Volume 27, Issue 4

In this paper, we study the functional limits of occupation time fluctuations of a kind of site-dependent binary branching particle systems. Our result extends Theorem 2.2 in [Stochastic Process. Appl. 116 (2006), 1–18] and gets some new information on the behaviors of occupation time processes.

In this note, we give a necessary and sufficient condition for determining which integers can be written as a sum of two integral squares for certain quadratic fields by using the integral Brauer–Manin obstruction. The condition is computable and originally from the reciprocity law.

Let Λ be the von Mangoldt function and rG(n)=∑m1+m2=nΛ(m1)Λ(m2)$ {r_G(n) = \sum _{m_1 + m_2 = n} \Lambda (m_1) \Lambda (m_2)} $ be the counting function for the Goldbach numbers. Let N ≥ 2 be an integer. We prove that ∑n≤NrG(n)(1-nN)kΓ(k+1)=N2Γ(k+3)-2∑ρΓ(ρ)Γ(ρ+k+2)Nρ+1+∑ρ1∑ρ2Γ(ρ1)Γ(ρ2)Γ(ρ1+ρ2+k+1)Nρ1+ρ2+𝒪k(N1/2),$\sum _{n \le N} r_G(n) \frac{(1 - \frac{n}{N})^k}{\Gamma (k + 1)} = \frac{N^{2}}{\Gamma (k + 3)} - 2 \sum _{\rho } \frac{\Gamma (\rho )}{\Gamma (\rho + k + 2)} N^{\rho +1} + \sum _{\rho _1} \sum _{\rho _2} \frac{\Gamma (\rho _1) \Gamma (\rho _2)}{\Gamma (\rho _1 + \rho _2 + k + 1)} N^{\rho _1 + \rho _2} + \mathcal {O}_{k}(N^{1/2}), $ for k > 1, where ρ, with or without subscripts, runs over the non-trivial zeros of the Riemann zeta-function ζ( s ).

Let G be a finite group and Le(G)={x∈G∣xe=1}$L_e(G)=\lbrace x \in G \mid x^e=1\rbrace $, where e is a positive integer dividing |G|$\vert G\vert $. How do bounds on |Le(G)|$\vert L_e(G)\vert $ influence the structure of G ? Meng and Shi [Arch. Math. (Basel) 96 (2011), 109–114] have answered this question for |Le(G)|≤2e$\vert L_e(G)\vert \le 2e$. We generalize their contributions, considering the inequality |Le(G)|≤e2$\vert L_e(G)\vert \le e^2$ and finding a new class of groups of whose we study the structural properties.

The main goal of this article is to prove a second main theorem for entire curves into Hilbert modular surfaces. As an application of our main theorem, we obtain a condition ensuring that entire curves in a Hilbert modular surface of general type are contained in the exceptional divisors. We also show a second main theorem for a simple normal crossing divisor which is tangent to a holomorphic distribution of codimension one on a smooth projective algebraic manifold.

We prove a sub-convex estimate for the sup-norm of L 2 -normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over ℚ. More precisely we show that when the L 2 norm of an eigenfunction f is one,∥f∥∞≪εk12-133+ε$ \Vert f \Vert _\infty \ll _\varepsilon k^{\frac{1}{2} - \frac{1}{33} +\varepsilon } $ for any ε>0${\varepsilon >0}$ and for all k sufficiently large.

There are two families of Donaldson invariants for the complex projective plane, corresponding to the SU(2)-gauge theory and the SO(3)-gauge theory with non-trivial Stiefel–Whitney class. In 1997 Moore and Witten conjectured that the regularized u -plane integral on ℂP2$\mathrm {P}^2$ gives the generating functions for these invariants. In earlier work, the second two authors proved the conjecture for the SO(3)-gauge theory. Here we complete the proof of the conjecture by confirming the claim for the SU(2)-gauge theory. As a consequence, we find that the SU(2)-Donaldson invariants for ℂP2$\mathrm {P}^2$ are explicit linear combinations of the Hurwitz class numbers which arise in the theory of imaginary quadratic fields and orders.

For suitable pairs of diagonal quadratic forms in eight variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares.

We study the subelliptic heat kernels of the CR three-dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three-dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gradient bounds using a new type of curvature-dimension inequality.

This paper is devoted to the study of the L p -mapping properties of the commutators μΩ;b,1${\mu _{\Omega ;b,1}}$, μΩ,λ;b,1*${ \mu _{\Omega ,\lambda ;b,1}^{*}}$ and μΩ,S;b,1${ \mu _{\Omega ,S;b,1}}$, which are formed respectively by a BMO(ℝn)${(\mathbb {R}^n)}$ function b ( x ) and a class of rough Marcinkiewicz integral operators μΩ${\mu _\Omega }$, μΩ,λ*${\mu _{\Omega ,\lambda }^{*}}$ and μΩ,S${\mu _{\Omega ,S}}$ related to the Littlewood–Paley g -function, the Littlewood–Paley gλ*${g_\lambda ^*}$-function and the Lusin area integral, respectively, with the kernel condition which is different from the condition Ω ∈ H 1 ( S n -1 ).

