Given a Fourier transformable measure in two dimensions, we find a formula for the intensity of its Fourier transform along circles. In particular, we obtain a formula for the diffraction measure along a circle in terms of the autocorrelation measure. We then look at some applications of this formula.

Let T be a bounded linear operator on a separable Banach space that satisfies geometric properties similar to those of $\ell ^p,\, p>1$. We prove that the smallest and the largest norm of weak cluster points of all maximizing sequences for T can only take the values $0$ or $1$. The three classes of bounded linear operators emerging from the dichotomy of these extremal norm values coincide with the partition, created by considering the norm-attaining property and if the essential norm equals the norm.

Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with K-theory yields a rich combinatorial theory of inhomogeneous deformations, where Schur functions are replaced by their K-analogs, the symmetric Grothendieck functions. We initiate a theory of the Kromatic symmetric function $\overline {X}_G$, a K-theoretic analog of the chromatic symmetric function $X_G$ of a graph G. The Kromatic symmetric function is a generating series for graph colorings in which vertices receive any nonempty set of colors such that neighboring color sets are disjoint. Our main result lifts a theorem of Gasharov (1996), showing that when G is a claw-free incomparability graph, $\overline {X}_G$ is a positive sum of symmetric Grothendieck functions. This suggests a topological interpretation of Gasharov’s theorem. Kromatic symmetric functions of path graphs are not positive in any of several K-analogs of the e-basis, demonstrating that the Stanley–Stembridge conjecture (1993) does not have such a lift to K-theory and so is unlikely to be amenable to a topological perspective. We define a vertex-weighted extension of $\overline {X}_G$ which admits a deletion–contraction relation. Finally, we give a K-analog for $\overline {X}_G$ of the monomial-basis expansion of $X_G$.

Dilute Temperley–Lieb algebras are variants of Temperley–Lieb algebras arising in statistical mechanics in the study of solvable lattice models. In this article, we prove that the (co)homology of dilute Temperley–Lieb algebras vanishes in all positive degrees.

In 1970, Apostol introduced the Möbius function of order k for an integer $k\ge 1$. In 2001, it was generalized by Bege to the Möbius function $\mu _{k,m}$ of two parameters for any integers $m\ge k\ge 1$. In this article, we will establish three kinds of asymptotic formulas related to $\mu _{k,m}$ for $m\ge k\ge 2$ in a unified elementary method. These formulas are related to the shifted convolution sums of Fourier coefficients of holomorphic cusp forms, Bergelson and Richter’s dynamical generalization of the prime number theorem, and the Titchmarsh divisor problem.

We prove two results, generalizing certain theorems by Jin and Moosa (2020, Trans. Amer. Math. Soc., 373, 4863–4887), on the internality of the system of differential equations

$$ \begin{align*} \begin{cases} \begin{aligned} x' &= f(x)\\ y' &= g(x)y,\\ \end{aligned} \end{cases} \end{align*} $$

Let K be an absolutely abelian number field, and t be a positive integer relatively prime to $[K: \mathbb {Q}]$. We provide a method to use the t-rank of the class group of K to obtain a lower bound on the t-rank of class groups of any cyclotomic fields (or real cyclotomic fields) that contain this K. As an application, for any odd integer $t> 1$, we provide a family of cyclotomic fields whose class groups have t-rank at least $2$. Furthermore, we prove that the class number of $\mathbb {Q}(\zeta _{p^e} + \zeta _{p^e}^{-1})$ cannot be equal to $2$, where p is a prime, $e \geq 1$ is an integer, and $\zeta _{p^e}$ denotes a primitive $p^e$-th root of unity. We also provide an upper bound for the class number of a real cyclotomic field.

We determine the algebraic and transcendental lattices of a general cubic fourfold with a symplectic automorphism of prime order. We prove that cubic fourfolds admitting a symplectic automorphism of order at least three are rational, and we exhibit two families of rational cubic fourfolds that are not equivariantly rational with respect to their group of automorphisms. As an application, we determine the cohomological action of symplectic birational transformations of manifolds of OG10 type that are induced by prime order symplectic automorphisms of cubic fourfolds.

Symmetrically self-similar graphs are an important type of fractal graph. Their Green’s functions satisfy order one iterative functional equations. We show that when the branching number of a generating cell is two, either the graph is a star consisting of finitely many one-sided lines meeting at an origin vertex, in which case the Green’s function is algebraic, or the Green’s function is differentially transcendental over $\mathbb {C}(z)$. The proof strategy relies on a result in a recent preprint of Di Vizio, Fernandes, and Mishna. The result adds evidence to a conjecture of Krön and Teufl about the spectra of the difference Laplacian of this family of graphs.

A pair of Dehn fillings on a compact, orientable 3-manifold M with a torus boundary $\partial M$ is said to be purely cosmetic if the resulting 3-manifolds are orientation-preservingly homeomorphic. In this article, we show that if $\partial M$ is incompressible, then there are only finitely many pairs of purely cosmetic fillings.

Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha $ is equal to $\pi /\sin ((2+\alpha )\pi /p)$ when $-1<\alpha <p-2$. In this article, we provide a proof of this conjecture for $0\leq \alpha \leq \frac {6p^3-29p^2+17p-2+2p\sqrt {6p^2-11p+4}}{(3p-1)^2}$, and this range of $\alpha $ improves the best known result when $\alpha>\frac {1}{47}$ and $\alpha \not =1$.

We show that there are locally compact spaces that can be condensed on separable spaces, but not on compact separable spaces. We also show that for every cardinal $\kappa ,$ there is a locally compact topological group of cardinality $2^\kappa $ that can be condensed on a compact space but not on a compact topological group. These answer some questions of Arhangel’skii and Buzyakova.

We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves with rational components in terms of the combinatorics of the corresponding graphs.

We develop several $\ell ^p$-operator norm inequalities for $k\times k$ block matrices defined on the $\ell ^p$-sum of Banach spaces. Using these inequalities, we obtain p-numerical radius and spectral radius bounds for $k\times k$ block matrices. We deduce a p-numerical radius bound for the Kronecker product $A\otimes B$, where $A\in {M}_k(\mathbb {C})$ is a $k\times k$ complex matrix and $B\in \mathcal {L}(\mathbb {H})$ is a bounded linear operator on a complex Hilbert space $\mathbb {H}$. This improves and extends Holbrook’s bound $w(A\otimes B)\leq w(A)\|B\|.$ If $\|A\|_{\ell ^p}$ and $w_p(A)$ denote the $\ell ^p$-operator norm and p-numerical radius of $A\in {M}_k(\mathbb {C})$, respectively, then it is shown that

$$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$

$\mu _p(A)$

$\ell ^p$

$\frac {1}{2}\|A\|_p= w_p(A)$

It is known that the condition $|\arg f'(z)|<\pi /2$, $|z|<1$, is not sufficient for an analytic function $f(z)=z+a_2z^2+\cdots $ in $|z|<1$, to be starlike with respect to the origin. We look for the largest $\alpha>0$ such that the condition $|\arg f'(z)|<\alpha \pi /2$ in $|z|<1$ is a sufficient condition for f to be a univalent, starlike, convex, or Bazilevic̆ function.

Let K be a genus one two-bridge knot. Let p be a prime number and let ${\mathbb {Z}}_{p}$ denote the ring of p-adic integers. In the spirit of arithmetic topology, we observe that if $p\neq 2$ and p divides (or $p=2$ and $2^3$ divides) the size of the 1st homology group of some odd-th cyclic branched cover of the knot K, then its group $\pi _1(S^3-K)$ admits a liminal $\mathrm { SL}_2{\mathbb {Z}}_p$-character. In addition, we discuss the existence of liminal $\mathrm {SL}_2{\mathbb {Z}}_{p}$-representations and give a remark on a general two-bridge knot. In the course of the argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols.

Given an action by a finite quantum group $\mathbb {G}$ on a von Neumann algebra M, we prove that a number of familiar $W^*$ properties are equivalent for M and the fixed-point algebra $M^{\mathbb {G}}$ (i.e., hold or not simultaneously for the two algebras); these include being hyperfinite, atomic, diffuse, and of type I, $II$, or $III$. Moreover, in all cases, the canonical central projections of M and $M^{\mathbb {G}}$ cutting out the summand with the respective property coincide. The result generalizes its classical-$\mathbb {G}$ analog due to Jones–Takesaki.

We examine “stronger” versions of the Rost nilpotence principle, which hold in almost all cases where Rost nilpotence is known, and show that a direct sum of motives for which these stronger principles hold also satisfy it.

We consider the family of infinite positive Borel measures $\mu $ in the unit disc, with finite degree of contact at the unit circle, and the set $E_{\mu }$ of points in the unit circle for which every neighborhood contains infinite mass of $\mu $. Assuming that $E_{\mu }$ is a Carleson set, we show that all Blaschke sequences are zero sets for the corresponding Dirichlet space $D_{\mu }$.

The Unitary Dual Problem is one of the most important open problems in mathematics: classify the irreducible unitary representations of a given group. It is known for a real reductive Lie group that $A_{\mathfrak {q}}(\lambda )$ modules are unitary and that any unitarizable Harish-Chandra module of strongly regular infinitesimal character is isomorphic to an $A_{\mathfrak {q}}(\lambda )$. Thus, it is of interest to study representations of singular infinitesimal character. For a compact real form and any alcove of the form $w(-\lambda + \underline {A}_\circ ),$ where $\lambda $ is dominant (possibly singular) and $\underline {A}_\circ $ is the dominant fundamental alcove, the signature character of the canonical invariant Hermitian form on the irreducible Verma module of infinitesimal character in that alcove is the “negative” of a Hall–Littlewood polynomial summand at $q=-1$ times a version of the Weyl denominator. (Signature characters for other real forms and alcoves of other forms may also be expressed using Hall–Littlewood polynomial summands.) Such formulas give hope that the Unitary Dual Problem is tractable in the singular case.