Volume 36, Issue 1

Mukhin found in 1984 an important necessary and sufficient condition for the validity of the local limit theorem. Revisiting the succinct proof given in [A. B. Mukhin, Some necessary and sufficient conditions for the validity of local limit theorems, Dokl. Akad. Nauk USSR 1984, 8, 7–8], we could only prove rigorously a weaker necessary and sufficient condition, with a significantly different formulation. This is the object of this short note.

We obtained a non-commutative Khinchine-type inequality under assumption that Rademacher random variables are dependent under condition that the sum of them is equal to some integer M .

Given a singular connection D on a vector bundle E over an irreducible smooth projective curve X , defined over an algebraically closed field, we show that there is a unique maximal subsheaf of E on which D induces a nonsingular connection. Given a generically smooth map ϕ : Y → X {\phi:Y\rightarrow X} between irreducible smooth projective curves, and a singular connection ( V , D ) {(V,D)} on Y , the direct image ϕ * ⁢ V {\phi_{*}V} has a singular connection. Let 𝐑 ⁢ ( ϕ * ⁢ 𝒪 Y ) {\mathbf{R}(\phi_{*}{\mathcal{O}}_{Y})} be the unique maximal subsheaf on which the singular connection on ϕ * ⁢ 𝒪 Y {\phi_{*}{\mathcal{O}}_{Y}} – corresponding to the trivial connection on 𝒪 Y {{\mathcal{O}}_{Y}} – induces a nonsingular connection. We prove that the homomorphism of étale fundamental groups ϕ * : π 1 et ⁢ ( Y , y 0 ) → π 1 et ⁢ ( X , ϕ ⁢ ( y 0 ) ) {\phi_{*}:\pi_{1}^{\rm et}(Y,y_{0})\rightarrow\pi_{1}^{\rm et}(X,\phi(y_{0}))} induced by ϕ is surjective if and only if 𝒪 X ⊂ 𝐑 ⁢ ( ϕ * ⁢ 𝒪 Y ) {{\mathcal{O}}_{X}\subset\mathbf{R}(\phi_{*}{\mathcal{O}}_{Y})} is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism ϕ * {\phi_{*}} is surjective if and only if 𝒪 X = 𝐑 ⁢ ( ϕ * ⁢ 𝒪 Y ) {{\mathcal{O}}_{X}=\mathbf{R}(\phi_{*}{\mathcal{O}}_{Y})} . For any nonsingular connection D on a vector bundle V over X , there is a natural map V ↪ 𝐑 ⁢ ( ϕ * ⁢ ϕ * ⁢ V ) {V\hookrightarrow\mathbf{R}(\phi_{*}\phi^{*}V)} . When the characteristic of the base field is zero, we prove that the map ϕ is genuinely ramified if and only if V = 𝐑 ⁢ ( ϕ * ⁢ ϕ * ⁢ V ) {V=\mathbf{R}(\phi_{*}\phi^{*}V)} .

We study geometric properties of caustics of pseudo-spherical surfaces, that is, surfaces with constant negative Gaussian curvature -1 in the Euclidean 3-space ℝ 3 {\mathbb{R}^{3}} . We investigate the Gaussian and the mean curvature of caustics of pseudo-spherical surfaces. Moreover, a certain condition required for the caustics to be minimal surfaces is derived.

We give an explicit construction of lifting maps from integral and half-integral modular forms to vector-valued modular forms for Weil representations associated with arbitrary isotropic subgroups and finite quadratic modules of even and odd signature. This construction provides an explicit and general extension of previous work by Scheithauer and Zhang.

In this paper, we introduce and study a new type of convergences using statistical convergence via the power series method and measurable convergence. We also study their relationship with other convergences. Further, we demonstrate Korovkin-type approximation theorems for double sequences of positive linear operators using these newly specified convergences, and we also provide illustrations that demonstrate how our proven theorems are better than their classical counterparts. Finally, we have determined rates of statistical product measurable convergence using the power series approach and the modulus of continuity.

We study the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction. In both L 1 {L^{1}} and L ∞ {L^{\infty}} frameworks, we will first show the global existence of classical solutions with small initial data, and then prove that the global solution will scatter to free linear waves and study the rigidity aspect of the scattering problem. We also show the inverse scattering result: The scattering field can determine the global solution uniquely, in the L 1 {L^{1}} framework.

The idea of studying trisections of closed smooth 4-manifolds via (singular) triangulations, endowed with a suitable vertex-labelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to standard simply-connected 4-manifolds, via the so-called simple crystallizations . In the present paper we propose generalizations of these ideas by taking into consideration a possible extension of trisections to compact PL 4-manifolds with connected boundary, which is related to Birman’s special Heegaard sewing , and by analyzing gem-induced trisections , i.e. trisections that can be induced not only by simple crystallizations, but also by any 5-colored graph encoding a PL 4-manifold with empty or connected boundary. This last notion gives rise to that of G-trisection genus , as an analogue, in this context, of the well-known trisection genus. We give conditions on a 5-colored graph ensuring one of its gem-induced trisections – if any – to realize the G-trisection genus, and prove how to determine it directly from the graph. As a consequence, we detect a class of closed simply-connected 4-manifolds, comprehending all standard ones, for which both G-trisection genus and trisection genus coincide with the second Betti number and also with half the value of the graph-defined PL invariant regular genus . Moreover, the existence of gem-induced trisections and an estimation of the G-trisection genus via surgery description is obtained, for each compact PL 4-manifold admitting a handle decomposition lacking in 3-handles.

