Volume 34, Issue 2

Rudnick and Wigman (2008) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the d -dimensional torus is O⁢(E/𝒩){O(E/\mathcal{N})}, as E→∞{E\to\infty}, where E is the energy and 𝒩{\mathcal{N}} is the dimension of the eigenspace corresponding to E . Previous results have established this with stronger asymptotics when d=2{d=2} and d=3{d=3}. In this brief note we prove an upper bound of the form O⁢(E/𝒩1+α⁢(d)-ϵ){O(E/\mathcal{N}^{1+\alpha(d)-\epsilon})}, for any ϵ>0{\epsilon>0} and d≥4{d\geq 4}, where α⁢(d){\alpha(d)} is positive and tends to zero with d . The power saving is the best possible with the current method (up to ϵ) when d≥5{d\geq 5} due to the proof of the ℓ2{\ell^{2}}-decoupling conjecture by Bourgain and Demeter.

For a commutative cancellative monoid M , we introduce the notion of the length density of both a nonunit x∈M{x\in M}, denoted LD⁡(x){\operatorname{LD}(x)}, and the entire monoid M , denoted LD⁡(M){\operatorname{LD}(M)}. This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L⁢(x){L(x)}, ℓ⁢(x){\ell(x)}, and ρ⁢(x){\rho(x)}. We consider some general properties of LD⁡(x){\operatorname{LD}(x)} and LD⁡(M){\operatorname{LD}(M)} and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD⁡(M){\operatorname{LD}(M)} is rational and there is a nonunit element x∈M{x\in M} with LD⁡(M)=LD⁡(x){\operatorname{LD}(M)=\operatorname{LD}(x)} (such a monoid is said to have accepted length density). While it is well known that the much studied asymptotic versions of L⁢(x){L(x)}, ℓ⁢(x){\ell(x)}, and ρ⁢(x){\rho(x)} (denoted L¯⁢(x){\overline{L}(x)}, ℓ¯⁢(x){\overline{\ell}(x)}, and ρ¯⁢(x){\overline{\rho}(x)}) always exist, we show the somewhat surprising result that LD¯⁢(x)=limn→∞⁡LD⁡(xn){\overline{\operatorname{LD}}(x)=\lim_{n\rightarrow\infty}\operatorname{LD}(x^% {n})} may not exist. We also give some finiteness conditions on M that force the existence of LD¯⁢(x){\overline{\operatorname{LD}}(x)}.

Let T be a bilinear Calderón–Zygmund singular integral operator and let T*{T^{*}} be its corresponding truncated maximal operator. For any b∈BMO⁡(ℝn){b\in\operatorname{BMO}({\mathbb{R}^{n}})} and b→=(b1,b2)∈BMO⁡(ℝn)×BMO⁡(ℝn){\vec{b}=(b_{1},b_{2})\in\operatorname{BMO}({\mathbb{R}^{n}})\times% \operatorname{BMO}({\mathbb{R}^{n}})}, let Tb,j*{T^{*}_{b,j}} (j=1,2{j=1,2}) and Tb→*{T^{*}_{\vec{b}}} be the commutators in the j-th entry and the iterated commutators of T*{T^{*}}, respectively. In this paper, for all 1<p1,p2<∞{1<p_{1},p_{2}<\infty}, 1p=1p1+1p2{\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}}, we show that Tb,j*{T^{*}_{b,j}} and Tb→*{T^{*}_{\vec{b}}} are compact operators from Lp1⁢(w1)×Lp2⁢(w2){L^{p_{1}}(w_{1})\times L^{p_{2}}(w_{2})} to Lp⁢(vw→){L^{p}(v_{\vec{w}})} if b,b1,b2∈CMO⁡(ℝn){b,b_{1},b_{2}\in\operatorname{CMO}(\mathbb{R}^{n})} and w→=(w1,w2)∈Ap→{\vec{w}=(w_{1},w_{2})\in A_{\vec{p}}}, vw→=w1p/p1⁢w2p/p2{v_{\vec{w}}=w_{1}^{p/p_{1}}w_{2}^{p/p_{2}}}. Here CMO⁡(ℝn){\operatorname{CMO}(\mathbb{R}^{n})} denotes the closure of 𝒞c∞⁢(ℝn){\mathcal{C}_{c}^{\infty}(\mathbb{R}^{n})} in the BMO⁡(ℝn){\operatorname{BMO}(\mathbb{R}^{n})} topology and Ap→{A_{\vec{p}}} is the multiple weights class.

