Q}$\mathbb{Q}$. By explicitly calculating the discriminant of f(x)$f(x)$, we prove that, when \gcd (n,mB)=C=1$\gcd (n,mB)=C=1$, there exist infinitely many values of A$A$ such that the set \{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$\{1,\unicode[STIX]{x1D703},\unicode[STIX]{x1D703}^{2},\ldots ,\unicode[STIX]{x1D703}^{n-1}\}$ is an integral basis for the ring of integers of \mathbb{Q}(\unicode[STIX]{x1D703})$\mathbb{Q}(\unicode[STIX]{x1D703})$, where f(\unicode[STIX]{x1D703})=0$f(\unicode[STIX]{x1D703})=0$.

For a group G$G$, let \unicode[STIX]{x1D6E4}(G)$\unicode[STIX]{x1D6E4}(G)$ denote the graph defined on the elements of G$G$ in such a way that two distinct vertices are connected by an edge if and only if they generate G$G$. Let \unicode[STIX]{x1D6E4}^{\ast }(G)$\unicode[STIX]{x1D6E4}^{\ast }(G)$ be the subgraph of \unicode[STIX]{x1D6E4}(G)$\unicode[STIX]{x1D6E4}(G)$ that is induced by all the vertices of \unicode[STIX]{x1D6E4}(G)$\unicode[STIX]{x1D6E4}(G)$ that are not isolated. We prove that if G$G$ is a 2-generated noncyclic abelian group, then \unicode[STIX]{x1D6E4}^{\ast }(G)$\unicode[STIX]{x1D6E4}^{\ast }(G)$ is connected. Moreover, \text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=2$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=2$ if the torsion subgroup of G$G$ is nontrivial and \text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=\infty$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(G))=\infty$ otherwise. If F$F$ is the free group of rank 2, then \unicode[STIX]{x1D6E4}^{\ast }(F)$\unicode[STIX]{x1D6E4}^{\ast }(F)$ is connected and we deduce from \text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(\mathbb{Z}\times \mathbb{Z}))=\infty$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(\mathbb{Z}\times \mathbb{Z}))=\infty$ that \text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(F))=\infty$\text{diam}(\unicode[STIX]{x1D6E4}^{\ast }(F))=\infty$.

In this paper, we study the probability distribution of the word map w(x_{1},x_{2},\ldots ,x_{k})=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{k}^{n_{k}}$w(x_{1},x_{2},\ldots ,x_{k})=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{k}^{n_{k}}$ in a compact Lie group. We show that the probability distribution can be represented as an infinite series. Moreover, in the case of the Lie group \text{SU}(2)$\text{SU}(2)$, our computations give a nice convergent series for the probability distribution.

Let E$E$ be an elliptic curve over \mathbb{Q}$\mathbb{Q}$ without complex multiplication. Let p\geq 5$p\geq 5$ be a prime in \mathbb{Q}$\mathbb{Q}$ and suppose that E$E$ has good ordinary reduction at p$p$. We study the dual Selmer group of E$E$ over the compositum of the \text{GL}_{2}$\text{GL}_{2}$ extension and the anticyclotomic \mathbb{Z}_{p}$\mathbb{Z}_{p}$-extension of an imaginary quadratic extension as an Iwasawa module.

We analyse local aspects of chaos for nonautonomous periodic dynamical systems in the context of generating autonomous dynamical systems and the possibility of disturbing them.

For G$G$ a finite non-Abelian group we write c(G)$c(G)$ for the probability that two randomly chosen elements commute and k(G)$k(G)$ for the largest integer such that any k(G)$k(G)$-colouring of G$G$ is guaranteed to contain a monochromatic quadruple (x,y,xy,yx)$(x,y,xy,yx)$ with xy\neq yx$xy\neq yx$. We show that c(G)\rightarrow 0$c(G)\rightarrow 0$ if and only if k(G)\rightarrow \infty$k(G)\rightarrow \infty$.

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant H_{2,2}$H_{2,2}$ and Zalcman functional J_{2,3}$J_{2,3}$ over the class {\mathcal{C}}{\mathcal{V}}${\mathcal{C}}{\mathcal{V}}$ of analytic functions f$f$ normalised such that \text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.

Let K$K$ be a field that admits a cyclic Galois extension of degree n\geq 2$n\geq 2$. The symmetric group S_{n}$S_{n}$ acts on K^{n}$K^{n}$ by permutation of coordinates. Given a subgroup G$G$ of S_{n}$S_{n}$ and u\in K^{n}$u\in K^{n}$, let V_{G}(u)$V_{G}(u)$ be the K$K$-vector space spanned by the orbit of u$u$ under the action of G$G$. In this paper we show that, for a special family of groups G$G$ of affine type, the dimension of V_{G}(u)$V_{G}(u)$ can be computed via the greatest common divisor of certain polynomials in K[x]$K[x]$. We present some applications of our results to the cases K=\mathbb{Q}$K=\mathbb{Q}$ and K$K$ finite.

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space B$B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when B$B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group G$G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if G$G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if G$G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite k_{\unicode[STIX]{x1D714}}$k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus \mathbb{T}^{\unicode[STIX]{x1D714}}$\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.

We consider a one-parameter family of dynamical systems W:[0,1]\rightarrow [0,1]$W:[0,1]\rightarrow [0,1]$ constructed from a pair of monotone increasing diffeomorphisms W_{i}$W_{i}$ such that W_{i}^{-1}:$W_{i}^{-1}:$[0,1]\rightarrow [0,1]$[0,1]\rightarrow [0,1]$(i=0,1)$(i=0,1)$. We characterise the set of symbolic itineraries of W$W$ using an attractor \overline{\unicode[STIX]{x1D6FA}}$\overline{\unicode[STIX]{x1D6FA}}$ of an iterated closed relation, in the terminology of McGehee, and prove that there is a member of the family for which \overline{\unicode[STIX]{x1D6FA}}$\overline{\unicode[STIX]{x1D6FA}}$ is symmetrical.

We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if A\subseteq \mathbb{F}_{p}$A\subseteq \mathbb{F}_{p}$ satisfies |A|\leq p^{64/117}$|A|\leq p^{64/117}$ then \max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$ Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018, arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy E^{+}(P)$E^{+}(P)$ of some subset P\subseteq A+A$P\subseteq A+A$. Our main novelty comes from reducing the estimation of E^{+}(P)$E^{+}(P)$ to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’, Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.

The word w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$w=[x_{i_{1}},x_{i_{2}},\ldots ,x_{i_{k}}]$ is a simple commutator word if k\geq 2,i_{1}\neq i_{2}$k\geq 2,i_{1}\neq i_{2}$ and i_{j}\in \{1,\ldots ,m\}$i_{j}\in \{1,\ldots ,m\}$ for some m>1$m>1$. For a finite group G$G$, we prove that if i_{1}\neq i_{j}$i_{1}\neq i_{j}$ for every j\neq 1$j\neq 1$, then the verbal subgroup corresponding to w$w$ is nilpotent if and only if |ab|=|a||b|$|ab|=|a||b|$ for any w$w$-values a,b\in G$a,b\in G$ of coprime orders. We also extend the result to a residually finite group G$G$, provided that the set of all w$w$-values in G$G$ is finite.