Volume 37, Issue 5

In this paper, we give a description of differential operators on tensor products A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} , where A and B are finitely generated 𝕂 {\mathbb{K}} -algebras. We prove that any differential operator on A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} can be written as a finite sum of D 1 ⊗ D 2 {D_{1}\otimes D_{2}} , where D 1 {D_{1}} and D 2 {D_{2}} are differential operators on A and B , respectively. Moreover, we introduce a series of new invariants, the ( k , s ) {(k,s)} -th Tjurina algebra A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} for an isolated hypersurface singularity ( V , 𝟎 ) = ( V ⁢ ( f ) , 𝟎 ) ⊆ ( ℂ r , 𝟎 ) {(V,\boldsymbol{0})=(V(f),\boldsymbol{0})\subseteq(\mathbb{C}^{r},\boldsymbol{% 0})} . We formulate a sharp upper estimate for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most 1, and we give lower and upper bounds for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most n .

In [H. H. Nguyen and R. Van Peski, Universality for cokernels of random matrix products, Adv. Math. 438 2024, Paper No. 109451], Nguyen and Van Peski raised the question of whether the surjective flag of ℤ p {{\mathbb{Z}}_{p}} -modules modeled by cok ⁢ ( M 1 ⁢ … ⁢ M k ) ↠ ⋯ ↠ cok ⁢ ( M 1 ) \mathrm{cok}(M_{1}\dots M_{k})\twoheadrightarrow\cdots\twoheadrightarrow% \mathrm{cok}(M_{1}) for independent random matrices M 1 , … , M k ∈ Mat n ⁡ ( ℤ p ) M_{1},\dots,M_{k}\in\operatorname{Mat}_{n}({\mathbb{Z}}_{p}) satisfies the Cohen–Lenstra heuristic. We answer the question affirmatively when M 1 , … , M k {M_{1},\dots,M_{k}} follow the Haar measure, and our proof demonstrates how classical ideas in Cohen–Lenstra heuristic adapt naturally to the flag setting. We also prove an analogue for non-square matrices.

In this paper, we obtain the reverse Bakry–Émery-type estimates for a class of hypoelliptic diffusion operator by coupling method. The (right and reverse) Poincaré inequalities and the (right and reverse) logarithmic Sobolev inequalities are presented as consequences of such estimates. Wang–Harnack inequality, Hamilton’s gradient estimate and Liouville property are also presented by the reverse logarithmic Sobolev inequality.

We study the compactness of commutator of a locally integrable function and an oscillatory singular integral operator defined by a real-valued polynomial and a kernel function satisfying the Hölder condition. We show that such a commutator is compact from L p {L^{p}} to L q {L^{q}} if and only if the locally integrable function is the sum of a constant and a function in certain Lebesgue space.

We show that a pair of newforms ( f , g ) (f,g) can be uniquely determined by the product of the central 𝐿-values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two twisted 𝐿-functions, L ⁢ ( 1 / 2 , f × χ ) ⁢ L ⁢ ( 1 / 2 , g × χ ⁢ ψ ) L(1/2,f\times\chi)L(1/2,g\times\chi\psi) , where ( f , g ) (f,g) is a pair of newforms. The average is taken over the primitive Dirichlet characters 𝜒 and 𝜓 of distinct prime moduli.

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers especially when their partial quotients exhibit specific growth rates. For any positive function Φ, the Wang–Wu theorem (2008) comprehensively describes the Hausdorff dimension of the set E 1 ( Φ ) : = { x ∈ [ 0 , 1 ) : a n ( x ) ≥ Φ ( n ) for infinitely many n ∈ N } . \mathcal{E}_{1}(\Phi):=\{x\in[0,1):a_{n}(x)\geq\Phi(n)\ \text{for infinitely many}\ n\in\mathbb{N}\}. Various generalisations of this set exist, such as substituting one partial quotient with the product of consecutive partial quotients in the aforementioned set which has connections with the improvements to Dirichlet’s theorem, and many other sets of similar nature. Establishing the upper bound of the Hausdorff dimension of such sets is significantly easier than proving the lower bound. In this paper, we present a unified approach to get an optimal lower bound for many known setups, including results by Wang–Wu [ Adv. Math. (2008)], Huang–Wu–Xu [ Israel J. Math. (2020)], Tan–Zhou [ Nonlinearity (2023)], and several others. We also provide a new theorem derived as an application of our main result. We do this by finding an exact Hausdorff dimension of the set S m ⁢ ( A 0 , … , A m − 1 ) ⁢ = def ⁢ { x ∈ [ 0 , 1 ) : c i ⁢ A i n ≤ a n + i ⁢ ( x ) < 2 ⁢ c i ⁢ A i n ,  0 ≤ i ≤ m − 1 , for infinitely many ⁢ n ∈ N } , S_{m}(A_{0},\ldots,A_{m-1})\overset{\mathrm{def}}{=}\{x\in[0,1):c_{i}A_{i}^{n}\leq a_{n+i}(x)<2c_{i}A_{i}^{n},\,0\leq i\leq m-1,\,\text{for infinitely many}\ n\in\mathbb{N}\}, where each partial quotient grows exponentially and the base is given by a parameter A i > 1 A_{i}>1 . For proper choices of A i A_{i} , this set serves as a subset for sets under consideration, providing an optimal lower bound of Hausdorff dimension in all of them. The crux of the proof lies in introducing multiple probability measures consistently distributed over the Cantor-type subset of S m ⁢ ( A 0 , … , A m − 1 ) S_{m}(A_{0},\ldots,A_{m-1}) .

In the first part of the paper, we obtain a necessary and sufficient condition on a complex-valued function 𝐹 on R 2 \mathbb{R}^{2} such that F ⁢ ( f ) ∈ M θ r , s ⁢ ( R n ) F(f)\in{\boldsymbol{M}}^{r{,}s}_{\theta}(\mathbb{R}^{n}) for all f ∈ M θ r , 1 ⁢ ( R n ) f\in{\boldsymbol{M}}^{r{,}1}_{\theta}(\mathbb{R}^{n}) , 1 ≤ r < ∞ 1\leq r<\infty , 1 ≤ s < 2 1\leq s<2 , where M θ r , s ⁢ ( R n ) {\boldsymbol{M}}^{r{,}s}_{\theta}(\mathbb{R}^{n}) is the affine modulation space associated with the fractional Fourier transform. In this context, we obtain that if 𝐹 acts on M θ r , 1 ⁢ ( R n ) {\boldsymbol{M}}^{r{,}1}_{\theta}(\mathbb{R}^{n}) by composition, then 𝐹 is necessarily real analytic and F ⁢ ( 0 ) = 0 F(0)=0 . As M θ r , 1 ⁢ ( R n ) {\boldsymbol{M}}^{r{,}1}_{\theta}(\mathbb{R}^{n}) is neither a Banach algebra nor closed under complex conjugation, we prove sufficiency of the condition for a subclass of M θ r , 1 ⁢ ( R n ) {\boldsymbol{M}}^{r{,}1}_{\theta}(\mathbb{R}^{n}) under some additional hypothesis on 𝐹. We provide an example to show that we cannot drop the additional hypothesis. Thus we characterize the functions that operate in the affine modulation space setting. In the second part, we study the factorization problem in the context of M θ r , s ⁢ ( R n ) {\boldsymbol{M}}^{r{,}s}_{\theta}(\mathbb{R}^{n}) . We obtain factorization of M θ r , s ⁢ ( R 2 ⁢ n ) {\boldsymbol{M}}^{r{,}s}_{\theta}(\mathbb{R}^{2n}) , viewing it as an L 1 ⁢ ( R 2 ⁢ n ) L^{1}(\mathbb{R}^{2n}) module with respect to convolution and 𝜆-twisted convolution associated with the fractional Fourier transform. The problems mentioned above originated in the work of Helson, Kahane, Katznelson, Rudin, Cohen, and Hewitt respectively.

