New Zealand Journal of Mathematics

Nearly self-conjugate integer partitions

John Campbell, Shane Chern — Sun, 25 Jun 2023 00:00:00 +1200

We investigate integer partitions $\lambda$ of $n$ that are nearly self-conjugate in the sense that there are $n - 1$ overlapping cells among the Ferrers diagram of $\lambda$ and its transpose, by establishing a correspondence, through the method of combinatorial telescoping, to partitions of $n$ in which (i).~there exists at least one even part; (ii).~any even part is of size $2$; (iii).~the odd parts are distinct; and (iv).~no odd part is of size $1$. In particular, this correspondence confirms a conjecture that had been given in the OEIS.

Amendment to "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]

Jiarul Hoque, Shyamapada Modak — Sun, 25 Jun 2023 00:00:00 +1200

We give a counterexample in this amendment to show that there is an error in consideration of the statement "{\it if $f : X \to Y$ and ${\bf J}$ is an ideal on $Y$, then $f^{-1}({\bf J}) = \{f^{-1}(J) : J \in {\bf J}\}$ is an ideal on $X$}" by Hamlett in his paper "Lindelöf with respect to an ideal" [New Zealand J. Math. 42, 115-120, 2012]. We also modify it here in a new way and henceforth put forward correctly all the results that were based on the said statement derived therein.

The 2-fold pure extensions need not split

Aliakbar Alijani — Sun, 25 Jun 2023 00:00:00 +1200

In this paper, we give an example of locally compact abelian groups $A$ and $C$ such that ${\rm Pext}^{2}(C,A)\neq 0$.

The conjugate locus in convex 3-manifolds

Thomas Waters, Matthew Cherrie — Sat, 01 Jul 2023 00:00:00 +1200

In this paper we study the conjugate locus in convex manifolds. Our main tool is Jacobi fields, which we use to define a special coordinate system on the unit sphere of the tangent space; this provides a natural coordinate system to study and classify the singularities of the conjugate locus. We pay particular attention to 3-dimensional manifolds, and describe a novel method for determining conjugate points. We then make a study of a special case: the 3-dimensional (quadraxial) ellipsoid. We emphasise the similarities with the focal sets of 2-dimensional ellipsoids.

Corrigendum to: Two new proofs of the fact that triangle groups are distinguished by their finite quotients

Marston Conder — Tue, 03 Oct 2023 00:00:00 +1300

This brief corrigendum corrects some minor errors in the paper `"Two new proofs of the fact that triangle groups are distinguished by their finite quotients", published in the New Zealand Journal of Mathematics 52 (2022), 827--844.

Group Actions on Product Systems

Valentin Deaconu, Leonard Huang — Thu, 19 Oct 2023 00:00:00 +1300

We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{\ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.

Yamabe solitons in contact geometry

Rahul Poddar, S. Balasubramanian, Ramesh Sharma — Wed, 20 Dec 2023 00:00:00 +1300

It is shown that the scalar curvature of a Yamabe soliton as a Sasakian manifold is constant and the soliton vector field is Killing. The same conclusion is shown to hold for a Yamabe soliton as a $K$-contact manifold $M^{2n+1}$ if any one of the following conditions hold: (i) its scalar curvature is constant along the soliton vector field $V$, (ii) $V$ is an eigenvector of the Ricci operator with eigenvalue $2n$, (iii) $V$ is gradient.

A note on the regularity criterion for the micropolar fluid equations in homogeneous Besov spaces

Qiang Li, Mianlu Zou — Wed, 20 Dec 2023 00:00:00 +1300

This paper gives a further investigation on the regularity criteria for three-dimensional micropolar equations in Besov spaces. More precisely, it is proved that the weak solution $(u, \omega)$ is regular if the velocity $u$ satisfies

$$\int_{0}^{T}\| \nabla_{h}u_{h}\|_{\dot{B}_{p,\frac{2p}{3}}^{0}}^{q} d t<\infty,\ with\ \ \frac{3}{p}+\frac{2}{q}=2,\ \frac{3}{2}<p\leq\infty,$$
or $$\int_{0}^{T}\| \nabla_{h}u\|_{\dot{B}_{\infty ,\infty}^{-1}}^{\frac{8}{3}} d t<\infty,$$
or $$\int_{0}^{T}\|\nabla_{h} u_{h}\|_{\dot{B}_{\infty,\infty}^{-\alpha}}^{\frac{2}{2-\alpha}} d t<\infty,\ with\ 0< \alpha< 1. $$