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

Let PG⁡(r,q){\operatorname{PG}(r,q)} be the r -dimensional projective space over the finite field GF⁡(q){\operatorname{GF}(q)}. A set 𝒳{\mathcal{X}} of points of PG⁡(r,q){\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG⁡(r,q){\operatorname{PG}(r,q)} the set Π∩𝒳{\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG⁡(3,q3){\operatorname{PG}(3,q^{3})} of size 3⁢(q+1)⁢(q2+1){3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q -order subgeometries, and a cutting blocking set of PG⁡(5,q){\operatorname{PG}(5,q)} of size 7⁢(q+1){7(q+1)} from seven lines of a Desarguesian line spread of PG⁡(5,q){\operatorname{PG}(5,q)}. In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q -ary linear code having dimension 4 and 6.

This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia groups and their analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.

Let 𝔽q{\mathbb{F}_{q}} be the finite field of q=pm≡1(mod4){q=p^{m}\equiv 1~{}(\bmod~{}4)} elements with p being an odd prime and m being a positive integer. For c,y∈𝔽q{c,y\in\mathbb{F}_{q}} with y∈𝔽q*{y\in\mathbb{F}_{q}^{*}} non-quartic, let Nn⁢(c){N_{n}(c)} and Mn⁢(y){M_{n}(y)} be the numbers of zeros of x14+⋯+xn4=c{x_{1}^{4}+\cdots+x_{n}^{4}=c} and x14+⋯+xn-14+y⁢xn4=0{x_{1}^{4}+\cdots+x_{n-1}^{4}+yx_{n}^{4}=0}, respectively. In 1979, Myerson used Gauss sums and exponential sums to show that the generating function ∑n=1∞Nn⁢(0)⁢xn{\sum_{n=1}^{\infty}N_{n}(0)x^{n}} is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions ∑n=1∞Nn⁢(c)⁢xn and ∑n=1∞Mn+1⁢(y)⁢xn\sum_{n=1}^{\infty}N_{n}(c)x^{n}\quad\text{and}\quad\sum_{n=1}^{\infty}M_{n+1}% (y)x^{n} are rational functions in x . We also obtain the explicit expressions of these generating functions. Our result extends Myerson’s theorem gotten in 1979.

In this paper, the authors introduce weak martingale Hardy-type spaces associated with a quasi-Banach function lattice. The authors then establish the atomic characterizations of these weak martingale Hardy-type spaces. As applications, the authors give the sufficient conditions for the boundedness of σ-sublinear operators from weak martingale Hardy-type spaces to a quasi-Banach function lattice. Furthermore, the authors clarify the relation among different weak martingale Hardy-type spaces in the framework of a rearrangement-invariant quasi-Banach function space. Finally, the authors apply these results to the weighted Lorentz space and the generalized grand Lebesgue space.

We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that for any prime p the number of character codegrees of a finite group G that are divisible by p is at most k , then the number of prime divisors of |G|{|G|} is bounded in terms of k . We prove this conjecture for solvable groups.

