Volume 37, Issue 3

We characterize the square-integrable representations of (connected, simply connected) solvable Lie groups in terms of the generalized orbits of the coadjoint action. We prove that the normal representations corresponding, via the Pukánszky correspondence, to open coadjoint orbits are type I, not necessarily square-integrable representations. We show that the quasi-equivalence classes of type I square-integrable representations are in bijection with the simply connected open coadjoint orbits, and the existence of an open coadjoint orbit guarantees the existence of a compact open subset of the space of primitive ideals of the group. When the nilradical has codimension 1, we prove that the isolated points of the primitive ideal space are always of type I. This is not always true for codimension greater than 2, as shown by specific examples of solvable Lie groups that have dense, but not locally closed, coadjoint orbits.

Consider the chiral non-Hermitian random matrix ensemble with parameters n and v , and let ( ζ i ) 1 ≤ i ≤ n {(\zeta_{i})_{1\leq i\leq n}} be its n eigenvalues with positive x -coordinate. In this paper, we establish deviation probabilities and moderate deviation probabilities for the spectral radius ( n n + v ) 1 2 ⁢ max 1 ≤ i ≤ n ⁡ | ζ i | 2 {(\frac{n}{n+v})^{\frac{1}{2}}\max_{1\leq i\leq n}|\zeta_{i}|^{2}} , as well as ( n n + v ) 1 2 ⁢ min 1 ≤ i ≤ n ⁡ | ζ i | 2 {(\frac{n}{n+v})^{\frac{1}{2}}\min_{1\leq i\leq n}|\zeta_{i}|^{2}} .

We classify nef vector bundles on a smooth hyperquadric of dimension ≥ 4 {\geq 4} with first Chern class two over an algebraically closed field of characteristic zero.

Let p , q , N ≥ 2 {N\geq 2} be three positive integers and let D = { 0 , 1 , … , N - 1 } ⊕ N p ⁢ { 0 , 1 , … , N - 1 } {D=\{0,1,\ldots,N-1\}\oplus N^{p}\{0,1,\ldots,N-1\}} be a product form digit set. It is well known that if q ∤ p {q\nmid p} , then the self-similar measure μ N q , D {\mu_{N^{q},D}} generated by the iterated function system { ( N q ) - 1 ⁢ ( x + d ) } d ∈ D , x ∈ ℝ {\{(N^{q})^{-1}(x+d)\}_{d\in D,x\in\mathbb{R}}} is a spectral measure with a spectrum Λ ⁢ ( N q , C ) = { ∑ i = 0 finite c i ⁢ N q ⁢ i : c i ∈ C } , \Lambda(N^{q},C)=\Bigg{\{}\sum_{i=0}^{\text{finite}}c_{i}N^{qi}:c_{i}\in C% \Bigg{\}}, where C = N q - p - 1 ⁢ D {C=N^{q-p-1}D} . In this paper, based on the properties of cyclic groups in number theory, we give some conditions on real number t under which the scaling set t ⁢ Λ ⁢ ( N q , C ) {t\Lambda(N^{q},C)} is also a spectrum of μ N q , D {\mu_{N^{q},D}} .

Let 𝒜 {\mathcal{A}} be a complex unital Banach algebra and let R ⊆ 𝒜 {R\subseteq\mathcal{A}} be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property, R D {R^{D}} is constructed as an extension of R to axiomatically study the accumulation of σ R ⁢ ( a ) {\sigma_{R}(a)} for any a ∈ 𝒜 {a\in\mathcal{A}} . At last, several illustrative examples on Banach algebra and operator algebra are provided.

In this article, we study the geodesic orbit Randers spaces of the form ( G / H , F ) {(G/H,F)} , such that G is one of the compact classical Lie groups SO ⁢ ( n ) {{\mathrm{S}}{\mathrm{O}}(n)} , SU ⁢ ( n ) {{\mathrm{S}}{\mathrm{U}}(n)} , Sp ⁢ ( n ) {{\mathrm{S}}{\mathrm{p}}(n)} , and H is a diagonally embedded product H 1 × ⋯ × H s {H_{1}\times\cdots\times H_{s}} , where H i {H_{i}} is of the same type as G . Such spaces include spheres, Stiefel manifolds, Grassmann manifolds, and flag manifolds. The present work is a contribution to the study of geodesic orbit Randers spaces ( G / H , F ) {(G/H,F)} with H semisimple. We construct new examples of non-Riemannian Randers g.o. metrics in homogeneous bundles over generalized Stiefel manifolds which are not naturally reductive. Also, we obtain the specific expressions of these Randers g.o. metrics.

