In this paper, we give a complete description of closed ideals of the Banach algebra $\mathcal {B}^{s}_{p}\cap \lambda _{\alpha }$ , where $\mathcal {B}^{s}_{p}$ denotes the analytic Besov space and $\lambda _{\alpha }$ is the separable analytic Lipschitz space. Our result extends several previous results in Bahajji-El Idrissi and El-Fallah (2020, Studia Mathematica 255, 209–217), Bouya (2009, Canadian Journal of Mathematics 61, 282–298), and Shirokov (1982, Izv. Ross. Akad. Nauk Ser. Mat. 46, 1316–1332).

Let G be a simple algebraic group of adjoint type over an algebraically closed field k of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs $(M,{\mathcal O})$ where M is the identity component of the centralizer of a semisimple element in G and ${\mathcal O}$ is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.

In this article, we give a representation of bounded complex linear operators that preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is, moreover, positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group antihomomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realized by the idempotent preserving operators.

Let $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$ , where n is a positive integer and $F_n$ denotes the nth Fibonacci number. Leonetti and Sanna proved that $\mathcal {A}$ has natural density equal to zero, and asked for a more precise upper bound. We prove that

$$ \begin{align*} \#\big(\mathcal{A} \cap [1, x]\big) \ll \frac{x \log \log \log x}{\log \log x} \end{align*} $$

We study a 3D ternary system which combines an interface energy with a long-range interaction term. Several such systems were derived as a sharp-interface limit of the Nakazawa–Ohta density functional theory of triblock copolymers. Both the binary case in 2D and 3D, and the ternary case in 2D, are quite well understood, whereas very little is known about the ternary case in 3D. In particular, it is even unclear whether minimizers are made of finitely many components. In this paper, we provide a positive answer to this, by proving that the number of components in a minimizer is bounded from above by a computable quantity depending only on the total masses and the interaction coefficients. There are two key difficulties, namely, the impossibility to decouple the long-range interaction from the perimeter term, and the absence of a quantitative isoperimetric inequality with two mass constraints in 3D. Therefore, the actual shape of minimizers is unknown, even for small masses, making the construction of suitable competing configurations significantly more delicate.

Let $\{R_{k}\}_{k=1}^{\infty }$ be a sequence of expanding integer matrices in $M_{n}(\mathbb {Z})$ , and let $\{D_{k}\}_{k=1}^{\infty }$ be a sequence of finite digit sets with integer vectors in ${\mathbb Z}^{n}$ . In this paper, we prove that under certain conditions in terms of $(R_{k},D_{k})$ for $k\ge 1$ , the Moran measure

$$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$

$(R,D)$

A subset ${\mathcal D}$ of a domain $\Omega \subset {\mathbb C}^d$ is determining for an analytic function $f:\Omega \to \overline {{\mathbb D}}$ if whenever an analytic function $g:\Omega \rightarrow \overline {{\mathbb D}}$ coincides with f on ${\mathcal D}$, equals to f on whole $\Omega $. This note finds several sufficient conditions for a subset of the symmetrized bidisk to be determining. For any $N\geq 1$, a set consisting of $N^2-N+1$ many points is constructed which is determining for any rational inner function with a degree constraint. We also investigate when the intersection of the symmetrized bidisk intersected with some special algebraic varieties can be determining for rational inner functions.

We investigate the leading digit distribution of the kth largest prime factor of n (for each fixed $k=1,2,3,\dots $ ) as well as the sum of all prime factors of n. In each case, we find that the leading digits are distributed according to Benford’s law. Moreover, Benford behavior emerges simultaneously with equidistribution in arithmetic progressions uniformly to small moduli.

We study the boundedness and compactness of weighted composition operators acting on weighted Bergman spaces and weighted Dirichlet spaces by using the corresponding Carleson measures. We give an estimate for the norm and the essential norm of weighted composition operators between weighted Bergman spaces as well as the composition operators between weighted Hilbert spaces.

