Using properties of Ramanujan’s theta functions, we give an elementary proof of Hirschhorn’s conjecture on 2^n$2^n$-dissection of Euler’s product E(q):=(q;q)_\infty $E(q):=(q;q)_\infty$.

In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by \gamma _{t2}(G)$\gamma _{t2}(G)$, is the minimum cardinality of a semitotal dominating set in G. Goddard, Henning and McPillan [‘Semitotal domination in graphs’, Utilitas Math. 94 (2014), 67–81] characterised the trees and graphs of minimum degree 2 with semitotal domination number half their order. In this paper, we characterise all graphs whose semitotal domination number is half their order.

A cycle C of a graph G is dominating if V(C)$V(C)$ is a dominating set and V(G)\backslash V(C)$V(G)\backslash V(C)$ is an independent set. Wu et al. [‘Degree sums and dominating cycles’, Discrete Mathematics 344 (2021), Article no. 112224] proved that every longest cycle of a k-connected graph G on n\geq 3$n\geq 3$ vertices with k\geq 2$k\geq 2$ is dominating if the degree sum is more than (k+1)(n+1)/3$(k+1)(n+1)/3$ for any k+1$k+1$ pairwise nonadjacent vertices. They also showed that this bound is sharp. In this paper, we show that the extremal graphs G for this condition satisfy (n-2)/3K_1\vee (n+1)/3K_2 \subseteq G \subseteq K_{(n-2)/3}\vee (n+1)/3K_2$(n-2)/3K_1\vee (n+1)/3K_2 \subseteq G \subseteq K_{(n-2)/3}\vee (n+1)/3K_2$ or 2K_1\vee 3K_{(n-2)/3}\subseteq G \subseteq K_2\vee 3K_{(n-2)/3}.$2K_1\vee 3K_{(n-2)/3}\subseteq G \subseteq K_2\vee 3K_{(n-2)/3}.$

Let k\geq 4$k\geq 4$ be an integer. We prove that the set \mathcal {O}$\mathcal {O}$ of all nonzero generalised octagonal numbers is a k-additive uniqueness set for the set of multiplicative functions. That is, if a multiplicative function f_k$f_k$ satisfies the condition

\begin{align*} f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \end{align*} $$\begin{align*} f_k(x_1+x_2+\cdots+x_k)=f_k(x_1)+f_k(x_2)+\cdots+f_k(x_k) \end{align*}$$

for arbitrary x_1,\ldots ,x_k\in \mathcal {O}$x_1,\ldots ,x_k\in \mathcal {O}$, then f_k$f_k$ is the identity function f_k(n)=n$f_k(n)=n$ for all n\in \mathbb {N}$n\in \mathbb {N}$. We also show that f_2$f_2$ and f_3$f_3$ are not determined uniquely.

We establish a q-analogue of a supercongruence related to a supercongruence of Rodriguez-Villegas, which extends a q-congruence of Guo and Zeng [‘Some q-analogues of supercongruences of Rodriguez-Villegas’, J. Number Theory 145 (2014), 301–316]. The important ingredients in the proof include Andrews’ _4\phi _3$_4\phi _3$ terminating identity.

We determine the characteristic polynomials of the matrices [q^{\,j-k}+t]_{1\le \,j,k\le n}$[q^{\,j-k}+t]_{1\le \,j,k\le n}$ and [q^{\,j+k}+t]_{1\le \,j,k\le n}$[q^{\,j+k}+t]_{1\le \,j,k\le n}$ for any complex number q\not =0,1$q\not =0,1$. As an application, for complex numbers a,b,c$a,b,c$ with b\not =0$b\not =0$ and a^2\not =4b$a^2\not =4b$, and the sequence (w_m)_{m\in \mathbb Z}$(w_m)_{m\in \mathbb Z}$ with w_{m+1}=aw_m-bw_{m-1}$w_{m+1}=aw_m-bw_{m-1}$ for all m\in \mathbb Z$m\in \mathbb Z$, we determine the exact value of \det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.

A monic polynomial f(x)\in {\mathbb Z}[x]$f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if f(x)$f(x)$ is irreducible over {\mathbb Q}${\mathbb Q}$ and \{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of {\mathbb Q}(\theta )${\mathbb Q}(\theta )$, where f(\theta )=0$f(\theta )=0$. We prove that there exist exactly three distinct monogenic trinomials of the form x^4+bx^2+d$x^4+bx^2+d$ whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.