Recently, Guo and Tang independently established some q -supercongruences from Rahman’s quadratic transformation. In this paper, by applying the method of creative microscoping devised by Guo and Zudilin together with Rahman’s quadratic transformation again, we provide proofs for eight conjectures on q -supercongruences proposed by Guo and by Tang.

The Absolute Subspace Theorem, a vast generalization and a quantitative improvement of Schmidt’s Subspace Theorem, was first established by Evertse and Schlickewei and then strengthened remarkably by Evertse and Ferretti. We study quantitative generalizations and extensions of subspace theorems in various contexts. We establish a generalization of Evertse and Ferretti’s Absolute Subspace Theorem for hyperplanes in general position. We obtain improved (non-absolute) Quantitative Subspace Theorems for hyperplanes in general position and in subgeneral position. We show a Semi-quantitative Subspace Theorem for hyperplanes in non-subdegenerate position.

Let 𝔤 = 𝔤 0 ¯ ⊕ 𝔤 1 ¯ {{\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}} be a basic classical Lie superalgebra over an algebraically closed field 𝐤 {{\mathbf{k}}} of characteristic p > 2 {p>2} . Denote by 𝒵 {\mathcal{Z}} the center of the universal enveloping algebra U ⁢ ( 𝔤 ) {U({\mathfrak{g}})} . Then 𝒵 {\mathcal{Z}} turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction Frac ⁡ ( 𝒵 ) {\operatorname{Frac}(\mathcal{Z})} is isomorphic to Frac ⁡ ( ℨ ) {\operatorname{Frac}(\mathfrak{Z})} for the center ℨ {\mathfrak{Z}} of U ⁢ ( 𝔤 0 ¯ ) {U({\mathfrak{g}}_{\bar{0}})} . Consequently, both Zassenhaus varieties for 𝔤 {{\mathfrak{g}}} and 𝔤 0 ¯ {{\mathfrak{g}}_{\bar{0}}} are birationally equivalent via a subalgebra 𝒵 ~ ⊂ 𝒵 {\widetilde{\mathcal{Z}}\subset\mathcal{Z}} , and Spec ⁡ ( 𝒵 ) {\operatorname{Spec}(\mathcal{Z})} is rational under the standard hypotheses.

We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L . We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.

In this note, we define and study Asai gamma factors over finite fields. We also prove some results about local Asai L-functions over p-adic fields for level zero representations.

In this paper, we study representations of non-finitely graded Lie algebras 𝒲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} related to Virasoro algebra, where ϵ = ± 1 {\epsilon=\pm 1} . Precisely speaking, we completely classify the free 𝒰 ⁢ ( 𝔥 ) {\mathcal{U}(\mathfrak{h})} -modules of rank one over 𝒲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} , and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules.

This paper investigates the existence and multiplicity of solutions to fractional elliptic systems on conical spaces. Specifically, we focus on the challenges posed by complex geometric configurations, including cones with rough bases, and their implications for the treatment of lateral boundary conditions. By utilizing the fibering map approach and iterative method, we aim to address these challenges and provide new insights into the field. Notably, these issues have not been previously explored in existing literature, highlighting the originality and significance of our study.

The main aim of this paper is to study the arithmetic Bohr radius for holomorphic functions defined on a Reinhardt domain in ℂ n {\mathbb{C}^{n}} with positive real part. The present investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611–2619]. A part of our study in the present paper includes a connection between the classical Bohr radius and the arithmetic Bohr radius of unit ball in the Minkowski space ℓ q n {\ell^{n}_{q}} , 1 ≤ q ≤ ∞ {1\leq q\leq\infty} . Further, we determine the exact value of a Bohr radius in terms of arithmetic Bohr radius.

In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori 𝕋 d {\mathbb{T}^{d}} , where d ≥ 3 {d\geq 3} . The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter, The proof of the l 2 l^{2} decoupling conjecture, Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong, Strichartz estimates for N -body Schrödinger operators with small potential interactions, Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate.