In a recent work, Nathanson proved that, for $K=\mathbb {R}$ or $\mathbb {C}$, the set of $B_h$-vectors in $K^n$ is open and dense. Leonetti later extended this to $B_h[g]$-vectors. In this note, we isolate a finite linear functional principle behind these results. Namely, if $\mathcal F$ is a finite family of pairwise distinct linear functionals on $K^n$, then the set of points at which no value is assumed by more than g members of $\mathcal F$ is open and dense in $K^n$. As consequences, we recover the theorems of Nathanson and Leonetti for $B_h[g]$-vectors and obtain the following variant: for every linear form $L(x_1,\ldots ,x_h)=c_1x_1+\cdots +c_hx_h$, with nonzero integer coefficients, the corresponding set of $L[g]$-vectors, evaluated on distinct variables, is open and dense in $K^n$.

In this article, we study how certain homological properties of modules over a commutative Noetherian ring interact with each other. Specifically, we investigate how the finiteness of the homological dimensions of the Ext modules between two finitely generated modules affects the homological dimensions of the modules themselves. One of our main results shows that if all Ext modules up to a suitably chosen bound have finite projective dimension, then one module has finite projective or injective dimension exactly when the other module does.

A Besicovitch–Rado–Kinney (BRK) set in $\mathbb {R}^n$ contains a hypersphere of every radius. In $\mathbb {F}_q^n$, BRK-type sets of degree $\ell $ analogously contain a family of $(n-1)$-dimensional surfaces, parameterized by a dilation factor and determined by a fixed homogeneous polynomial of degree $\ell $. We define $(n,d)$-BRK-type sets of degree $\ell $, which contain a family of d-dimensional sets parameterized by an $(n-d)$-dimensional dilation factor and determined by fixed homogeneous polynomials of degree $\ell $. We use the polynomial method to obtain a lower bound $|S| \gtrsim _{n, \ell } q^n$ on $(n,d)$-BRK-type sets S of degree $\ell $. We obtain an improved lower bound $|S| \geq \frac {(q-1)^n}{(\ell + 1 - 2\ell /q)^n}$ by implementing the method of multiplicities; this is the same bound obtained by Trainor on BRK-type sets of degree $\ell $, and we obtain this bound independently of d.

In this note, we show that the category of Latin (resp. commutative) medial quandles is equivalent to the category of affine modules over a certain Laurent polynomial ring (resp. the dyadic rationals). As applications, we describe free objects in these categories and obtain a structure theorem for finitely generated medial commutative quandles. We also characterize racks whose duals are commutative. Collectively, this solves two open problems of Bardakov and Elhamdadi (2026, Idempotents and powers of ideals in quandle rings).

Let $p\geq 3$ be a prime. The hyper-algebraic elements in the p-adic Mal’cev–Neumann field ${\mathbb{L}} _p$ form an algebraically closed subfield ${\mathbb{L}} _p^{\text {ha}}$. In this article, we clarify the relations among the fields ${\mathbb{L}} _p^{\text {ha}}$, $\overline {{\mathbb{Q}} }_p$, and ${\mathbb{C}} _p$. We introduce two arithmetic invariants (hyper-tame index and hyper-inertia index) of hyper-algebraic elements and study the relation between these invariants and classical arithmetic invariants of p-adic algebraic numbers. Finally, we give a criterion for hyper-algebraic elements to be tamely ramified over ${\mathbb{Q}} _p$.

Let $a,b,$ and n be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of n distinct positive integers. We denote by $[S^a]$ the $n\times n$ matrix having the ath power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$-entry. For $x\in S$, let ${G_{S}(x)=\{d\in S: d < x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}}$. In this article, we show that $[S^a]$ divides $[S^b]$ in the ring of $n\times n$ matrices over the integers if $a|b$ and S is gcd closed (i.e., $\gcd(x_i, x_j)\in S$ for all integers i and j with $1\le i,j\le n$) and satisfies the condition $\mathcal G$ (That is, for any $x\in S$, either $G_S(x)$ contains at most one element, or $G_S(x)$ contains at least two elements and satisfies that ${\rm lcm}(y_1,y_2)=x$ and $\gcd(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$). This confirms a conjecture of Hong proposed in (2026, Bull. Aust. Math. Soc., 113, 231–243).

