In the spirit of Blomer’s work on Hecke eigenvalues along monic quadratic polynomials, we investigate power sums of Hecke eigenvalues over values of locally determined quadratic forms.
The algebraic group structure of stabilizers of points in Schubert varieties is investigated. We show that these stabilizers are always strongly solvable algebraic groups and discuss their connectedness properties. Furthermore, we provide a complete characterization of horospherical Schubert varieties and prove that every horospherical Schubert variety is nonsingular. We also introduce a new class of spherical varieties, which we call doubly-spherical varieties, and show that all nearly toric Schubert varieties belong to this family.
Baum, Fulton and MacPherson have proved Riemann–Roch for singular varieties, using localized Chern character. We also get another localized Chern character by using Fulton–MacPherson’s bivariant theory. In this note we give a quick survey on a general case dealing with more general multiplicative cohomology theories in bivariant theory, and then we compare these two localized Chern characters. We also give a general question about characteristic classes of singular varieties and a bivariant-theoretical question motivated by the comparison of localized Chern characters.
We show that the cone of non-decreasing functions in a weighted Lebesgue space is generated by the integration operator. This representation of the cone allows one to apply all the techniques for estimating operators in Lebesgue spaces to estimating these operators on the cone. The absence of a transition to associated spaces allows one to consider operators from the cone into quasi-normed spaces.
An interesting extension of the notion of affine diameters is the one given by Sancho de San Roman (Rev. Mat. Hisp.-Amer. 16, 151–171 (1956)) the triangular width. The triangular width of K at the point x in the boundary of K is the area of the triangle with maximal area inscribed in K and with x as one of its vertices. In the same article, Sancho de San Roman also proved that if K is a convex body with constant triangular width and the locus of the centroids of all the maximal triangles is a point, then K must be an ellipse. In this article, we give some results related to the notion of triangular width. As an application of the characterization of the ellipse by Sancho de San Roman we prove another characterization of the ellipse: if a convex floating body of a smooth convex body K is a translated copy of -1/2K then K is an ellipse.
We propose an analog of the Satake–Baily–Borel compactification and Borel’s extension theorem for arbitrary period maps. The proposed analog is constructed as a proper topological completion of the period map. It is conjectured that the construction is projective algebraic, and the conjecture is reduced to a certain extension problem.
Abstract For a fixed element $$g\in \textrm{SL}\hspace{0.55542pt}(2,\mathbb {C})$$ g ∈ SL ( 2 , C ) and a word $$w=[x^n\!,y^m]$$ w = [ x n , y m ] we consider the automorphism group $$\textrm{Aut}\hspace{0.55542pt}(S_{g})$$ Aut ( S g ) of the affine threefold $$S_{g}=\{(x,y)\in \textrm{SL}\hspace{0.55542pt}(2,\mathbb {C})^2 \,{ |}\, w(x,y)=g\}$$ S g = { ( x , y ) ∈ SL ( 2 , C ) 2 | w ( x , y ) = g } . We prove that the Makar-Limanov invariant $$\textrm{ML}(S_{g})=\mathscr {O}(S_{g})$$ ML ( S g ) = O ( S g ) and $$\textrm{Aut}\hspace{0.55542pt}(S_{g})$$ Aut ( S g ) is Jordan.
Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights κ _1 and κ _2 for the full modular group Γ =SL(2,ℤ) , and let λ _sym^if×sym^jg(n) be the n-th normalized coefficient of the Dirichlet expansion of the Rankin–Selberg L-function L(sym^if×sym^jg,s) associated to f and g. In this paper, we mainly investigate the averages of shifted convolution sums related to the general divisor problem of λ _sym^if×sym^jg(n) , for any given positive integers i,j⩾ 1 . Similar results can also be obtained in the setting of the Hecke–Maass cusp forms on Γ . These results extend the previous results in this direction.
Let G be a plane bipartite graph admitting a perfect matching, and let R(G) denote its resonance graph. It is known that any connected resonance graph can be isometrically embedded into hypercubes as a finite distributive lattice. Let n_0(G) denote the number of finite faces of G without forbidden edges on their peripheries. We show that any connected R(G) has isometric dimension at least n_0(G) , and the lower bound is attained if and only if G is a plane weakly elementary bipartite graph such that the infinite face of each nontrivial elementary component of G is forcing. We also design an algorithm to produce a binary coding on the vertex set of R(G) which induces an isometric embedding of R(G) as a finite distributive lattice into a hypercube of dimension n_0(G) .
