Band 33, Heft 2

Let 𝑀 be a connected compact PL 4-manifold with boundary. In this article, we give several lower bounds for regular genus and gem-complexity of the manifold 𝑀. In particular, we prove that if 𝑀 is a connected compact 4-manifold with ℎ boundary components, then its gem-complexity k⁢(M)k(M) satisfies the inequalities k⁢(M)≥3⁢χ⁢(M)+7⁢m+7⁢h-10k(M)\geq 3\chi(M)+7m+7h-10 and k⁢(M)≥k⁢(∂⁡M)+3⁢χ⁢(M)+4⁢m+6⁢h-9k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9, and its regular genus G⁢(M)\mathcal{G}(M) satisfies the inequalities G⁢(M)≥2⁢χ⁢(M)+3⁢m+2⁢h-4\mathcal{G}(M)\geq 2\chi(M)+3m+2h-4 and G⁢(M)≥G⁢(∂⁡M)+2⁢χ⁢(M)+2⁢m+2⁢h-4\mathcal{G}(M)\geq\mathcal{G}(\partial M)+2\chi(M)+2m+2h-4, where 𝑚 is the rank of the fundamental group of the manifold 𝑀. These lower bounds enable to strictly improve previously known estimations for regular genus and gem-complexity of a PL 4-manifold with boundary. Further, the sharpness of these bounds is also shown for a large class of PL 4-manifolds with boundary.

In this paper we study smooth orientation-preserving free actions of the cyclic group ℤ/m{\mathbb{Z}/m} on a class of (n-1){(n-1)}-connected 2⁢n{2n}-manifolds, ♯⁡♯⁡g⁢(Sn×Sn)Σ{\mathbin{\sharp}g(S^{n}\times S^{n})\mathbin{\sharp}\Sigma}, where Σ is a homotopy 2⁢n{2n}-sphere. When n=2{n=2}, we obtain a classification up to topological conjugation. When n=3{n=3}, we obtain a classification up to smooth conjugation. When n≥4{n\geq 4}, we obtain a classification up to smooth conjugation when the prime factors of m are larger than a constant C⁢(n){C(n)}.

This paper studies the Prasad conjecture for the special orthogonal group SO3,3\mathrm{SO}_{3,3}. Then we use the local theta correspondence between Sp4\mathrm{Sp}_{4} and O⁢(V)\mathrm{O}(V) to study the Sp4\mathrm{Sp}_{4}-distinction problems over a quadratic field extension E/FE/F and dim⁡V=4\dim V=4 or 6. Thus we can verify the Prasad conjecture for a square-integrable representation of Sp4⁢(E)\mathrm{Sp}_{4}(E).

Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

In this paper, we consider a finite-dimensional vector space 𝒫{{\mathcal{P}}} over the Galois field GF⁡(p){\operatorname{GF}(p)}, with p being an odd prime, and the family ℬkx{{\mathcal{B}}_{k}^{x}} of all k -sets of elements of 𝒫{\mathcal{P}} summing up to a given element x . The main result of the paper is the characterization, for x=0{x=0}, of the permutations of 𝒫{\mathcal{P}} inducing permutations of ℬk0{{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space 𝒫{\mathcal{P}} if p does not divide k , and as the invertible affinities of the affine space 𝒫{\mathcal{P}} if p divides k . The same question is answered also in the case where the elements of the k -sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.

We study the Diophantine equations obtained by equating a polynomial and the factorial function, and prove the finiteness of integer solutions under certain conditions. For example, we show that there exist only finitely many l such that l!{l!} is represented by NA⁢(x){N_{A}(x)}, where NA{N_{A}} is a norm form constructed from the field norm of a field extension K/𝐐{K/\mathbf{Q}}. We also deal with the equation NA⁢(x)=l!S{N_{A}(x)=l!_{S}}, where l!S{l!_{S}} is the Bhargava factorial. In this paper, we also show that the Oesterlé–Masser conjecture implies that for any infinite subset S of 𝐙{\mathbf{Z}} and for any polynomial P⁢(x)∈𝐙⁢[x]{P(x)\in\mathbf{Z}[x]} of degree 2 or more the equation P⁢(x)=l!S{P(x)=l!_{S}} has only finitely many solutions (x,l){(x,l)}. For some special infinite subsets S of 𝐙{\mathbf{Z}}, we can show the finiteness of solutions for the equation P⁢(x)=l!S{P(x)=l!_{S}} unconditionally.