From a complex perspective, Birch and Bombieri (1985) proposed the study of an important class of n -dimensional exponential sums for n = 4 {n=4} , which was then generalized by Katz (1987) to general positive integer n . Our previous paper (2022) enhanced the preceding work from a complex point of view and expanded the subject to a p -adic point of view for any positive integer n . In this paper, we investigate the algebraic degree of the above exponential sum as an algebraic integer.

In this paper, employing some identities due to Newman, we present a new method for discovering infinite families of congruences and strange congruences for c ⁢ ( n ) {c(n)} which is defined by ∑ n = 0 ∞ c ⁢ ( n ) ⁢ q n = ∏ k = 1 ∞ ( 1 - q k ) r ⁢ ( 1 - q k ⁢ t ) s . \sum_{n=0}^{\infty}c(n)q^{n}=\prod_{k=1}^{\infty}(1-q^{k})^{r}(1-q^{kt})^{s}. Here r , s , t {r,s,t} are some integers with t > 1 {t>1} . By our method, in order to discover infinite families of congruences modulo M for c ⁢ ( n ) {c(n)} , it suffices to compute the ranks of ( a , b ) {(a,b)} -Fibonacci sequences modulo M . As applications, we establish many infinite families of congruences and strange congruences for some Ramanujan’s mock theta functions and certain partition functions, such as, a partition function related to mock theta function ω ⁢ ( q ) {\omega(q)} and broken k -diamond partitions.

In this paper, we study long time dynamics of radial threshold solutions for the focusing, generalized energy-critical Hartree equation and classify all radial threshold solutions. The main arguments are the spectral theory of the linearized operator, the modulational analysis and the concentration compactness rigidity argument developed by T. Duyckaerts and F. Merle to classify all threshold solutions for the energy critical NLS and NLW in [T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 2009, 6, 1787–1840, T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP 2008 2008, Art ID rpn002], later by D. Li and X. Zhang in [D. Li and X. Zhang, Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions, J. Funct. Anal. 256 2009, 6, 1928–1961, D. Li and X. Zhang, Dynamics for the energy critical nonlinear wave equation in high dimensions, Trans. Amer. Math. Soc. 363 2011, 3, 1137–1160] in higher dimensions. The new ingredient here is to solve the nondegeneracy of positive bubble solutions with nonlocal structure in H ˙ 1 ⁢ ( ℝ N ) {\dot{H}^{1}(\mathbb{R}^{N})} (i.e. the spectral assumption in [C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, Forum Math. 27 2015, 1, 373–447]) by the nondegeneracy result of positive bubble solution in L ∞ ⁢ ( ℝ N ) {L^{\infty}(\mathbb{R}^{N})} in [X. Li, C. Liu, X. Tang and G. Xu, Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations, preprint 2023, https://arxiv.org/abs/2304.04139] and the Moser iteration method in [S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of ℝ n \mathbb{R}^{n} , Appunti. Sc. Norm. Super. Pisa (N. S.) 15, Edizioni della Normale, Pisa, 2017], which is related to the spectral analysis of the linearized operator with nonlocal structure, and plays a key role in the construction of the special threshold solutions, and the classification of all threshold solutions.

We show that certain modular induced representations of GL 2 ⁢ ( F q ) \mathrm{GL}_{2}(\mathbb{F}_{q}) can be written as cokernels of operators acting on symmetric power representations of GL 2 ⁢ ( F q ) \mathrm{GL}_{2}(\mathbb{F}_{q}) . When the induction is from the Borel subgroup or the anisotropic torus, the operators involve multiplication by newly defined twisted Dickson polynomials or twisted Serre operators, respectively. Our isomorphisms are explicitly defined using differential operators. As a corollary, we improve some periodicity results for quotients in the theta filtration.

Let G be a finite group of order divisible by a prime p and let P ∈ Syl p ⁡ ( G ) {P\in{\operatorname{Syl}}_{p}(G)} . We prove a recent conjecture by Hung stating that | Irr p ′ ⁡ ( G ) | ≥ exp ⁡ ( P / P ′ ) - 1 p - 1 + 2 ⁢ p - 1 - 1 {|{\operatorname{Irr}}_{p^{\prime}}(G)|\geq\frac{\exp(P/P^{\prime})-1}{p-1}+2% \sqrt{p-1}-1} . Let a ≥ 2 {a\geq 2} be an integer and suppose that p a {p^{a}} does not exceed the exponent of the center of P . We also show that the number of conjugacy classes of elements of G for which p a {p^{a}} is the exact p -part of their order is at least p a - 1 {p^{a-1}} .