In Gauthier, Manolaki, and Nestoridis (2021, Advances in Mathematics 381, 107649), in order to correct a false Mergelyan-type statement given in Gamelin and Garnett (1969, Transactions of the American Mathematical Society 143, 187–200) on uniform approximation on compact sets K in $\mathbb C^d$ , the authors introduced a natural function algebra $A_D(K)$ which is smaller than the classical one $A(K)$ . In the present paper, we investigate when these two algebras coincide and compare them with the classes of all plausibly approximable functions by polynomials or rational functions or functions holomorphic on open sets containing the compact set K. Finally, we introduce a notion of O-hull of K and strengthen known results.

We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables:

1. (1) The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$.

2. (2) There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by

   $$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$
   where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$.

3. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.

Let $\alpha < \omega _1$ be a prime component, and let $X$ and $Y$ be metric spaces. In [8], it was shown that if $C_p(X)$ and $C_p(Y)$ are linearly homeomorphic, then the scattered heights $\kappa (X)$ and $\kappa (Y)$ of $X$ and $Y$ satisfy $\kappa (X) \leq \alpha $ if and only if $\kappa (Y) \leq \alpha $. We will prove that this also holds if $C_p^*(X)$ and $C_p^*(Y)$ are linearly homeomorphic and that these results do not hold for arbitrary Tychonov spaces. We will also prove that if $C_p^*(X)$ and $C_p^*(Y)$ are linearly homeomorphic, then $\kappa (X) < \alpha $ if and only if $\kappa (Y) < \alpha $, which was shown in [9] for $\alpha = \omega $. This last statement is not always true for linearly homeomorphic $C_p(X)$ and $C_p(Y)$. We will show that if $\alpha = \omega ^{\mu }$ where $\mu < \omega _1$ is a successor ordinal, it is true, but for all other prime components, this is not the case. Finally, we will prove that if $C_p^*(X)$ and $C_p^*(Y)$ are linearly homeomorphic, then $X$ is scattered if and only if $Y$ is scattered. This result does not directly follow from the above results. We will clarify why the results for linearly homeomorphic spaces $C_p^*(X)$ and $C_p^*(Y)$ do require a different and more complex approach than the one that was used for linearly homeomorphic spaces $C_p(X)$ and $C_p(Y)$.

We improve to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb {F}_q$ -rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we slice the given hypersurface with a random plane.

Let $ \mathcal {B} $ be the class of analytic functions $ f $ in the unit disk $ \mathbb {D}=\{z\in \mathbb {C} : |z|<1\} $ such that $ |f(z)|<1 $ for all $ z\in \mathbb {D} $. If $ f\in \mathcal {B} $ of the form $ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then $ \sum _{n=0}^{\infty }|a_nz^n|\leq 1 $ for $ |z|=r\leq 1/3 $ and $ 1/3 $ cannot be improved. This inequality is called Bohr inequality and the quantity $ 1/3 $ is called Bohr radius. If $ f\in \mathcal {B} $ of the form $ f(z)=\sum _{n=0}^{\infty }a_nz^n $, then $ |\sum _{n=0}^{N}a_nz^n|<1\;\; \mbox {for}\;\; |z|<{1}/{2} $ and the radius $ 1/2 $ is the best possible for the class $ \mathcal {B} $. This inequality is called Bohr–Rogosinski inequality and the corresponding radius is called Bohr–Rogosinski radius. Let $ \mathcal {H} $ be the class of all complex-valued harmonic functions $ f=h+\bar {g} $ defined on the unit disk $ \mathbb {D} $, where $ h $ and $ g $ are analytic in $ \mathbb {D} $ with the normalization $ h(0)=h^{\prime }(0)-1=0 $ and $ g(0)=0 $. Let $ \mathcal {H}_0=\{f=h+\bar {g}\in \mathcal {H} : g^{\prime }(0)=0\}. $ For $ \alpha \geq 0 $ and $ 0\leq \beta <1 $, let

$$ \begin{align*} \mathcal{W}^{0}_{\mathcal{H}}(\alpha, \beta)=\{f=h+\overline{g}\in\mathcal{H}_{0} : \mathrm{Re}\left(h^{\prime}(z)+\alpha zh^{\prime\prime}(z)-\beta\right)>|g^{\prime}(z)+\alpha zg^{\prime\prime}(z)|,\;\; z\in\mathbb{D}\} \end{align*} $$

be a class of close-to-convex harmonic mappings in $ \mathbb {D} $. In this paper, we prove the sharp Bohr–Rogosinski radius for the class $ \mathcal {W}^{0}_{\mathcal {H}}(\alpha , \beta ) $.