A monic polynomial f(x)\in {\mathbb Z}[x]$f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if f(x)$f(x)$ is irreducible over {\mathbb Q}${\mathbb Q}$ and \{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of {\mathbb Q}(\theta )${\mathbb Q}(\theta )$, where f(\theta )=0$f(\theta )=0$. We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly 96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

Let \mathbb {F}$\mathbb {F}$ be a field and (s_0,\ldots ,s_{n-1})$(s_0,\ldots ,s_{n-1})$ be a finite sequence of elements of \mathbb {F}$\mathbb {F}$. In an earlier paper [G. H. Norton, ‘On the annihilator ideal of an inverse form’, J. Appl. Algebra Engrg. Comm. Comput. 28 (2017), 31–78], we used the \mathbb {F}[x,z]$\mathbb {F}[x,z]$ submodule \mathbb {F}[x^{-1},z^{-1}]$\mathbb {F}[x^{-1},z^{-1}]$ of Macaulay’s inverse system \mathbb {F}[[x^{-1},z^{-1}]]$\mathbb {F}[[x^{-1},z^{-1}]]$ (where z is our homogenising variable) to construct generating forms for the (homogeneous) annihilator ideal of (s_0,\ldots ,s_{n-1})$(s_0,\ldots ,s_{n-1})$. We also gave an \mathcal {O}(n^2)$\mathcal {O}(n^2)$ algorithm to compute a special pair of generating forms of such an annihilator ideal. Here we apply this approach to the sequence r of the title. We obtain special forms generating the annihilator ideal for (r_0,\ldots ,r_{n-1})$(r_0,\ldots ,r_{n-1})$ without polynomial multiplication or division, so that the algorithm becomes linear. In particular, we obtain its linear complexities. We also give additional applications of this approach.

Previously [‘Radicals and idempotents I, II’, Comm. Alg. 49(1) (2021), 73–84 and 50(11) (2022), 4791–4804], we have studied the interaction between radicals of rings and idempotents in general or those of particular types, for example, left semicentral. Here we carry out similar investigations for q-central idempotents, that is, those idempotents e satisfying the condition (ea-eae)(be-ebe) = 0$(ea-eae)(be-ebe) = 0$ for all a, b.

Let G be a finite group and let \chi $\chi$ be an irreducible character of G. The number |G:\mathrm {ker}\chi |/\chi (1)$|G:\mathrm {ker}\chi |/\chi (1)$ is called the codegree of the character \chi $\chi$. We provide several relations between the structure of G and the codegrees of the characters in a given subset of \mathrm {Irr}(G)$\mathrm {Irr}(G)$, where \mathrm {Irr}(G)$\mathrm {Irr}(G)$ is the set of all complex irreducible characters of G. For example, we show that if the codegrees of all strongly monolithic characters of G are odd, then G is solvable, analogous to the well-known fact that if all irreducible character degrees of a finite group are odd, then that group is solvable.

Let \alpha $\alpha$ be a complex valued 2$2$-cocycle of finite order of a finite group G.$G.$ The nth Frobenius–Schur indicator of an irreducible \alpha $\alpha$-character of G is defined and its properties are investigated. The indicator is interpreted in general for n =2$n =2$ and it is shown that it can be used to determine whether an irreducible \alpha $\alpha$-character is real-valued under the assumption that the order of \alpha $\alpha$ and its cohomology class are both 2$2$. A formula, involving the real \alpha $\alpha$-regular conjugacy classes of G,$G,$ is found to count the number of real-valued irreducible \alpha $\alpha$-characters of G under the additional assumption that these characters are class functions.

Let V be a finite dimensional vector space over the field with p elements, where p is a prime number. Given arbitrary \alpha ,\beta \in \mathrm {GL}(V)$\alpha ,\beta \in \mathrm {GL}(V)$, we consider the semidirect products V\rtimes \langle \alpha \rangle $V\rtimes \langle \alpha \rangle$ and V\rtimes \langle \beta \rangle $V\rtimes \langle \beta \rangle$, and show that if V\rtimes \langle \alpha \rangle $V\rtimes \langle \alpha \rangle$ and V\rtimes \langle \beta \rangle $V\rtimes \langle \beta \rangle$ are isomorphic, then \alpha $\alpha$ must be similar to a power of \beta $\beta$ that generates the same subgroup as \beta $\beta$; that is, if H and K are cyclic subgroups of \mathrm {GL}(V)$\mathrm {GL}(V)$ such that V\rtimes H\cong V\rtimes K$V\rtimes H\cong V\rtimes K$, then H and K must be conjugate subgroups of \mathrm {GL}(V)$\mathrm {GL}(V)$. If we remove the cyclic condition, there exist examples of nonisomorphic, let alone nonconjugate, subgroups H and K of \mathrm {GL}(V)$\mathrm {GL}(V)$ such that V\rtimes H\cong V\rtimes K$V\rtimes H\cong V\rtimes K$. Even if we require that noncyclic subgroups H and K of \mathrm {GL}(V)$\mathrm {GL}(V)$ be abelian, we may still have V\rtimes H\cong V\rtimes K$V\rtimes H\cong V\rtimes K$ with H and K nonconjugate in \mathrm {GL}(V)$\mathrm {GL}(V)$, but in this case, H and K must at least be isomorphic. If we replace V by a free module U over {\mathbb {Z}}/p^m{\mathbb {Z}}${\mathbb {Z}}/p^m{\mathbb {Z}}$ of finite rank, with m>1$m>1$, it may happen that U\rtimes H\cong U\rtimes K$U\rtimes H\cong U\rtimes K$ for nonconjugate cyclic subgroups of \mathrm {GL}(U)$\mathrm {GL}(U)$. If we completely abandon our requirements on V, a sufficient criterion is given for a finite group G to admit nonconjugate cyclic subgroups H and K of \mathrm {Aut}(G)$\mathrm {Aut}(G)$ such that G\rtimes H\cong G\rtimes K$G\rtimes H\cong G\rtimes K$. This criterion is satisfied by many groups.