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces.

Let p be an odd prime. Let f 1 {f_{1}} and f 2 {f_{2}} be weight 2 cuspidal Hecke eigenforms with isomorphic residual Galois representations at p . Greenberg–Vatsal and Emerton–Pollack–Weston showed that if p is a good ordinary prime for the two forms, the Iwasawa invariants of their p -primary Selmer groups and p -adic L -functions over the cyclotomic ℤ p {\mathbb{Z}_{p}} -extension of ℚ {\mathbb{Q}} are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, let K be an imaginary quadratic field where p splits. Suppose that the generalized Heegner hypothesis holds with respect to both ( f 1 , K ) {(f_{1},K)} and ( f 2 , K ) {(f_{2},K)} . We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDP p -adic L -functions of f 1 {f_{1}} and f 2 {f_{2}} .

Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ ( n ) {\tau(n)} is odd if and only if n is an odd square, we only need to consider τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} for primes p and natural numbers n ≥ 1 {n\geq 1} . This is a rather delicate question. In this direction, we show that for any ϵ > 0 {\epsilon>0} and integer n ≥ 1 {n\geq 1} , the largest prime factor of τ ⁢ ( p 2 ⁢ n ) {\tau(p^{2n})} , denoted by P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) {P(\tau(p^{2n}))} , satisfies P ⁢ ( τ ⁢ ( p 2 ⁢ n ) ) > ( log ⁡ p ) 1 8 ⁢ ( log ⁡ log ⁡ p ) 3 8 - ϵ P(\tau(p^{2n}))>(\log p)^{\frac{1}{8}}(\log\log p)^{\frac{3}{8}-\epsilon} for almost all primes p . This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients.

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group S n {S_{n}} has three natural extensions, namely the braid group B n {B_{n}} , the twin group T n {T_{n}} and the triplet group L n {L_{n}} . The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander–Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both T n {T_{n}} and L n {L_{n}} when n ≥ 4 {n\geq 4} . We also determine subquotients of principal congruence subgroups of T n {T_{n}} , and identify the pure twin group P ⁢ T n {PT_{n}} and the pure triplet group P ⁢ L n {PL_{n}} with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that T n / P ⁢ T n ′ {T_{n}/PT_{n}^{\prime}} , T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} and L n / P ⁢ L n ′ {L_{n}/PL_{n}^{\prime}} are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ {T_{n}/T_{n}^{\prime\prime}} .

Let M ⁢ ( d , χ ) {M(d,\chi)} , with ( d , χ ) = 1 {(d,\chi)=1} , be the moduli space of semistable sheaves on ℙ 2 {\mathbb{P}^{2}} supported on curves of degree d and with Euler characteristic χ. The cohomology ring H * ⁢ ( M ⁢ ( d , χ ) , ℤ ) {H^{*}(M(d,\chi),\mathbb{Z})} of M ⁢ ( d , χ ) {M(d,\chi)} is isomorphic to its Chow ring A * ⁢ ( M ⁢ ( d , χ ) ) {A^{*}(M(d,\chi))} by Markman’s result. Pi and Shen have described a minimal generating set of A * ⁢ ( M ⁢ ( d , χ ) ) {A^{*}(M(d,\chi))} consisting of 3 ⁢ d - 7 {3d-7} generators, which they also showed to have no relation in A ≤ d - 2 ⁢ ( M ⁢ ( d , χ ) ) {A^{\leq d-2}(M(d,\chi))} . We compute the two Betti numbers b 2 ⁢ ( d - 1 ) {b_{2(d-1)}} and b 2 ⁢ d {b_{2d}} of M ⁢ ( d , χ ) {M(d,\chi)} , and as a corollary we show that the generators given by Pi and Shen have no relations in A ≤ d - 1 ⁢ ( M ⁢ ( d , χ ) ) {A^{\leq d-1}(M(d,\chi))} , but do have three linearly independent relations in A d ⁢ ( M ⁢ ( d , χ ) ) {A^{d}(M(d,\chi))} .

Let W be a finite reflection group associated with a root system R in ℝ d {\mathbb{R}^{d}} . Let C + {C_{+}} denote a positive Weyl chamber distinguished by a choice of R + {R_{+}} , a set of positive roots. We define and investigate Hardy and BMO spaces on C + {C_{+}} in the framework of boundary conditions given by a homomorphism η ∈ Hom ⁡ ( W , ℤ ^ 2 ) {\eta\in\operatorname{Hom}(W,\widehat{\mathbb{Z}}_{2})} which attaches the ± {\pm} signs to the facets of C + {C_{+}} . Specialized to orthogonal root systems, atomic decompositions in H η 1 {H^{1}_{\eta}} and h η 1 {h^{1}_{\eta}} are obtained and the duality problem is also treated.

We discuss de Branges–Rovnyak spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ ( b ) {Y=S|_{\mathcal{H}(b)}} , the restriction of the shift operator S on H 2 {H^{2}} to ℋ ⁢ ( b ) {\mathcal{H}(b)} , is a strict 2 ⁢ m {2m} -isometry and proved that such spaces ℋ ⁢ ( b ) {\mathcal{H}(b)} are equal to local Dirichlet spaces of order m . Here we give a characterization of local Dirichlet spaces of order m in terms of the m -th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ ( b ) {\mathcal{H}(b)} coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213].