It is shown that the Fourier sine transform, $\mathcal{F}_S [f(t)](\omega )$ on $\mathbb {R}_0^+$, of any given real-valued function $f(t)$ that does not vanish at $t=0$ or has a nonvanishing even-order derivative at $t=0$, has a definite sign at least for $\omega> \omega _0$, where $\omega _0$ can be estimated. Similarly, the cosine transform, $\mathcal{F}_C [f(t)](\omega )$, of functions with a nonvanishing odd-order derivative at zero also has a definite sign for sufficiently large $\omega $. Several examples are given.

In this paper, we generalise to the family of Fermat quartics $X^4 + Y^4 = 2^m, m \in \mathbb {Z}$, a result of Aigner [‘Über die Möglichkeit von $x^4 + y^4 = z^4$ in quadratischen Körpern’, Jahresber. Deutsch. Math.-Ver. 43 (1934), 226–228], which proves that there is only one quadratic field, namely $\mathbb {Q}(\sqrt {-7})$, that contains solutions to the Fermat quartic $X^4 + Y^4 = 1$. The $m \equiv 0 \pmod 4$ case is due to Aigner. The $m \equiv 2 \pmod 4$ case follows from a result of Emory [‘The Diophantine equation $X^4 + Y^4 = D^2Z^4$ in quadratic fields’, Integers 12 (2012), Article no. A65, 8 pages]. This paper focuses on the two cases $m \equiv 1, 3 \pmod 4$, classifying for $m \equiv 1 \pmod 4$ the infinitely many quadratic number fields that contain solutions, and proving for $m \equiv 3 \pmod 4$ that $\mathbb {Q}(\sqrt {2})$ and $\mathbb {Q}(\sqrt {-2})$ are the only quadratic number fields that contain solutions.

Existing multivariate versions of the logistic probability distribution generally lack some of the useful properties of the univariate logistic distribution, such as its bounded score function or the tractability of its density function, or lack the rotational symmetry necessary for many applications. This paper clarifies some of the properties of such distributions and proposes a multivariate distribution closely related to the univariate logistic that has a tractable density, including the necessary normalising constant, bounded score function and elliptical symmetry. Some properties of its marginal distributions are explored, particularly in the bivariate case.

For each $n\geq 1$, let $FT_n$ be the free tree monoid of rank n and $E_n$ the full extensive transformation monoid over the finite chain $\{1, 2, \ldots , n\}$. It is shown that the monoids $FT_n$ and $E_{n+1}$ satisfy the same identities. Therefore, $FT_n$ is finitely based if and only if $n\leq 3$.

The lattice walks in the plane starting at the origin $\mathbf {0}$ with steps in $\{-1,0,1\}^{2}\setminus \{\mathbf {0}\}$ are called king walks. We investigate enumeration and divisibility for higher dimensional king walks confined to certain regions. Specifically, we establish an explicit formula for the number of $(r+s)$-dimensional king walks of length n ending at $(a_1,\ldots ,a_r,b_1,\ldots ,b_s)$ which never dip below $x_i=0$ for $i=1,\ldots ,r$. We also derive divisibility properties for the number of $(r+s)$-dimensional king walks of length p (an odd prime) through group actions.

Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form

$$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$

where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$, $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$, we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$.

A subset of a finite set of filling curves on a surface is not necessarily filling. However, when a filling set spans homology and curves intersect pairwise at most once, it is shown that one can always add a curve and subtract a different curve to obtain a filling set that spans homology. A motivation for filling sets of curves that span homology comes from the Thurston spine and the Steinberg module of the mapping class group.

A subgroup X of a group G is said to be transitively normal if X is normal in any subgroup Y of G such that $X\leq Y$ and X is subnormal in Y. We investigate the structure of generalised soluble groups with dense transitively normal subgroups, that is, groups in which every nonempty open interval in their subgroup lattice contains a transitively normal subgroup. In particular, it will be proved that nonperiodic generalised soluble groups with dense transitively normal subgroups are abelian.

We investigate semigroups S which have the property that every subsemigroup of $S\times S$ which contains the diagonal $\{ (s,s)\colon s\in S\}$ is necessarily a congruence on S. We call such an S a DSC semigroup. It is well known that all finite groups are DSC, and easy to see that every DSC semigroup must be simple. Building on this, we show that for broad classes of semigroups, including periodic, stable, inverse and several well-known types of simple semigroups, the only DSC members are groups. However, it turns out that there exist nongroup DSC semigroups, which we obtain by utilising a construction introduced by Byleen for the purpose of constructing interesting congruence-free semigroups. Such examples can additionally be regular or bisimple.

Brun’s constant is the summation of the reciprocals of all twin primes, given by

$$ \begin{align*}B=\sum_{p \in P_2}{\bigg( \frac{1}{p} + \frac{1}{p+2}\bigg)}.\end{align*} $$

While rigorous unconditional bounds on B are known, we present the first rigorous bound on Brun’s constant under the assumption of GRH, yielding $B < 2.1594$.

We strengthen two results of Moretó. We prove that the index of the Fitting subgroup is bounded in terms of the degrees of the irreducible monomial Brauer characters of the finite solvable group G and it is also bounded in terms of the average degree of the irreducible Brauer characters of G that lie over a linear character of the Fitting subgroup.

If G is a graph, then $X\subseteq V(G)$ is a general position set if for every two vertices $v,u\in X$ and every shortest $(u,v)$-path P, no inner vertex of P lies in X. We propose three algorithms to compute a largest general position set in G: an integer linear programming algorithm, a genetic algorithm and a simulated annealing algorithm. These approaches are supported by examples from different areas of graph theory.

We extend the definition of the X-Rokhlin property to countable discrete groups and prove some permanence properties. If the action of a countable discrete group on X is free and minimal and the action of this group on the separable simple $C^*$-algebra has the X-Rokhlin property, then the reduced crossed product is simple.