Let R be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when R is commutative, is provided. If, however, R has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if R is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If R is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of a is greater than $2$.

We obtain a system of identities relating boundary coefficients and spectral data for the one-dimensional Schrödinger equation with boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter. These identities can be thought of as a kind of mini version of the Gelfand–Levitan integral equation for boundary coefficients only.

An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify the reversible and strongly reversible elements in the quaternionic special linear group $ \mathrm {SL}(n,\mathbb {H})$ and quaternionic projective linear group $ \mathrm {PSL}(n,\mathbb {H})$. We prove that an element of $ \mathrm {SL}(n,\mathbb {H})$ (resp. $ \mathrm {PSL}(n,\mathbb {H})$) is reversible if and only if it is a product of two skew-involutions (resp. involutions).

Fix $\alpha >0$. Then by Fejér's theorem $(\alpha (\log n)^{A}\,\mathrm {mod}\,1)_{n\geq 1}$ is uniformly distributed if and only if $A>1$. We sharpen this by showing that all correlation functions, and hence the gap distribution, are Poissonian provided $A>1$. This is the first example of a deterministic sequence modulo $1$ whose gap distribution and all of whose correlations are proven to be Poissonian. The range of $A$ is optimal and complements a result of Marklof and Strömbergsson who found the limiting gap distribution of $(\log (n)\, \mathrm {mod}\,1)$, which is necessarily not Poissonian.

The embedding problem of Markov chains examines whether a stochastic matrix$\mathbf{P} $ can arise as the transition matrix from time 0 to time 1 of a continuous-time Markov chain. When the chain is homogeneous, it checks if $ \mathbf{P}=\exp{\mathbf{Q}}$ for a rate matrix $ \mathbf{Q}$ with zero row sums and non-negative off-diagonal elements, called a Markov generator. It is known that a Markov generator may not always exist or be unique. This paper addresses finding $ \mathbf{Q}$, assuming that the process has at most one jump per unit time interval, and focuses on the problem of aligning the conditional one-jump transition matrix from time 0 to time 1 with $ \mathbf{P}$. We derive a formula for this matrix in terms of $ \mathbf{Q}$ and establish that for any $ \mathbf{P}$ with non-zero diagonal entries, a unique $ \mathbf{Q}$, called the ${\unicode{x1D7D9}}$-generator, exists. We compare the ${\unicode{x1D7D9}}$-generator with the one-jump rate matrix from Jarrow, Lando, and Turnbull (1997), showing which is a better approximate Markov generator of $ \mathbf{P}$ in some practical cases.

For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny’s constant. We prove that for any partially specified stochastic matrix for which the problem is well defined, there is a minimizing completion that is as sparse as possible. We also find the minimum value of Kemeny’s constant in two special cases: when the diagonal has been specified and when all specified entries lie in a common row.

Let A be an F-central simple algebra of degree $m^2=\prod _{i=1}^k p_i^{2\alpha _i}$ and G be a subgroup of the unit group of A such that $F[G]=A$. We prove that if G is a central product of two of its subgroups M and N, then $F[M]\otimes _F F[N]\cong F[G]$. Also, if G is locally nilpotent, then G is a central product of subgroups $H_i$, where $[F[H_i]:F]=p_i^{2\alpha _i}$, $A=F[G]\cong F[H_1]\otimes _F \cdots \otimes _F F[H_k]$ and $H_i/Z(G)$ is the Sylow $p_i$-subgroup of $G/Z(G)$ for each i with $1\leq i\leq k$. Additionally, there is an element of order $p_i$ in F for each i with $1\leq i\leq k$.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the ‘weakly adelic Mumford–Tate hypothesis’ and prove the generalised André–Pink–Zannier conjecture under this assumption, using Pila–Zannier strategy.

This paper focuses on the fundamental aspects of super-resolution, particularly addressing the stability of super-resolution and the estimation of two-point resolution. Our first major contribution is the introduction of two location-amplitude identities that characterize the relationships between locations and amplitudes of true and recovered sources in the one-dimensional super-resolution problem. These identities facilitate direct derivations of the super-resolution capabilities for recovering the number, location, and amplitude of sources, significantly advancing existing estimations to levels of practical relevance. As a natural extension, we establish the stability of a specific $l_{0}$ minimization algorithm in the super-resolution problem.

