Abstract In this article, we study how certain homological properties of modules over a commutative Noetherian ring interact with each other. Specifically, we investigate how the finiteness of the homological dimensions of the Ext modules between two finitely generated modules affects the homological dimensions of the modules themselves. One of our main results shows that if all Ext modules up to a suitably chosen bound have finite projective dimension, then one module has finite projective or injective dimension exactly when the other module does.
Abstract In this article, we study normalized univalent harmonic mappings associated with quasi-subordination. For the class S H q 0 ( S ) $\mathcal {S}_{H_q}^0(\mathcal {S})$ script upper S Subscript upper H Sub Subscript q Superscript 0 Baseline left parenthesis script upper S right parenthesis , we establish the harmonic analog of the Bieberbach conjecture together with the corresponding sharp growth estimate. Under suitable additional assumptions on the quasi-subordination data, we obtain integral mean estimates for this class. We also derive analogous results for several significant subclasses of S H 0 $\mathcal {S}^{0}_{H}$ script upper S Subscript upper H Superscript 0 , including the class of mappings with m -fold symmetric analytic and co-analytic parts, as well as the class of mappings admitting minimal surface lifts. By leveraging the flexibility of quasi-subordination, these findings extend and unify several earlier results in the study of univalent harmonic mappings.
In this article, we reformulate LXYZ’s $L^p$ affine Sobolev inequality chain (including Lutwak–Yang–Zhang’s $L^p$ affine Sobolev inequality and Xiao’s p -affine capacity inequality) in the setting of Grassmann manifolds. For this purpose, the Grassmannian $ p $ -affine capacity is introduced.
Abstract Let ${\mathbb {F}_q}$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod p representations of ${G_q} = \text {GL}_2({{\mathbb {F}_q}})$ to ${G_p} = \text {GL}_2({\mathbb {F}_p})$ . We first study the restrictions of principal series which are obtained by induction from a Borel subgroup ${B_q}$ . We then analyze the restrictions of inductions from an anisotropic torus ${T_q}$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of f . The proofs depend on writing down explicit orbit decompositions of ${G_p} \backslash {G_q} / H,$ where $H = {B_q}$ or ${T_q}$ using the fact that ${G_q}/H$ is an explicit orbit in a certain projective line, along with Mackey theory.
Abstract 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.
Abstract 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.
Abstract 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 $$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$ where $\mu _p(A)$ is a positive real number that involves the $\ell ^p$ -operator norms of the Cartesian decomposition of A . In addition, a complete characterization of the equality case $\frac {1}{2}\|A\|_p= w_p(A)$ is given.
Abstract Let f and g be two distinct Hecke–Maass cusp forms of weight zero for the full modular group S L ( 2 , Z ) $SL(2,\mathbb {Z})$ upper S upper L left parenthesis 2 comma double struck upper Z right parenthesis with Laplacian eigenvalues 1 4 + t 1 2 $\frac {1}{4}+t_{1}^{2}$ one fourth plus t 1 squared and 1 4 + t 2 2 $\frac {1}{4}+t_{2}^{2}$ one fourth plus t 2 squared , and denote by λ f ( n ) $\lambda _{f}(n)$ lamda Subscript f Baseline left parenthesis n right parenthesis and λ g ( n ) $\lambda _{g}(n)$ lamda Subscript g Baseline left parenthesis n right parenthesis the n -th normalized Fourier coefficients attached to f and g , respectively. In the present article, we establish a quantitative result regarding the sign changes of the sequence { λ f ( n ) λ g ( n ) } n ⩾ 1 $\{\lambda _{f}(n)\lambda _{g}(n)\}_{n\geqslant 1}$ left brace lamda Subscript f Baseline left parenthesis n right parenthesis lamda Subscript g Baseline left parenthesis n right parenthesis right brace Subscript n greater than or slanted equals 1 in the interval ( x , 2 x ] $(x,2x]$ left parenthesis x comma 2 x right bracket , supported at a certain primitive integral binary quadratic form with fixed negative discriminant D < 0 $D<0$ upper D less than 0 , for sufficiently large x .
Let double struck upper G $\mathbb {G}$ 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 German g $ \mathfrak {g}$ g . Let pi $\pi $ pi be an irreducible unitary representation of G in a Hilbert space script upper H Subscript pi $\mathcal {H}_{\pi }$ H pi , f be a linear form on German g $ \mathfrak {g}$ g , and German h $\mathfrak {h}$ h be a subordinate subalgebra to f. Consider chi Subscript f $\chi _f$ chi f , the character of upper H equals exp left parenthesis German h right parenthesis $H= \exp (\mathfrak {h})$ H = exp ( h ) associated with f. The goal of this article is to describe some of the fine structure of left parenthesis script upper H Subscript pi Superscript negative infinity Baseline right parenthesis Superscript upper H comma chi Super Subscript f Superscript $(\mathcal {H}_{\pi }{-\infty }){H,\chi _f}$ ( H pi - infinity ) H , chi f , the space of chi Subscript f $\chi _f$ chi f -semi-invariant distributions associated with pi $\pi $ pi .
We compute extension groups in the category of duals of p-adic Banach space representations of GL(2)(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 etale cohomology of the finite-level Drinfeld spaces.
In Chen and Zhang (2022, Adv. Math., 405, 108516), the authors noted that the 2 $2$ 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 investigate groups of rotations of $\mathbb {R}3$ and $\mathbb {R}4$ that act transitively on the unit sphere.
Let a >= 1. Denote by N-a(n) the number of solutions of x(1) + x(2) + center dot center dot center dot + x(n) = ax(1)x(2)center dot...center dot x(n), x(1) >= x(2) >= center dot center dot center dot >= x(n) >= 1. For fixed a > 1, we obtain the asymptotic behavior of & sum;(a <= n <= x) N-a (n) as x -> infinity. We also obtain a symptotics for & sum;(a <= n <= x) n N-a(n), & sum;(a <= n <= x) 1/n N-a (n), & sum;(a <= n <= x )& sum;(d divided by n) N-a(d), and & sum;(a <= n <= x )& sum;(d divided by n) N-a(d) tau (n/d). Furthermore, we prove that for every epsilon > 0 and sufficiently large n, the inequality N-a (n) < n(1+epsilon) holds.
Let T be a bounded linear operator on a separable Banach space that satisfies geometric properties similar to those of l(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.
A characterization of Muckenhoupt weights in terms of a Harnack-type property is presented along with several applications.
Let n >= 1, r >= 0, and s >= 0 be integers satisfying 4 + r + 3s <= 3n+1.Given linear polynomials fi(x) = mix + ni for 1 <= i <= r + s, where the coefficients mi, ni 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 Q(root f1(D)),...,Q(root fr(D)) and Q(root-fr+1(D)),..., Q(root-fr+s(D)) is simultaneously less than n. For a positive integer k, let g1, ... , gk e Q[x] be polynomials taking integer values at integers with gi (0) = 0. We also prove that there exist positive integers a, d, such that each a + gi (d) is a fundamental discriminant \/ \/ and the 3-rank of the class group of each quadratic field Q(root a+g1(d)),...,Q(root a+gk(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 Q(root a + gi (d)).
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.
It is known that the condition | arg f '(z)| 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 Bazilevic function.
