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)

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

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

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

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.

If a complex X is a subcomplex of a diagrammatically reducible (DR) 2-complex Y that has locally indicable fundamental group, then X has locally indicable fundamental group. This is a consequence of the Corson–Trace characterization of diagrammatic reducibility. In this article, we use a Corson–Trace-like characterization of diagrammatic reducibility away from a subcomplex to obtain a considerable stronger result. We apply this to the question of local indicability in the context of Whitehead’s asphericity conjecture. We show that an injective labeled oriented tree that is DR of degree 2, and all its quotients are as well, is locally indicable.

We study the relationship between the ultraproduct of a crossed product C*algebra ${(\mathcal {A}\rtimes _{r}G)^{\omega }}$ and the crossed product of an ultraproduct C*algebra $\mathcal {A}^{\omega }\rtimes _{r}G$ for a fixed free ultrafilter $\omega $ on $\mathbb {N}$. For a general locally compact amenable group G, we show that $\mathcal {A}^{\omega }_{\alpha }\rtimes _{r}G$ embeds into $(\mathcal {A}\rtimes _{r}G)^{\omega }$. This problem was first introduced by Reiji Tomatsu in the framework of von Neumann algebras in Tomatsu (2017, Illinois J. Math., 61, 275–286).

In Chen and Zhang (2022, Adv. Math., 405, 108516), the authors noted that the $2$-Bergman space has an orthonormal basis which can be defined as certain extremal functions. They posed a question of whether similar extremal functions form a Schauder Basis for the p-Bergman space. We provide evidence supporting this conjecture, demonstrating its validity for a selection of complete Reinhardt domains.

We compute extension groups in the category of duals of p-adic Banach space representations of $\mathrm {GL}_2(\mathbb {Q}_p)$. Focusing on representations arising from the p-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the dual p-adic Banach space representations attached to reducible Galois representations and supercuspidal Galois isotypic components of the p-adic étale cohomology of the finite-level Drinfeld spaces.

Let $\mathbb {G}$ be a unipotent algebraic group defined over a p-adic field of characteristic zero. The set of its rational points G is a p-adic Lie group with Lie algebra $ \mathfrak {g}$. Let $\pi $ be an irreducible unitary representation of G in a Hilbert space $\mathcal {H}_{\pi }$, f be a linear form on $ \mathfrak {g}$, and $\mathfrak {h}$ be a subordinate subalgebra to f. Consider $\chi _f$, the character of $H= \exp (\mathfrak {h})$ associated with f. The goal of this article is to describe some of the fine structure of $(\mathcal {H}_{\pi }^{-\infty })^{H,\chi _f}$, the space of $\chi _f$-semi-invariant distributions associated with $\pi $.

The notion of constructible functions in the setting of tame real geometry has been introduced by Cluckers and Miller in their work on parametric integration of globally subanalytic functions. A function on a globally subanalytic set is called constructible if it is a finite sum of finite products of globally subanalytic functions and the logarithm of positive globally subanalytic functions. We show that the class of constructible functions is stable under taking derivatives.

For an appropriate function f on the plane, we study the operator

$$\begin{align*}Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-\phi(t)) \frac{e^{2\pi i \gamma(t)}}{\psi(t)} dt, \end{align*}$$

$\phi $

$\psi $

$\gamma $

$\psi (t)\to 0$

$|\gamma (t)|\to \infty $

$t\to 0$

$\psi $

$\gamma $

$L^2(\mathbb {R}^2)$

$L^p(\mathbb {R}^2)$

$1 < p < \infty $

We provide a symbolic classification of generalized Seifert fiber spaces, which were introduced by Mitsuishi and Yamaguchi in the classification of collapsing Alexandrov $3$-spaces. Additionally, we show that the canonical double branched cover of a non-manifold generalized Seifert fiber space is a Seifert manifold and compute its symbolic invariants in terms of those of the original space.

We study the distribution of large prime factors of a random element u of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1,$ the large prime factors of u tend to a Poisson–Dirichlet process, while if the sequence has any positive level of distribution the correlation functions of large prime factors tend to a Poisson–Dirichlet process against test functions of restricted support. For sequences with positive level of distribution, we also estimate the probability that the largest prime factor of u is greater than $u^{1-\epsilon }$, showing that this probability is $O(\epsilon )$. Examples of sequences described include shifted primes and values of single-variable irreducible polynomials. The proofs involve (i) a characterization of the Poisson–Dirichlet process due to Arratia–Kochman–Miller and (ii) an upper bound sieve.

We study geometric and topological properties of Hessenberg varieties of codimension one in the type A flag variety. Our main results: (1) give a formula for the Poincaré polynomial, (2) characterize when these varieties are irreducible, and (3) show that all are reduced schemes. We prove that the singular locus of any nilpotent codimension one Hessenberg variety is also a Hessenberg variety. A key tool in our analysis is a new result applying to all (type A) Hessenberg varieties without any restriction on codimension, which states that their Poincaré polynomials can be computed by counting the points in the corresponding variety defined over a finite field. The results below were motivated by earlier work of the authors studying the precise relationship between Hessenberg and Schubert varieties, and we obtain a corollary extending the results from that paper to all codimension one (type A) Schubert varieties.

Let $a\ge 1$. Denote by $N_a(n)$ the number of solutions of

$$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$

$a>1,$

$\sum \limits _{a\le n\le x}N_a(n)$

$x\to \infty $

$\sum \limits _{a\le n\le x}nN_a(n)$

$\sum \limits _{a\le n\le x}\frac {1}{n}N_a(n)$

$\sum \limits _{a\le n\le x}\sum \limits _{d\mid n} N_a(d),$

$\sum \limits _{a\le n\le x}\sum \limits _{d\mid n} N_a(d)\,\tau \!\left (\frac {n}{d}\right )$

$\varepsilon>0$

$ N_a(n)<n^{1+\varepsilon } $