For a finite group $G$, let $\unicode[STIX]{x1D6E5}(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we obtain a necessary and sufficient condition which guarantees that the complement of the character graph $\unicode[STIX]{x1D6E5}(G)$ of a finite group $G$ is a nonbipartite Hamiltonian graph.

In this paper, we discover a sufficient and necessary condition under which two geodesic segments joining the base point and another point in an infinite-dimensional Teichmüller space are the same.

In this work, we generalise the study of Favard's Problems to the weighted diameters of a complete set of conjugate algebraic integers, that is, the roots of an irreducible monic polynomial with coefficients in ℤ.

Let R be an integral domain with quotient field Q. We investigate quasi- and Q-projective ideals, and properties of domains all ideals of which are quasi-projective. It is shown that the so-called l½-generated ideals are quasi-projective, moreover, projective. A module M is quasi-projective if and only if, for a projective ideal P of R, the tensor product M ⊗RP is quasi-projective. Domains whose all ideals are quasi-projective are characterised as almost maximal Prüfer domains. Q is quasi-projective if and only if every proper submodule of Q is complete in its R-topology.

An new extension of Hölder's inequality is derived. This is shown to follow from a generalisation of Steffensen's inequality.

The topologies of the density type in Euclidean space of dimension higher than one are introduced. Definitions are based on a notion of density point connected with a set of sequences of real numbers. Our purpose is to study properties of these topologies and connections between them.

The recently introduced two-parameter eight-state Uq [gl(3|1)] supersymmetric fermion model is extended to include boundary terms. Nine classes of boundary conditions are constructed, all of which are shown to be integrable via the graded boundary quantum inverse scattering method. The boundary systems are solved by using the coordinate Bethe ansatz and the Bethe ansatz equations are given for all nine cases.

Let G1 and G2 be graphs and h1, h2 be hamiltonian paths (h-paths) in G1 and G2 respectively. The hamiltonian product (G1, h2)*(G2, h2) was defined by Holton. If a hamiltonian cycle exists in G2, it can give rise to 2nh-paths. Peckham conjectured that (G1, h1)*(G2, h2)≡(G1, h1)*(G2, h3) where h2 and h3 are any two of these 2nh-paths of G2. He has proved the validity of this conjecture for those h2, h3 where h3 is obtainable from h2 by a rotation along the h-cycle of G2. Here we disprove this conjecture for those h2, h3 where one is obtained from the other by a reflection of the h-cycle.

We prove several commutativity theorems for unital rings with polynomial constraints on certain subsets, which improve and generalise the recent results of Grosen, and Ashraf and Quadri.

Let $R=\mathbb{F}_{p}+u\mathbb{F}_{p}+u^{2}\mathbb{F}_{p}+\cdots +u^{d-1}\mathbb{F}_{p}$ , where $u^{d}=u$ and $p$ is a prime with $d-1$ dividing $p-1$ . A relation between the support weight distribution of a linear code $\mathscr{C}$ of type $p^{dk}$ over $R$ and the dual code $\mathscr{C}^{\bot }$ is established.

We give necessary and sufficient conditions on the semigroup S for the Jacobson radical to be S-invariant.

This paper gives criteria, necessary or sufficient for a vector-valued function F = (f1, f2, …, fk) to be invex. Here each fi is of the -class (that is, each fi is a function whose gradient mapping is locally Lipschitz in a neighbourhood of x0) and the invexity of F means that F(x) − F(x0) ⊂ ˚F′(X) + Q for a fixed convex cone Q of Rk and every x near x0 (˚F′ being the Jacobian matrix of F at x0).

Using the inductive structure of a Fermat variety by Shioda and Katsura [‘On Fermat varieties’, Tohoku Math. J. (2) 31(1) (1979), 97–115], we estimate the refined motivic dimension of certain Fermat varieties. As an application of our computation, we present an elementary proof of the generalised Hodge conjecture for those varieties.

In this paper, we show that the character of any monotonically Lindelöf generalized ordered (GO) space is not greater than ω1, which gives a negative answer to a question posed by Levy and Matveev [‘Some questions on monotone Lindelöfness’, Questions Answers Gen. Topology26 (2008), 13–27, Question 51].

A basic problem in the theory of Lie algebra extensions concerns a given homomorphism X of a Lie algebra L into the Lie algebra of outer derivations of a Lie algebra B. In analogy with the theory of group extensions, Mori and HochschiId developed the concept of an obstruction to X being the homomorphism defined by some Lie algebra extension of B by L. This note considers an alternative approach to this theory, which is particularly simple when applied to the problem of realizing arbitrary three-cohomology classes of L as obstructions. The approach is analogous to one for groups, which was given recently by Gruenberg.

Let $E$ and $D$ be open subsets of $\mathbb{R}^{n+1}$ such that $\overline{D}$ is a compact subset of $E$, and let $v$ be a supertemperature on $E$. We call a temperature $u$ on $D$extendable by$v$ if there is a supertemperature $w$ on $E$ such that $w=u$ on $D$ and $w=v$ on $E\backslash \overline{D}$. Such a temperature need not be a thermic minorant of $v$ on $D$. We show that either there is a unique temperature extendable by $v$, or there are infinitely many. Examples of temperatures extendable by $v$ include the greatest thermic minorant $GM_{v}^{D}$ of $v$ on $D$, and the Perron–Wiener–Brelot solution of the Dirichlet problem $S\!_{v}^{D}$ on $D$ with boundary values the restriction of $v$ to $\unicode[STIX]{x2202}D$. In the case where these two examples are distinct, we give a formula for producing infinitely many more. Clearly $GM_{v}^{D}$ is the greatest extendable thermic minorant, but we also prove that there is a least one, which is not necessarily equal to $S\!_{v}^{D}$.

Because of its practical and theoretical importance in rheology, numerous algorithms have been proposed and utilised to solve the convolution equation $g(x)=(\text{sech}\,\star h)(x)\;(x\in \mathbb{R})$ for $h$ , given $g$ . There are several approaches involving the use of series expansions of derivatives of $g$ , which are then truncated to a small number of terms for practical application. Such truncations can only be expected to be valid if the infinite series converge. In this note, we examine two specific truncations and provide a rigorous analysis to obtain sufficient conditions on $g$ (and equivalently on $h$ ) for the convergence of the series concerned.

The dimension of an ordered set is invariant with respect to any subdivision of its completion. This may be applied to support the conjecture (still open) that the problem to determine the (order) dimension of an N-free ordered set is NP-complete.

We study a particular oscillating continued fraction, and find its two limit points.

Let $n$ be a positive integer. We obtain new Menon’s identities by using the actions of some subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$ on the set $\mathbb{Z}/n\mathbb{Z}$. In particular, let $p$ be an odd prime and let $\unicode[STIX]{x1D6FC}$ be a positive integer. If $H_{k}$ is a subgroup of $(\mathbb{Z}/p^{\unicode[STIX]{x1D6FC}}\mathbb{Z})^{\times }$ with index $k=p^{\unicode[STIX]{x1D6FD}}u$ such that $0\leqslant \unicode[STIX]{x1D6FD}<\unicode[STIX]{x1D6FC}$ and $u\mid p-1$, then

$$\begin{eqnarray}\mathop{\sum }_{x\in H_{k}}(x-1,p^{\unicode[STIX]{x1D6FC}})=\frac{\unicode[STIX]{x1D711}(p^{\unicode[STIX]{x1D6FC}})}{k}\bigg(1+k(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})+u\frac{p^{\unicode[STIX]{x1D6FD}}-1}{p-1}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D711}(n)$