We study the central extensions of the infinite-dimensional simple n-Lie algebra defined by Filippov, known as the Jacobian algebra. We first show the existence of nontrivial central extensions. Then, we classify all central extensions by computing the second cohomology group.
Using representations of affine Lie algebras, we describe line bundles on a broad class of contractions of $$\overline{\textrm{M}}_{0,n}$$ M ¯ 0 , n , the moduli space of stable n -pointed rational curves, and show a variant of the cone and contraction theorem for these morphisms. These include the celebrated constructions of Kapranov, Keel, and Knudsen. Our main result suggests that while many so-called F-curves are not $$K_X$$ K X -negative, they exhibit behavior similar to $$K_X$$ K X -negative curves. This reveals a distinguished property of Knudsen’s construction $$f_{\textrm{Knu}}:\overline{\textrm{M}}_{0,n}\rightarrow \overline{\textrm{M}}_{0,n-1}\hspace{1.111pt}{\times }_{\overline{\textrm{M}}_{0,n-2}}\hspace{1.111pt}\overline{\textrm{M}}_{0,n-1}$$ f Knu : M ¯ 0 , n → M ¯ 0 , n - 1 × M ¯ 0 , n - 2 M ¯ 0 , n - 1 , allowing for the classification of all possible factorizations of $$f_{\textrm{Knu}}$$ f Knu , as well as further applications.
A Ψ -space Ψ (𝒜) is a Tychonoff space of the form ω∪𝒜 , where ω is a countable open discrete subspace, 𝒜 is an almost disjoint family in ω which becomes an uncountable closed discrete subspace in Ψ (𝒜) topologized by the finest topology which makes ω open and discrete and each A ∈𝒜 is a sequence in ω converging to the point A ∈Ψ (𝒜) . The modification of a Ψ -space replacing the points of ω with certain “building blocks” is called fat Ψ -space. Recently, Bonanzinga and Giacopello introduced a “ machine” that associates with a zero-dimensional space having a “bad” family of open sets another zero-dimensional space having an open cover exhibiting similarly “bad” properties, while preserving some of the “good” characteristics of the original space. The construction of the machine is a specific type of fat Ψ -space construction. In this paper, we prove that the machine offers a simple and efficient method to distinguish several well-known covering properties and give a partial negative answer to a question posed in 2000 by Bonanzinga and Matveev.
In a bounded connected open set Ω , we obtain an interior a priori estimate for solutions of certain elliptic differential equations, with norms in a BMO space related to weights in a “local" Muckenhoupt’s class.
We study topological spaces with open countable tightness introduced by Arhangel’skii. Two characterizations of such spaces are given. Using these characterizations we show that if a topological group G has open countable tightness, then so has its completion and any quotient. The direct locally convex sum of an uncountable family of locally convex spaces does not have open countable tightness. Let Y be a non-trivial metrizable abelian group. It is proved that for every Y-Tychonoff space X, the space C_p(X,Y) has open countable tightness. Generalizing a result of M. Sakai it is shown that if X is a strongly Y-Tychonoff space, then the space C_k(X,Y) has open countable tightness iff every moving off family of compact subsets of X has a countable subfamily which is strongly moving off.
For a finite group G , let ν _p(G) be the number of Sylow p -subgroups of G , let σ _p(G) be their common order, and γ (G)=∫ _0^1∑ _p∈π (G)ν _p(G)x^σ _p(G) dx =∑ _p∈π (G)ν _p(G)/σ _p(G)+1. A conjecture attributed to Anabanti and Asboei asserts that, for every finite nonsolvable group G , the equality γ (G)=9/2 holds if and only if G≅ A_5 . We disprove this assertion by an explicit central extension of A_5 . More generally, we prove an exact compensation formula for direct products A_5×N , where N is finite nilpotent. The formula reduces the equality γ (A_5×N)=9/2 to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of N . Taking N=𝖢_2×𝖢_7×𝖢_11×𝖢_13×𝖢_17×𝖢_19×𝖢_29×𝖢_71×𝖢_83, the loss in the old 2 -Sylow contribution is exactly compensated by the new normal Sylow subgroups. Hence A_5×N is nonsolvable, is not isomorphic to A_5 , has solvable radical N , and nevertheless satisfies γ (A_5×N)=9/2 . Several further exact compensation certificates are also recorded.
We verify Batyrev’s conjecture for symmetric spaces and the non-symmetric triple space.