The second crucial contribution of this paper is the theoretical proof of a two-point resolution limit in multi-dimensional spaces. The resolution limit is expressed as

$$\begin{align*}\mathscr R = \frac{4\arcsin \left(\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2}} \right)}{\Omega} \end{align*}$$

${\frac {\sigma }{m_{\min }}}{\leqslant }{\frac {1}{2}}$

${\frac {\sigma }{m_{\min }}}$

${\mathrm {SNR}}$

$\Omega $

${\frac {\pi }{\Omega }}$

$2$

${\mathscr {R}}$

The walk matrix associated to an $n\times n$ integer matrix $\mathbf{X}$ and an integer vector $b$ is defined by ${\mathbf{W}} \,:\!=\, (b,{\mathbf{X}} b,\ldots, {\mathbf{X}}^{n-1}b)$. We study limiting laws for the cokernel of $\mathbf{W}$ in the scenario where $\mathbf{X}$ is a random matrix with independent entries and $b$ is deterministic. Our first main result provides a formula for the distribution of the $p^m$-torsion part of the cokernel, as a group, when $\mathbf{X}$ has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the ${\mathbb{Z}}[x]$-module structure.

The motivation for this work arises from an open problem in spectral graph theory, which asks to show that random graphs are often determined up to isomorphism by their (generalised) spectrum. Sufficient conditions for generalised spectral determinacy can, namely, be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper.

Motivated by the recent work of Zhi-Wei Sun [‘Problems and results on determinants involving Legendre symbols’, Preprint, arXiv:2405.03626], we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi $ be the unique quadratic multiplicative character of the finite field $\mathbb {F}_q$. If the set $\{s_1,\ldots ,s_{(q-1)/2}\}=\{x^2:\ x\in \mathbb {F}_q\setminus \{0\}\}$, then we prove that

$$ \begin{align*}\det[t+\phi(s_i+s_j)+\phi(s_i-s_j)]_{1\le i,j\le (q-1)/2}=\bigg(\frac{q-1}{2}t-1\bigg)q^{{(q-3)}/{4}}.\end{align*} $$

This confirms a conjecture of Zhi-Wei Sun.

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.

We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$-th Bell number, which is much smaller than the previously best-known upper bounds when $n \geq 4$. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over ${\mathbb{F}}_2$ has tensor rank exactly equal to $12$. Our results also improve upon the best-known upper bound for the Waring rank of the determinant when $n \geq 17$, and lead to a new family of axis-aligned polytopes that tile ${\mathbb{R}}^n$.

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

We obtain asymptotic formulas for the number of matrices in the congruence subgroup

$$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$

We show that properties of pairs of finite, positive, and regular Borel measures on the complex unit circle such as domination, absolute continuity, and singularity can be completely described in terms of containment and intersection of their reproducing kernel Hilbert spaces of “Cauchy transforms” in the complex unit disk. This leads to a new construction of the classical Lebesgue decomposition and proof of the Radon–Nikodym theorem using reproducing kernel theory and functional analysis.

We solve the problem of finding the inverse connection formulae for the generalised Bessel polynomials and their reciprocals, the reverse generalised Bessel polynomials. The connection formulae express monomials in terms of the generalised Bessel polynomials. They enable formulae for the elements of change of basis matrices for both kinds of generalised Bessel polynomials to be derived and proved correct directly.

Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, Acta Arith. 115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting $a_1,\ldots ,a_{(q-1)/2}$ be all the nonzero squares in the finite field $\mathbb {F}_q$ containing q elements with $2\nmid q$, we give the explicit value of the determinant $D_{(q-1)/2}=\det [(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if $q=p$ is a prime greater than $3$, then

$$ \begin{align*}\bigg(\frac{\det D_{(p-1)/2}}{p}\bigg)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4,\\ (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}\end{align*} $$

where $(\frac {\cdot }{p})$ is the Legendre symbol and $h(-p)$ is the class number of $\mathbb {Q}(\sqrt {-p})$.

We introduce a generalization of immanants of matrices, using partition algebra characters in place of symmetric group characters. We prove that our immanant-like function on square matrices, which we refer to as the recombinant, agrees with the usual definition for immanants for the special case whereby the vacillating tableaux associated with the irreducible characters correspond, according to the Bratteli diagram for partition algebra representations, to the integer partition shapes for symmetric group characters. In contrast to previously studied variants and generalizations of immanants, as in Temperley–Lieb immanants and f-immanants, the sum that we use to define recombinants is indexed by a full set of partition diagrams, as opposed to permutations.

Brazil et al. [‘Maximal subgroups of infinite symmetric groups’, Proc. Lond. Math. Soc. (3) 68(1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group $G(X)$ defined on an infinite set X. It is easy to see that, in this case, $G(X)$ contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of $G(X)$. We provide infinitely many examples of such semigroups.