In this article, we consider the following problem: - Δ 𝔹 N ⁢ u - λ ⁢ u = | u | p - 1 ⁢ u , u ∈ H 1 ⁢ ( 𝔹 N ) , -\Delta_{\mathbb{B}^{N}}u-\lambda u=|u|^{p-1}u,\quad u\in H^{1}(\mathbb{B}^{N}), where 𝔹 N {\mathbb{B}^{N}} represents the Poincaré ball model of the hyperbolic space, 1 < p < 2 * - 1 = N + 2 N - 2 {1<p<2^{*}-1=\frac{N+2}{N-2}} , λ < ( N - 1 ) 2 4 {\lambda<\frac{(N-1)^{2}}{4}} , N ≥ 4 {N\geq 4} , and H 1 ⁢ ( 𝔹 N ) {H^{1}(\mathbb{B}^{N})} denotes the Sobolev space on 𝔹 N {\mathbb{B}^{N}} . In [D. Ganguly and S. Kunnath, Sign changing solutions of the Brezis–Nirenberg problem in the hyperbolic space, Calc. Var. Partial Differential Equations 50 2014, 1–2, 69–91], Ganguly and Sandeep establish the existence of radial sign-changing solutions for the critical case. In this work, we will discuss the existence and multiplicity of non-radial sign-changing solutions for the sub-critical problem. To establish the presence of sign-changing solutions that are not radial, we will demonstrate the existence of an isometric group action on 𝔹 N {\mathbb{B}^{N}} and operate inside a permissible space of functions that change signs. We shall also discuss the multiplicity of sign-changing solutions.

The goal of this paper is to give a characterization of the dual of the space of polyanalytic functions on the complement of a compact set K and vanishing at infinity. The class of polyanalytic functions generalizes holomorphic functions and serves as a middle ground between holomorphic functions in one complex variable and those in two complex variables. The duality result is also expressed in topological terms through a new class of infinite-order differential operators, which includes well-known families of operators like the Laplace and Helmholtz operators. Since the notion of polyanalytic function generalizes that of holomorphic function, the duality theorems established in this paper can be considered a non-trivial generalization of the Köthe–Grothendieck theorem.

Let ℓ ≥ 5 {\ell\geq 5} be a prime number and let 𝔽 ℓ {\mathbb{F}_{\ell}} denote the finite field with ℓ {\ell} elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to GL 2 ⁡ ( 𝔽 ℓ ) {\operatorname{GL}_{2}(\mathbb{F}_{\ell})} and absolute discriminant bounded above by X is asymptotically at least X ℓ / ( 12 ⁢ ( ℓ - 1 ) ⁢ # ⁢ GL 2 ⁡ ( 𝔽 ℓ ) ) log ⁡ X {\frac{X^{{\ell}/({12(\ell-1)\#\operatorname{GL}_{2}(\mathbb{F}_{\ell})})}}{% \log X}} . We also obtain a similar result for the number of surjective homomorphisms ρ : Gal ⁡ ( ℚ ¯ / ℚ ) → GL 2 ⁡ ( 𝔽 ℓ ) {\rho:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{% GL}_{2}(\mathbb{F}_{\ell})} ordered by the prime to ℓ {\ell} part of the Artin conductor of ρ.

The existence of polarized cylinders in normal projective varieties corresponds to admitting non-trivial 𝔾 a {\mathbb{G}_{a}} -action on their affine cones defined by the polarized pairs. Note that a criterion for polarized cylinders in smooth del Pezzo surfaces is studied. Hence, we observe the existing condition of polarized cylinders in del Pezzo surfaces with at worst Du Val singularities. Let S be a del Pezzo surface with at worst Du Val singularities of degree ≥ 3 {\geq 3} . We then construct an H -polar cylinder for any ample ℚ {\mathbb{Q}} -divisor H on S .