Building on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ is $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where Q is a projection of rank $n-1$ and $T\in \mathbb {M}_n(\mathbb {C})$ is a nilpotent of minimal distance to Q. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.

Let $R$ be a ring. A map $f\,:\,R\,\to \,R$ is additive if $f(a\,+\,b)\,=\,f(a)\,+\,f(b)$ for all elements $a$ and $b$ of $R$. Here, a map $f\,:\,R\,\to \,R$ is called unit-additive if $f(u\,+\,v)\,=\,f(u)\,+\,f(v)$ for all units $u$ and $v$ of $R$. Motivated by a recent result of $\text{Xu}$, $\text{Pei}$ and $\text{Yi}$ showing that, for any field $F$, every unit-additive map of ${{\mathbb{M}}_{n}}(F)$ is additive for all $n\,\ge \,2$, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to ${{\mathbb{Z}}_{2}}\,\text{or}\,R/J(R)\,\cong \,{{\mathbb{Z}}_{2}}\,$ with $2\,=\,0$ in $R$. Consequently, for any semilocal ring $R$, every unit-additive map of ${{\mathbb{M}}_{n}}(R)$ is additive for all $n\,\ge \,2$. These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unithomomorphic if $f(uv)\,=\,f(u)f(v)$ for all units $u$, $v$ of $R$. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

An efficient high order numerical method is presented to solve the mobile-immobile advection-dispersion model with the Coimbra time variable-order fractional derivative, which is used to simulate solute transport in watershed catchments and rivers. On establishing an efficient recursive algorithm based on the properties of Jacobi polynomials to approximate the Coimbra variable-order fractional derivative operator, we use spectral collocation method with both temporal and spatial discretisation to solve the time variable-order fractional mobile-immobile advection-dispersion model. Numerical examples then illustrate the effectiveness and high order convergence of our approach.

Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitianmatrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization formaps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

We study product theorems for matrix spaces. In particular, we prove the following theorems.

Theorem 1. For all $\varepsilon>0$, there is $\delta>0$ such that if $A\subset\mathrm{SL}_3(\mathbb{Z})$ is a finite set, then either $A$ intersects a coset of a nilpotent subgroup in a set of size at least $|A|^{1-\varepsilon}$, or $|A^3|>|A|^{1+\delta}$.

Theorem 2. Let $A$ be a finite subset of $\mathrm{SL}_2(\mathbb{C})$. Then either $A$ is contained in a virtually abelian subgroup, or $|A^3|>c|A|^{1+\delta}$ for some absolute constant $\delta>0$.

Here $A^3=\{a_1a_2a_3:a_i\in A,\ i=1,2,3\}$ is the $3$-fold product set of $A$.

In the Mathematical Review of ‘On a conjecture of Kontsevich and variants of Castelnuovo's lemma’ [Compositio Mathematica 115 (1999), 205–230], Gizatullin pointed out that the proof as written is incomplete for the complex case because the possibility of a certain equation vanishing identically was ignored. This corrigendum addresses the missing case. Note that it is simply another iteration of the argument already present in the article.

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space, partitioned into two classes termed ‘open’ and ‘closed’. It is possible to observe only which class the process is in. A burst of channel openings is defined to be a succession of open sojourns separated by closed sojourns all having duration less than t0. Let N(t) be the number of bursts commencing in (0, t]. Then are measures of the degree of temporal clustering of bursts. We develop two methods for determining the above measures. The first method uses an embedded Markov renewal process and remains valid when the underlying channel process is semi-Markov and/or brief sojourns in either the open or closed classes of state are undetected. The second method uses a ‘backward’ differential-difference equation.

The observed channel process when brief sojourns are undetected can be modelled by an embedded Markov renewal process, whose kernel is shown, by exploiting connections with bursts when all sojourns are detected, to satisfy a differential-difference equation. This permits a unified derivation of both exact and approximate expressions for the kernel, and leads to a thorough asymptotic analysis of the kernel as the length of undetected sojourns tends to zero.

Non-hermitian solutions of algebraic matrix Riccati equations (of the continuous and discrete types) are studied. Existence is proved of non-hermitian solutions with given upper bounds of the ranks of the skew-hermitian parts, under the sign controllability hypothesis.

The distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k × k matrices with the usual operator norm, is shown to be sec(π/(k:+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the space of k × k matrices. Another result of note is that the shortest distance between the set of non-zero projections and the set of nilpotents in the space of k × k matrices is .

We show that a certain category 𝓖 whose objects are pairs G ⊃ H of groups subject to simple axioms is equivalent to the category of ≥ 2-dimensional vector spaces and injective semi-linear maps; and deduce via the "Fundamental Theorem of Projective Geometry" that the category of ≥ 2-dimensional projective spaces is equivalent to the quotient of a suitable subcategory of 𝓖 by the least equivalence relation which identifies conjugation by any element of H with the identity automorphism of G.