Let \mathbb {Z}$\mathbb {Z}$ be the additive (semi)group of integers. We prove that for a finite semigroup S the direct product \mathbb {Z}\times S$\mathbb {Z}\times S$ contains only countably many subdirect products (up to isomorphism) if and only if S is regular. As a corollary we show that \mathbb {Z}\times S$\mathbb {Z}\times S$ has only countably many subsemigroups (up to isomorphism) if and only if S is completely regular.

We consider the existence problem of meromorphic solutions of the Fermat-type difference equation

\begin{align*} f(z)^p+f(z+c)^q=h(z), \end{align*} $$\begin{align*} f(z)^p+f(z+c)^q=h(z), \end{align*}$$

where p,q$p,q$ are positive integers, and h has few zeros and poles in the sense that N(r,h) + N(r,1/h) = S(r,h)$N(r,h) + N(r,1/h) = S(r,h)$. As a particular case, we consider h=e^g$h=e^g$, where g is an entire function. Additionally, we briefly discuss the case where h is small with respect to f in the standard sense T(r,h)=S(r,f)$T(r,h)=S(r,f)$.

We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.) 35(1) (2024), 60–75], proving that if (x_n) \subset X$(x_n) \subset X$ and (y_n) \subset Y$(y_n) \subset Y$ are two weakly null sequences such that (x_n \otimes y_n)$(x_n \otimes y_n)$ converges weakly in X \widehat {\otimes }_\pi Y$X \widehat {\otimes }_\pi Y$, then (x_n \otimes y_n)$(x_n \otimes y_n)$ is also weakly null.

We prove a local gradient estimate for positive eigenfunctions of the \mathcal {L}$\mathcal {L}$-operator on conformal solitons given by a general conformal vector field. As an application, we obtain a Liouville type theorem for \mathcal {L} u=0$\mathcal {L} u=0$, which improves the one of Li and Sun [‘Gradient estimate for the positive solutions of \mathcal Lu = 0$\mathcal Lu = 0$ and \mathcal Lu ={\partial u}/{\partial t}$\mathcal Lu ={\partial u}/{\partial t}$ on conformal solitons’, Acta Math. Sin. (Engl. Ser.) 37(11) (2021), 1768–1782]. We also consider applications where manifolds are special conformal solitons and obtain a better Liouville type theorem in the case of self-shrinkers.

We recover Reidemeister’s theorem using \mathcal {C}^{\infty }$\mathcal {C}^{\infty }$ functions and transversality.

Let \{\omega _n\}_{n\geq 1}$\{\omega _n\}_{n\geq 1}$ be a sequence of independent and identically distributed random variables on a probability space (\Omega , \mathcal {F}, \mathbb {P})$(\Omega , \mathcal {F}, \mathbb {P})$, each uniformly distributed on the unit circle \mathbb {T}$\mathbb {T}$, and let \ell _n=cn^{-\tau }$\ell _n=cn^{-\tau }$ for some c>0$c>0$ and 0<\tau <1$0<\tau <1$. Let I_{n}=(\omega _n,\omega _n+\ell _n)$I_{n}=(\omega _n,\omega _n+\ell _n)$ be the random interval with left endpoint \omega _n$\omega _n$ and length \ell _n$\ell _n$. We study the asymptotic property of the covering time N_n(x)=\sharp \{1\leq k\leq n: x\in I_k\}$N_n(x)=\sharp \{1\leq k\leq n: x\in I_k\}$ for each x\in \mathbb {T}$x\in \mathbb {T}$. We prove the quenched central limit theorem for the covering time, that is, \mathbb {P}$\mathbb {P}$-almost surely,

\begin{align*}\frac{N_n(x)-\mathbb{E}_{\mathbb{P}}(N_n(x))}{\sqrt{\sum_{k=1}^n \ell_k(1-\ell_k)}}\end{align*} $$\begin{align*}\frac{N_n(x)-\mathbb{E}_{\mathbb{P}}(N_n(x))}{\sqrt{\sum_{k=1}^n \ell_k(1-\ell_k)}}\end{align*}$$

converges in law to the standard normal distribution.