Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from {\mathcal{P}}(A)${\mathcal{P}}(A)$, the set of all subsets of a set A$A$, onto A$A$, then A$A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of {\mathcal{P}}(A)${\mathcal{P}}(A)$ and the set S(A)$S(A)$ of permutations on A$A$ are equal for any infinite set A$A$. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with {\mathcal{P}}(A)${\mathcal{P}}(A)$ replaced by S(A)$S(A)$, provable without AC.

Let A$A$ be a subset of \mathbb{N}$\mathbb{N}$ , the set of all nonnegative integers. For an integer h\geq 2$h\geq 2$ , let hA$hA$ be the set of all sums of h$h$ elements of A$A$ . The set A$A$ is called an asymptotic basis of order h$h$ if hA$hA$ contains all sufficiently large integers. Otherwise, A$A$ is called an asymptotic nonbasis of order h$h$ . An asymptotic nonbasis A$A$ of order h$h$ is called a maximal asymptotic nonbasis of order h$h$ if A\cup \{a\}$A\cup \{a\}$ is an asymptotic basis of order h$h$ for every a\notin A$a\notin A$ . In this paper, we construct a sequence of asymptotic nonbases of order h$h$ for each h\geq 2$h\geq 2$ , each of which is not a subset of a maximal asymptotic nonbasis of order h$h$ .

Let B_{k,\ell }(n)$B_{k,\ell }(n)$ denote the number of (k,\ell )$(k,\ell )$-regular bipartitions of n$n$. Employing both the theory of modular forms and some elementary methods, we systematically study the arithmetic properties of B_{3,\ell }(n)$B_{3,\ell }(n)$ and B_{5,\ell }(n)$B_{5,\ell }(n)$. In particular, we confirm all the conjectures proposed by Dou [‘Congruences for (3,11)-regular bipartitions modulo 11’, Ramanujan J.40, 535–540].

We investigate the monotonic characteristics of the generalised binomial coefficients (phinomials) based upon Euler’s totient function. We show, unconditionally, that the set of integers for which this sequence is unimodal is finite and, assuming the generalised Riemann hypothesis, we find all the exceptions.

Let a,b,c$a,b,c$ be a primitive Pythagorean triple and set a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$a=m^{2}-n^{2},b=2mn,c=m^{2}+n^{2}$ , where m$m$ and n$n$ are positive integers with m>n$m>n$ , \text{gcd}(m,n)=1$\text{gcd}(m,n)=1$ and m\not \equiv n~(\text{mod}~2)$m\not \equiv n~(\text{mod}~2)$ . In 1956, Jeśmanowicz conjectured that the only positive integer solution to the Diophantine equation (m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$(m^{2}-n^{2})^{x}+(2mn)^{y}=(m^{2}+n^{2})^{z}$ is (x,y,z)=(2,2,2)$(x,y,z)=(2,2,2)$ . We use biquadratic character theory to investigate the case with (m,n)\equiv (2,3)~(\text{mod}~4)$(m,n)\equiv (2,3)~(\text{mod}~4)$ . We show that Jeśmanowicz’ conjecture is true in this case if m+n\not \equiv 1~(\text{mod}~16)$m+n\not \equiv 1~(\text{mod}~16)$ or y>1$y>1$ . Finally, using these results together with Laurent’s refinement of Baker’s theorem, we show that Jeśmanowicz’ conjecture is true if (m,n)\equiv (2,3)~(\text{mod}~4)$(m,n)\equiv (2,3)~(\text{mod}~4)$ and n<100$n<100$ .

In 1939, Erdös and Mahler [‘Some arithmetical properties of the convergents of a continued fraction’, J. Lond. Math. Soc. (2)14 (1939), 12–18] studied some arithmetical properties of the convergents of a continued fraction. In particular, they raised a conjecture related to continued fractions and Liouville numbers. In this paper, we shall apply the theory of linear forms in logarithms to obtain a result in the direction of this problem.

Let r\geq 2$r\geq 2$ and s\geq 2$s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the r$r$-ary expansion and the s$s$-ary expansion of an irrational real number, viewed as infinite words on \{0,1,\ldots ,r-1\}$\{0,1,\ldots ,r-1\}$ and \{0,1,\ldots ,s-1\}$\{0,1,\ldots ,s-1\}$, and we show that this bound is best possible.

Two results related to the mixed joint universality for a polynomial Euler product \unicode[STIX]{x1D711}(s)$\unicode[STIX]{x1D711}(s)$ and a periodic Hurwitz zeta function \unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$\unicode[STIX]{x1D701}(s,\unicode[STIX]{x1D6FC};\mathfrak{B})$, when \unicode[STIX]{x1D6FC}$\unicode[STIX]{x1D6FC}$ is a transcendental parameter, are given. One is the mixed joint functional independence and the other is a generalised universality, which includes several periodic Hurwitz zeta functions.