We investigate horospherical homogeneous spaces – a class of spherical homogeneous spaces encompassing both flag varieties and algebraic tori – over algebraically closed fields of characteristic $p>0$, and establish their complete classification for $p> 2$.

We characterize the nuclearity of Toeplitz operators $T_\mu : F_\alpha ^p \to F_\alpha ^q$ with Borel measure symbols for $1\leq p,q\leq \infty $. For positive measures $\mu $ and $q\leq p$, we provide necessary and sufficient conditions in terms of the Berezin transform and establish a rigidity property for nuclearity across this range. In the case $p<q$, we obtain separate necessary and sufficient conditions, indicating that the Berezin transform alone is insufficient for a complete characterization. Our results extend to Fock spaces on $\mathbb {C}^n$.

Let f and g be two distinct Hecke–Maass cusp forms of weight zero for the full modular group $SL(2,\mathbb {Z})$ with Laplacian eigenvalues $\frac {1}{4}+t_{1}^{2}$ and $\frac {1}{4}+t_{2}^{2}$, and denote by $\lambda _{f}(n)$ and $\lambda _{g}(n)$ the n-th normalized Fourier coefficients attached to f and g, respectively. In the present article, we establish a quantitative result regarding the sign changes of the sequence $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\geqslant 1}$ in the interval $(x,2x]$, supported at a certain primitive integral binary quadratic form with fixed negative discriminant $D<0$, for sufficiently large x.

We introduce “book links” as a generalization of braids in open book decompositions; this new class of objects includes both braids and plats as special cases. We then prove a version of Markov’s theorem in this general setting by extending the theory of open book foliations.

We establish new uncertainty bounds for commutators and anti-commutators of observables in both finite- and infinite-dimensional quantum systems. Beginning with a short, elementary proof of Robertson’s standard uncertainty principle, we then derive inequalities whose lower bounds remain nonzero, regardless of the number of observables. The main result is a novel trace inequality comparing the block matrices of commutators and anti-commutators of several observables.

We introduce a slice-based quaternionic analog of wavelet sets in the right- quaternionic Hilbert space $L^{2}(\mathbb {R}^{d},\mathbb {H})$. Using a slice-quaternionic Fourier transform, we characterize boundedness and Parseval tightness of affine systems generated by $\widehat {\psi }=(2\pi )^{-d/2}\chi _{E}q$ via translation and dilation multiplicity conditions on E. These results extend the classical frame wavelet-set theory of Dai et al. (2003, J. Comput. Appl. Math. 155, 69–82) to the quaternionic setting under a natural slice-phase restriction. Several dyadic, directional, and multiwavelet examples illustrate the framework and provide the first systematic construction of quaternionic wavelet sets.

For $N>1$, we constructed a canonical connected fundamental domain for $\Gamma _0(N)$ in Nie and Parent (2024, Connected fundamental domains for congruence subgroups), utilizing an interesting function $W: {\mathbb Z}/N\to {{\mathbb N}}$. In this article, we further study the function W, prove some identities, and use it to match the cusps, with widths, produced by our connected fundamental domain with the known cusp classes of $\Gamma _0(N)$. Furthermore, we list the boundary arcs and the gluing patterns of our connected fundamental domain, a key step in understanding the modular curve $X_0(N)$ by this approach.

