Let $\alpha $ be a totally positive algebraic integer of degree d, with conjugates $\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$ . The absolute $S_k$ -measure of $\alpha $ is defined by $s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$ . We compute the lower bounds $\upsilon _k$ of $s_k(\alpha )$ for each integer in the range $2\leq k \leq 15$ and give a conjecture on the results for integers $k>15$ . Then we derive the lower bounds of $s_k(\alpha )$ for all real numbers $k>2$ . Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.

Let Q be a quiver of type $\tilde {A}_n$ . Let $\alpha =\alpha _1+\alpha _2+\cdots +\alpha _s$ be the canonical decomposition. For the polynomials $M_Q(\alpha ,q)$ that count the number of isoclasses of representations of Q over ${\mathbb F}_q$ with dimension vector $\alpha $ , we obtain a precise relation between the degree of $M_Q(\alpha ,q)$ and that of $\prod _{i=1}^{s} M_Q(\alpha _i,q)$ for an arbitrary dimension vector $\alpha $ .

Let N be a sufficiently large integer. We prove that, with at most $O(N^{23/48+\varepsilon })$ exceptions, all even positive integers up to N can be represented in the form $p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^4$ , where $p_1,p_2,p_3,p_4,p_5,p_6$ are prime numbers.

The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

$$ \begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*} $$

In other words, under EH, the $\gamma _q /\!\log q$ in these ranges converge to the one point distribution at $1$ . This theorem refines and extends a previous result of Ford, Luca and Moree for prime $q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.

We study pencils of hypersurfaces over finite fields $\mathbb {F}_q$ such that each of the $q+1$ members defined over $\mathbb {F}_q$ is smooth.

We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and $Z(G)$ is cyclic of order $|Z(G)|\in \{1, 2, 3, 4\}$ .

Let $p_t(a,b;n)$ denote the number of partitions of n such that the number of t-hooks is congruent to $a \bmod {b}$ . For $t\in \{2, 3\}$ , arithmetic progressions $r_1 \bmod {m_1}$ and $r_2 \bmod {m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males and Ono [‘Distributions on partitions arising from Hilbert schemes and hook lengths’, Forum Math. Sigma 10 (2022), Article no. e49] using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of t-cores and t-quotients.

Let E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of $E(K_n)$ is equal to the rank of $E(K)$ . We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.

In this note, we investigate some products of subgroups and vanishing conjugacy class sizes of finite groups. We prove some supersolubility criteria for groups with restrictions on the vanishing conjugacy class sizes of their subgroups.

We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$ . As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$ , $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$ , then R is an $n \times n$ matrix ring. This improves a result of Fuchs [‘A characterisation result for matrix rings’, Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R, we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so.

A group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if $\langle Q,g\rangle $ is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such that $G/Q$ is Černikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Černikov normal subgroup N such that $G/N$ is quasihamiltonian. This result should be compared with theorems of Černikov and Schlette stating that if a group G is Černikov over its centre, then G is abelian-by-finite and its commutator subgroup is Černikov.

We prove that any continuous function can be locally approximated at a fixed point $x_{0}$ by an uncountable family resistant to disruptions by the family of continuous functions for which $x_{0}$ is a fixed point. In that context, we also consider the property of quasicontinuity.

Given a finite group G, we denote by $L(G)$ the subgroup lattice of G and by ${\cal CD}(G)$ the Chermak–Delgado lattice of G. In this note, we determine the finite groups G such that $|{\cal CD}(G)|=|L(G)|-k$ , for $k=1,2$ .

Sufficient conditions are obtained for the oscillation of a general form of a linear second-order differential equation with discontinuous solutions. The innovations are that the impulse effects are in mixed form and the results obtained are applicable even if the impulses are small. The novelty of the results is demonstrated by presenting an example of an oscillating equation to which previous oscillation theorems fail to apply.

In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set $\mathcal L$ of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that $\mathcal L$ has cardinality $\mathfrak {c}$, the cardinality of the continuum, and is a dense $G_{\delta }$ subset of the set $\mathbb {R}$ of all real numbers. In 1962, Erdős proved that every real number is the sum of two Liouville numbers. In this paper, a set W of complex numbers is said to have the Erdős property if every real number is the sum of two numbers in W. The set W is said to be an Erdős–Liouville set if it is a dense subset of $\mathcal {L}$ and has the Erdős property. Each subset of $\mathbb {R}$ is assigned its subspace topology, where $\mathbb {R}$ has the euclidean topology. It is proved here that: (i) there exist $2^{\mathfrak {c}}$ Erdős–Liouville sets no two of which are homeomorphic; (ii) there exist $\mathfrak {c}$ Erdős–Liouville sets each of which is homeomorphic to $\mathcal {L}$ with its subspace topology and homeomorphic to the space of all irrational numbers; (iii) each Erdős–Liouville set L homeomorphic to $\mathcal {L}$ contains another Erdős–Liouville set $L'$ homeomorphic to $\mathcal {L}$. Therefore, there is no minimal Erdős–Liouville set homeomorphic to $\mathcal {L}$.

If G is a group with subgroup H and m, k are two fixed nonnegative integers, H is called an $(m,k)$ -subnormal subgroup of G if it has index at most m in a subnormal subgroup of G of defect less than or equal to k. We study the behaviour of uncountable groups of cardinality $\aleph $ where all subgroups of cardinality $\aleph $ are $(m,k)$ -subnormal.

Inspired by Xiao’s work on Hankel measures for Hardy and Bergman spaces [‘Pseudo-Carleson measures for weighted Bergman spaces’. Michigan Math. J. 47 (2000), 447–452], we introduce Hankel measures for Fock space $F^p_\alpha $ . Given $p\ge 1$ , we obtain several equivalent descriptions for such measures on $F^p_\alpha $ .

A $C^{*}$ -algebra A is said to detect nuclearity if, whenever a $C^{*}$ -algebra B satisfies $A\otimes _{\mathrm{min}} B = A\otimes _{\mathrm{max}} B,$ it follows that B is nuclear. In this note, we survey the main results associated with this topic and present the background and tools necessary for proving the main results. In particular, we show that the $C^{*}$ -algebra $A = C^{*}(\mathbb {F}_{\infty })\otimes _{\mathrm{min}} B(\ell ^{2})/K(\ell ^{2})$ detects nuclearity. This result is known to experts, but has never appeared in the literature.

In this note, we give an example of a domain whose d-balanced squeezing function is nonplurisubharmonic.

We determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.