Let R\subset S$R\subset S$ be an extension of integral domains, with R^{\ast }$R^{\ast }$ the integral closure of R$R$ in S$S$ . We study the set of intermediate rings between R$R$ and S$S$ . We establish several necessary and sufficient conditions for which every ring contained between R$R$ and S$S$ compares with R^{\ast }$R^{\ast }$ under inclusion. This answers a key question that figured in the work of Gilmer and Heinzer [‘Intersections of quotient rings of an integral domain’, J. Math. Kyoto Univ.7 (1967), 133–150].

Let S\subseteq \mathbb{P}^{d}$S\subseteq \mathbb{P}^{d}$ be an anticanonically embedded surface of degree d\geq 3$d\geq 3$ . In this note, we classify stable Ulrich bundles on S$S$ of rank two. We also study their moduli spaces.

In this paper, we show that every pair of large positive even integers can be represented in the form of a pair of Goldbach–Linnik equations, that is, linear equations in two primes and k$k$ powers of two. In particular, k=34$k=34$ powers of two suffice, in general, and k=18$k=18$ under the generalised Riemann hypothesis. Our result sharpens the number of powers of two in previous results, which gave k=62$k=62$, in general, and k=31$k=31$ under the generalised Riemann hypothesis.

We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let \mathbb{C}(X(N))$\mathbb{C}(X(N))$ be the field of meromorphic functions on the modular curve X(N)$X(N)$ of level N$N$. We construct a completely free element in the extension \mathbb{C}(X(N))/\mathbb{C}(X(1))$\mathbb{C}(X(N))/\mathbb{C}(X(1))$ by means of Siegel functions.

Let A_{2}$A_{2}$ be a free associative algebra or polynomial algebra of rank two over a field of characteristic zero. The main results of this paper are the classification of noninjective endomorphisms of A_{2}$A_{2}$ and an algorithm to determine whether a given noninjective endomorphism of A_{2}$A_{2}$ has a nontrivial fixed element for a polynomial algebra. The algorithm for a free associative algebra of rank two is valid whenever an element is given and the subalgebra generated by this element contains the image of the given noninjective endomorphism.

A group G$G$ is said to have the T$T$ -property (or to be a T$T$ -group) if all its subnormal subgroups are normal, that is, if normality in G$G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality  \aleph$\aleph$ whose proper subgroups of cardinality \aleph$\aleph$ have a transitive normality relation. It is proved that such a group  G$G$ is a T$T$ -group (and all its subgroups have the same property) provided that G$G$ has an ascending subnormal series with abelian factors. Moreover, it is shown that if G$G$ is an uncountable soluble group of cardinality \aleph$\aleph$ whose proper normal subgroups of cardinality  \aleph$\aleph$ have the T$T$ -property, then every subnormal subgroup of G$G$ has only finitely many conjugates.

In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if F(z)\in \mathbb{C}[[z]]$F(z)\in \mathbb{C}[[z]]$ is a power series with coefficients from a finite set, then F(z)$F(z)$ is either rational or it is transcendental over the field of meromorphic functions.

We complete the classification of the finite special linear groups \text{SL}_{n}(q)$\text{SL}_{n}(q)$ which are (2,3)$(2,3)$ -generated, that is, which are generated by an involution and an element of order 3$3$ . This also gives the classification of the finite simple groups \text{PSL}_{n}(q)$\text{PSL}_{n}(q)$ which are (2,3)$(2,3)$ -generated.

We give an explicit upper bound for the number of zeros of Hecke–Landau zeta-functions in a rectangle.

A special atom (respectively, supernilpotent atom) is a minimal element of the lattice \mathbb{S}$\mathbb{S}$ of all special radicals (respectively, a minimal element of the lattice \mathbb{K}$\mathbb{K}$ of all supernilpotent radicals). A semiprime ring R$R$ is called prime essential if every nonzero prime ideal of R$R$ has a nonzero intersection with each nonzero two-sided ideal of R$R$ . We construct a prime essential ring R$R$ such that the smallest supernilpotent radical containing R$R$ is not a supernilpotent atom but where the smallest special radical containing R$R$ is a special atom. This answers a question put by Puczylowski and Roszkowska.

A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality.

Necessary and sufficient conditions to characterise weakly r$r$-preinvex functions on an invex set are obtained and used to establish generalisations of the Hermite–Hadamard inequality for such functions.