In this article, we study well-posedness of the second-order integro-differential equations ($P_5$): $ \mu * (Mu)"(t) + \nu * {(Mu)}'(t) + \eta *(Mu)(t) = \tau * Au(t) + f(t), \ (t\in {\mathbb T}:=[0,2\pi ])$ with periodic boundary conditions $ Mu(0)=Mu(2\pi ), (Mu)'(0)=(Mu)'(2\pi )$, where $A,M$ are closed operators in a Banach space X such that $D(A)\subset D(M)$, $\mu , \nu , \eta ,$ and $\tau $ are finite scalar measures on ${\mathbb R}$. Using known results on Fourier multipliers, we find necessary and sufficient conditions on the measures $\mu , \nu , \eta , \tau $ and the operators $A,M$ to ensure well-posedness of ($P_5$) in $L^p({\mathbb T}; X)$, periodic Besov spaces $B_{p,q}^s({\mathbb T}; X),$ and Triebel–Lizorkin spaces $F_{p,q}^s({\mathbb T}; X)$.

We formulate an inverse problem for an uncoupled space–time fractional Schrödinger equation on closed manifolds. Our main goal is to determine the fractional powers and the Riemannian metric (up to an isometry) simultaneously from the knowledge of the associated source-to-solution map. Our argument relies on the asymptotic behavior of Mittag-Leffler functions, Weyl’s law for the eigenvalues of the Laplace–Beltrami operator, the unique continuation property of the space-fractional operator and the disjoint data metric determination result for the wave equation. We also provide a probabilistic formulation of our inverse problem.

A solution $(x,y,z) \in \mathbb {Z}^3-\left \{ (0,0,0) \right \}$ to a generalized Fermat equation (1)

$$ \begin{align} A\textsf{x}^{a} + B\textsf{y}^{b} + C\textsf{z}^{c} = 0 \end{align} $$

$\gcd (x,y,z) = 1$

$\chi = \tfrac {1}{a} + \tfrac {1}{b} + \tfrac {1}{c} - 1$

In this article, we develop a systematic calculus for the Duistermaat index, a symplectic invariant defined for triples of Lagrangian subspaces. Introduced nearly half a century ago, this index has lately been the subject of renewed attention, due to its central role in eigenvalue interlacing problems on quantum graphs (and more abstractly for self-adjoint extensions of symmetric operators). Here, we give an axiomatic characterization of the index that leads to elementary proofs of its fundamental properties. We also relate the index to other quantities often appearing in symplectic geometry, such as the Hörmander–Kashiwara–Wall index and the Maslov index. Among other things, this leads to a curious formula for the Morse index of a difference of Hermitian matrices.

We give an example of a unital $\mathrm {C}^*$-algebra $\mathbf {A}$ with a computable presentation and for which neither $K_0(\mathbf {A})$ nor $K_1(\mathbf {A})$ has a computable presentation.

This article is motivated by a conjecture proposed by Sinai Robins in 2024. The conjecture asserts that two convex, centrally symmetric sets of positive measure that are not multi-tilers must coincide up to rigid motions if and only if their Fourier transforms agree on the lattice $\mathbb {Z}^d$. In this article, we disprove the conjecture by constructing explicit counterexamples in dimensions $d \geq 4$.

Denote by $\mathbb F_q$ the finite field of order q and by $\mathbb F_{q^n}$ its extension of degree n. Some $a\in \mathbb F_{q^n}$ is called primitive if it generates the multiplicative group $\mathbb F_{q^n}^*$, and it is called $q^n/q$-normal if its $\mathbb F_q$-conjugates form an $\mathbb F_q$-basis of $\mathbb F_{q^n}$ if the latter is viewed as an $\mathbb F_q$-vector space. Furthermore, some $a\in \mathbb F_{q^n}$ is called $q^n/q$-completely normal if it is $q^n/q^d$-normal for all $d\mid n$. In this work, we prove a new construction of sets of completely normal elements and we establish, under conditions, the existence of elements that are simultaneously primitive and $q^n/q$-completely normal, covering some yet unresolved cases of a 30-year-old conjecture by Morgan and Mullen.