In this article, we provide an explicit upper bound for $h_K \mathcal {R}_K d_K^{-1/2}$ which depends on an effective constant in the error term of the Ideal Theorem.

Let F be a field of characteristic zero, and let $UT_2$ be the algebra of $2 \times 2$ upper triangular matrices over F. In a previous paper by Centrone and Yasumura, the authors give a description of the action of Taft’s algebras $H_m$ on $UT_2$ and its $H_m$ -identities. In this paper, we give a complete description of the space of multilinear $H_m$ -identities in the language of Young diagrams through the representation theory of the hyperoctahedral group. We finally prove that the variety of $H_m$ -module algebras generated by $UT_2$ has the Specht property, i.e., every $T^{H_m}$ -ideal containing the $H_m$ -identities of $UT_2$ is finitely based.

Given an open, bounded set $\Omega $ in $\mathbb {R}^N$, we consider the minimization of the anisotropic Cheeger constant $h_K(\Omega )$ with respect to the anisotropy K, under a volume constraint on the associated unit ball. In the planar case, under the assumption that K is a convex, centrally symmetric body, we prove the existence of a minimizer. Moreover, if $\Omega $ is a ball, we show that the optimal anisotropy K is not a ball and that, among all regular polygons, the square provides the minimal value.

Given an algebra R and G a finite group of automorphisms of R, there is a natural map $\eta _{R,G}:R\#G \to \mathrm {End}_{R^G} R$ , called the Auslander map. A theorem of Auslander shows that $\eta _{R,G}$ is an isomorphism when $R=\mathbb {C}[V]$ and G is a finite group acting linearly and without reflections on the finite-dimensional vector space V. The work of Mori–Ueyama and Bao–He–Zhang has encouraged the study of this theorem in the context of Artin–Schelter regular algebras. We initiate a study of Auslander’s result in the setting of nonconnected graded Calabi–Yau algebras. When R is a preprojective algebra of type A and G is a finite subgroup of $D_n$ acting on R by automorphism, our main result shows that $\eta _{R,G}$ is an isomorphism if and only if G does not contain all of the reflections through a vertex.

In 2005, N. Nikolski proved among other things that for any $r\in (0,1)$ and any $K\geq 1$, the condition number $CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $ of any invertible n-dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in $\left \{ r\leq |z|<1\right \}$, satisfies the inequality $CN(T)\leq CK(T)\Vert T \Vert n/r^{n}$ where $K(T)$ denotes the Kreiss constant of T and $C>0$ is an absolute constant. He also proved that for $r\ll 1/n,$ the latter bound is asymptotically sharp as $n\rightarrow \infty $. In this note, we prove that this bound is actually achieved by a family of explicit $n\times n$ Toeplitz matrices with arbitrary singleton spectrum $\{\lambda \}\subset \mathbb {D}\setminus \{0\}$ and uniformly bounded Kreiss constant. Independently, we exhibit a sequence of Jordan blocks with Kreiss constants tending to $\infty $ showing that Nikolski’s inequality is still asymptotically sharp as K and n go to $\infty $.

In this paper, we prove a noncommutative (nc) analog of Schwarz lemma for the nc Schur–Agler class and prove that the regular nc Schur–Agler class and the regular free Herglotz–Agler class are homeomorphic. Moreover, we give a characterization of regular free Herglotz–Agler functions. As an application, we will show that any regular free Herglotz–Agler functions can uniformly be approximated by regular Herglotz–Agler free polynomials.