Let $n\ge 1$, $r\ge 0$, and $s\ge 0$ be integers satisfying $4+r+3 s\le 3^{n+1}$. Given linear polynomials $f_{i}(x)=m_{i} x+n_{i}$ for $1 \le i \le r+s$, where the coefficients $m_{i} , n_{i}$ are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants $D>0$ such that the 3-rank of the class group of each quadratic fields $\mathbb {Q}(\sqrt {f_1(D)}), \ldots , \mathbb {Q}(\sqrt {f_r(D)})$ and $\mathbb {Q}(\sqrt {-f_{r+1}(D)}), \ldots , \mathbb {Q}(\sqrt {-f_{r+s}(D)})$ is simultaneously less than n. For a positive integer $k $, let $g_1,\dots ,g_k\in \mathbb {Q}[x]$ be polynomials taking integer values at integers with $g_i(0)=0$. We also prove that there exist positive integers $a,d$, such that each $a+g_i(d)$ is a fundamental discriminant and the 3-rank of the class group of each quadratic field $\mathbb {Q}(\sqrt {a+g_1(d)}), \ldots ,\mathbb {Q}(\sqrt {a+g_k(d)})$ is simultaneously less than n. Moreover, these discriminants can be chosen all positive or all negative, giving either all real or all imaginary quadratic fields $\mathbb {Q}(\sqrt {a+g_i(d)})$.

Let E be an elliptic curve over the finite field $\mathbb {F}_p$, and $P \in E(\mathbb {F}_p)$ be an ${\mathbb F}_p$-rational point. We obtain nontrivial estimates for multiplicative character sums associated with the division polynomials $\psi _n(P)$ twisted by several multiplicative functions, where $\psi _n(P)$ denotes the nth division polynomial evaluated at P.

In this article, we construct countably many isolated circular orders on the free products $G = F_{2n} \ast \mathbb {Z}_{m_1} \ast \cdots \ast \mathbb {Z}_{m_k}$ of cyclic groups. Moreover, we prove that these isolated circular orders are not the automorphic images of the others. By using these isolated circular orders, we also construct countably many isolated left orders on a certain central $\mathbb {Z}$-extension of G, which are not the automorphic images of the others.

A characterization of Muckenhoupt weights in terms of a Harnack-type property is presented along with several applications.

I give a short proof of a recent result due to Kiefer and Ryzhikov showing that a finite irreducible semigroup of $n\times n$ matrices has cardinality at most $3^{n^2}$.

If G is a connected semisimple Lie group with finite center and K is a maximal compact subgroup of G, then the Lie algebra of G admits a Cartan decomposition $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$. This allows us to define the Cartan motion group $H=\mathfrak {p}\rtimes K$. In this article, we study the regularity of K-finite matrix coefficients of unitary representations of H. We prove that the optimal exponent $\kappa (G)$ for which all such coefficients are $\kappa (G)$-Hölder continuous coincides with the optimal regularity of all K-finite coefficients of the group G itself. Our approach relies on stationary phase techniques that were previously employed by the author to study regularity in the setting of $(G,K)$. Furthermore, we provide a general framework to reduce the question of regularity from K-finite coefficients to K-bi-invariant coefficients.

We establish a characterization (under some natural conditions) of those orders in Dedekind domains which allow a transfer homomorphism to a monoid of zero-sum sequences. As a consequence, the inclusion map to the Dedekind domain is a transfer homomorphism, with the exception of a particular case. The arithmetic of Krull and Dedekind domains is well understood, and the existence of a transfer homomorphism implies that the order and the associated Dedekind domain share the same arithmetic properties. This is not the case for arbitrary orders in Dedekind domains.

We study the problem of continuity of derivations over Banach algebras. More specifically, we consider derivations over a class of Banach algebras that contain dense “$C^*$-like” subalgebras. The results we prove are then applied to $L^p$-crossed products and symmetrized $L^p$-crossed products. For example, it follows that every derivation over the $L^p$-crossed product $F^p(G,X,\alpha )$ is continuous, provided that G is infinite, finitely generated, has polynomial growth, and acts freely on the compact Hausdorff space X.

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 $X_G$.

We study Volterra-type operators on Bergman–Morrey spaces. First, we obtain sharp boundedness criteria between scales, identifying the exact Bloch-type regularity required of the symbol and showing that the companion operator is bounded precisely for bounded symbols. Next, we develop the holomorphic optimal domain, prove that it is a Banach space with bounded point evaluations and multiplier algebra $H^\infty $, and establish strict inclusions as the parameters vary. Then, we introduce the meromorphic optimal domain, characterize the constant-symbol case, and show that it is a Banach space with point evaluations bounded off the zero set of the derivative. Finally, we analyze the symbol classes $W_g$ and $V_g^{p}$, prove the structural identity $W_g=T_g(H^\infty )+\mathbb C$, and give equivalence and comparison criteria for optimal domains across symbols.

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.