We present some generalizations of the well-known correspondence, found by Exel, between partial actions of a group G on a set X and semigroup homomorphism of 𝒮⁡(G){\operatorname{\mathcal{S}}(G)} on the semigroup I⁢(X){I(X)} of partial bijections of X , with 𝒮⁡(G){\operatorname{\mathcal{S}}(G)} being an inverse monoid introduced by Exel. We show that any unital premorphism θ:G→S{\theta:G\to S}, where S is an inverse monoid, can be extended to a semigroup homomorphism θ*:T→S{\theta^{*}:T\to S} for any inverse semigroup T with 𝒮⁡(G)⊆T⊆P*⁢(G)×G{\operatorname{\mathcal{S}}(G)\subseteq T\subseteq P^{*}(G)\times G}, with P*⁢(G){P^{*}(G)} being the semigroup of non-empty subsets of G , and such that E⁢(S){E(S)} satisfies some lattice-theoretical condition. We also consider a topological version of this result. We present a minimal Hausdorff inverse semigroup topology on Γ⁢(X){\Gamma(X)}, the inverse semigroup of partial homeomorphisms between open subsets of a locally compact Hausdorff space X .

The homotopy braid group B^n{\widehat{B}_{n}} is the subject of the paper. First, the linearity of B^n{\widehat{B}_{n}} over the integers is proved. Then we prove that the group B^3{\widehat{B}_{3}} is torsion free.

We prove that a pseudoconvex complex space X of pure dimension n+1{n+1} is holomorphically convex provided that its singular set has only compact connected components (e.g., X has isolated singularities) and rankx⁡𝒪X=n{\operatorname{rank}_{x}\mathscr{O}_{X}=n} for all smooth points x outside a compact set of X . This extends a known result due to Ohsawa asserting that a weakly 1-complete, smooth complex surface is holomorphically convex if it admits a globally defined non-constant holomorphic function. We also revise an example of Markoe concerning non-invariance of holomorphic convexity under normalization.

In this paper, we give an explicit formula for the Shintani double zeta functions with any ramification in the most general setting of adeles over an arbitrary number field. Three applications of the explicit formula are given. First, we obtain a functional equation satisfied by the Shintani double zeta functions in addition to Shintani’s functional equations. Second, we establish the holomorphicity of a certain Dirichlet series generalizing a result by Ibukiyama and Saito. This Dirichlet series occurs in the study of unipotent contributions of the geometric side of the Arthur–Selberg trace formula of the symplectic group. Third, we prove an asymptotic formula of the weighted average of the central values of quadratic Dirichlet L -functions.

Let 𝒜{\mathscr{A}} be an abelian category having enough projective and injective objects, and let 𝒯{\mathscr{T}} be an additive subcategory of 𝒜{\mathscr{A}} closed under direct summands. A known assertion is that in a short exact sequence in 𝒜{\mathscr{A}}, the 𝒯{\mathscr{T}}-projective (resp. 𝒯{\mathscr{T}}-injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if 𝒯{\mathscr{T}} contains all projective (resp. injective) objects of 𝒜{\mathscr{A}}, then the above assertion holds true if and only if 𝒯{\mathscr{T}} is resolving (resp. coresolving). As applications, we get that a left and right Noetherian ring R is n -Gorenstein if and only if the Gorenstein projective (resp. injective, flat) dimension of any left R -module is at most n . In addition, in several cases, for a subcategory 𝒞{\mathscr{C}} of 𝒯{\mathscr{T}}, we show that the finitistic 𝒞{\mathscr{C}}-projective and 𝒯{\mathscr{T}}-projective dimensions of 𝒜{\mathscr{A}} are identical.

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang–Baxter equation, not necessarily involutive, by means of q -cycle sets. We entirely focus on the finite indecomposable ones, among which we especially study the class of simple solutions. In particular, we provide a group-theoretic characterization of these solutions, including their permutation groups. Finally, we deal with some open questions.

In this paper, we consider twisted Dirichlet series attached to quasimodular forms, study their analytic properties, and prove an analogue of Weil’s converse theorem for quasimodular forms over congruence subgroups. We also give some applications of our results to a certain q -series and sign changes of the Fourier coefficients of quasimodular forms.