Let C be a complex algebraic curve uniformized by a Fuchsian group Γ. In the first part of this paper we identify the automorphism group of the solenoid associated with Γ with the Belyaev completion of its commensurator Comm⁢(Γ){\mathrm{Comm}(\Gamma)} and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of Gal⁢(ℂ/ℚ){\mathrm{Gal}(\mathbb{C}/\mathbb{Q})} on algebraic curves. In turn, this fact yields a proof of the Galois invariance of the arithmeticity of Γ independent of Kazhhdan’s. In the second part we focus on the case in which Γ is arithmetic. The list of further Galois invariants we find includes: (i) the periods of Comm⁢(Γ){\mathrm{Comm}(\Gamma)}, (ii) the solvability of the equations X2+sin2⁡2⁢π2⁢k+1{X^{2}+\sin^{2}\frac{2\pi}{2k+1}} in the invariant quaternion algebra of Γ and (iii) the property of Γ being a congruence subgroup.

In this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.

Let X and Y be pathwise connected and paracompact Hausdorff spaces equipped with free involutions T:X→X{T:X\to X} and S:Y→Y{S:Y\to Y}, respectively. Suppose that there exists a sequence (Xi,Ti)⁢⟶hi⁢(Xi+1,Ti+1) for ⁢1≤i≤k,(X_{i},T_{i})\overset{h_{i}}{\longrightarrow}(X_{i+1},T_{i+1})\quad\text{for }% 1\leq i\leq k, where, for each i , Xi{X_{i}} is a pathwise connected and paracompact Hausdorff space equipped with a free involution Ti{T_{i}}, such that Xk+1=X{X_{k+1}=X}, and hi:Xi→Xi+1{h_{i}:X_{i}\to X_{i+1}} is an equivariant map, for all 1≤i≤k{1\leq i\leq k}. To achieve Borsuk–Ulam-type theorems, in several results that appear in the literature, the involved spaces X in the statements are assumed to be cohomological n -acyclic spaces. In this paper, by considering a more wide class of topological spaces X (which are not necessarily cohomological n -acyclic spaces), we prove that there is no equivariant map f:(X,T)→(Y,S){f:(X,T)\to(Y,S)} and we present some interesting examples to illustrate our results.

The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally, we give examples motivated from the theory of Galois coverings of curves.

In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire’s sense) invariant measure has, for each q>0{q>0}, zero lower q -generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system (X,T){(X,T)} (where X=Mℤ{X=M^{\mathbb{Z}}} is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q>1{q>1}, infinite upper q -correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s∈(0,1){s\in(0,1)} and each q>1{q>1}, zero lower s -generalized and infinite upper q -generalized dimensions.

We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich–Zagier series ℱt⁢(q){\mathscr{F}_{t}(q)} which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T⁢(3,2t){T(3,2^{t})}, t≥2{t\geq 2}, is a weight 32{\frac{3}{2}} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F⁢(q){F(q)}.

The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

We use Levinson’s method and the work of Blomer and Harcos on the GL2\mathrm{GL}_{2} shifted convolution problem to prove that at least 6.96 % of the nontrivial zeros of the 𝐿-function of a GL2\mathrm{GL}_{2} automorphic form lie on the critical line.

The main result of this paper is to give that if b∈Lip⁡(ℝn){b\in\operatorname{Lip}(\mathbb{R}^{n})}, hj∈BMO⁡(ℝn){h_{j}\in\operatorname{BMO}(\mathbb{R}^{n})}, j=1,…,k{j=1,\ldots,k}, k∈ℤ+{k\in\mathbb{Z}^{+}} and w∈Ap{w\in A_{p}}, 1<p<∞{1<p<\infty}, then the multilinear Calderón commutators TΩ,b,h→{T_{\Omega,b,\vec{h}}} with variable kernels are bounded on Lp⁢(w){L^{p}(w)}. In addition, the authors extend the above result to the Morrey space.

This article addresses two characterizations of BMO⁢(ℝn){\mathrm{BMO}(\mathbb{R}^{n})}-type space via the commutators of Hardy operators with homogeneous kernels on Lebesgue spaces: (i) characterization of the central BMO⁢(ℝn){\mathrm{BMO}(\mathbb{R}^{n})} space by the boundedness of the commutators; (ii) characterization of the central BMO⁢(ℝn){\mathrm{BMO}(\mathbb{R}^{n})}-closure of Cc∞⁢(ℝn){C_{c}^{\infty}(\mathbb{R}^{n})} space via the compactness of the commutators. This is done by exploiting the center symmetry of Hardy operator deeply and by a more explicit decomposition of the operator and the kernel function.

We apply Weyl 𝑛-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.

We develop a theory of syzygies in equivariant cohomology for tori as well as p -tori and coefficients in 𝔽p{\mathbb{F}_{p}}. A noteworthy feature is a new algebraic approach to the partial exactness of the Atiyah–Bredon sequence, which also covers all instances considered so far.

By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper, we determine the explicit signs of the epsilon factors for symplectic type characters of K×{K^{\times}}, where K/F{K/F} is a wildly ramified quadratic extension of a non-Archimedean local field F of characteristic zero.