Let 𝒜 be an essential complex hyperplane arrangement in ℂn${\mathbb {C}}^n$, and H denote the union of the hyperplanes. We develop the real-valued and circle-valued Morse theory on the space M=ℂn∖H$M={\mathbb {C}}^n\setminus H$ and prove, in particular, that M has the homotopy type of a space obtained from a manifold V fibered over a circle, by attaching |χ(M)|$\vert \chi (M)\vert $ cells of dimension n . We compute the Novikov homology H^*(M,ξ)$\hat{H}_*(M,\xi )$ for a large class of homomorphisms ξ:π1(M)→ℝ$\xi :\pi _1(M)\rightarrow {\mathbb {R}}$.

We study atomic Hardy H p spaces and their duals in the framework of an arbitrary proper open subset Ω of ℝ d .

We prove that for any closed subgroup H of a locally compact Hausdorff group G the following properties are mutually equivalent: (1) the coset space G / H is locally contractible, (2) G / H is finite-dimensional and locally connected, (3) G / H is a manifold. Assume that G is a locally compact group with compact space of connected components. If the natural action of G on a locally contractible coset space G / H is effective, then we prove that G is a Lie group.

In this paper we establish sharp L 2 and H p boundedness results for strongly singular operators and oscillating operators on Heisenberg groups.

This paper is devoted to studying initial value problems for semilinear wave equations with variable coefficients with subcritical exponents for n ( n ≥ 1) space dimensions. We prove that solutions to a certain kind of subcritical wave equations cannot be global if the initial values are positive somewhere and nonnegative no matter how small the initial data are, and also we give the sharp lifespan estimate of solutions for the problems. This solves a part of the famous Strauss' conjecture about semilinear wave equations in the case of subcritical exponents and variable coefficients Cauchy problems.

Let G be a real solvable exponential Lie group with Lie algebra 𝔤 and let f∈𝔤*${ f\in \mathfrak {g}^* }$. We take two polarizations 𝔭j${ \mathfrak {p}_j}$, j = 1,2, at f which meet the Pukanszky condition. Let Pj:=exp(𝔭j)${ P_j:=\exp ( \mathfrak {p}_j)}$, j = 1,2, be the associated subgroups in G . The linear functional f defines unitary characters χf(expX):=ei〈f,X〉${ \chi _f(\exp X):=e^{i \langle { f},{X}\rangle }}$, X∈𝔭j${X\in \mathfrak {p}_j }$, of P j . Let πj:=indPjGχf${\pi _j :=\hbox{ind}_{ P_j }^{G} \chi _f}$, j = 1,2, be the corresponding induced representations, which are unitary and irreducible. It is well known that τ1${ \tau _1 }$ and τ2${ \tau _2 }$ are unitarily equivalent. The description of the intertwining operator of such an equivalence is given via an abstract integral I𝔭2,𝔭1${ I_{\mathfrak {p}_2,\mathfrak {p}_1} }$. In this paper, we show that this formal integral gives us a concrete intertwining operator provided that it converges absolutely on a G - and 𝒰(𝔤)${\mathcal {U}(\mathfrak {g})}$-invariant dense subspace of the space ℋτ1∞${\mathcal {H}_{\tau _1}^\infty }$ of C ∞ vectors of π 1 (which is the case when the simple product P1P2${P_1P_2}$ is closed). Finally, given three Pukanszky polarizations Pi=exp(𝔭i)${ P_i=\exp ( \mathfrak {p}_i)}$, i = 1,2,3, at f of that kind, we accurately determine the composition formula of I𝔭3,𝔭2∘I𝔭2,𝔭1∘I𝔭1,𝔭3${ I_{\mathfrak {p}_3,\mathfrak {p}_2}\circ I_{\mathfrak {p}_2,\mathfrak {p}_1}\circ I_{\mathfrak {p}_1,\mathfrak {p}_3}}$ using Maslov's index.

We show that Artin's conjecture concerning p -adic solubility of Diophantine equations fails for infinitely many systems of r homogeneous diagonal equations whenever r ≥ 2.

In this paper, we define a simplicial complement ℙ={σ1,σ2,...,σs}${\mathbb {P}=\lbrace \sigma _1, \sigma _2,\ldots , \sigma _s\rbrace }$ to be a sequence of subsets of [m]${[m]}$ and prove that the simplicial complement ℙ corresponds to an unique simplicial complex Kℙ${K_\mathbb {P}}$ with vertices in [ m ]. Then, we define the homology algebra of a simplicial complement Hi,σ(Λ*,*[ℙ],d)${H_{i,\sigma }(\Lambda ^{*,*}[\mathbb {P}], d)}$ over a principal ideal domain 𝐤 and prove that H*,*(Λ[ℙ],d)${H_{*,*}(\Lambda [\mathbb {P}], d)}$ is isomorphic to the Tor-algebra of the corresponding face ring 𝐤(Kℙ)${\mathbf {k}(K_\mathbb {P})}$ by the Taylor resolution. This gives an algorithm for the cohomology algebraic structure of the moment-angle complex by the isomorphism H*(𝒵Kℙ,𝐤)≅ Tor *,*𝐤[𝐱](𝐤(Kℙ),𝐤)${H^*(\mathcal {Z}_{K_\mathbb {P}}, \mathbf {k}) \cong {\rm Tor}_{*,*}^{\mathbf {k}[\mathbf {x}]}(\mathbf {k}(K_\mathbb {P}), \mathbf {k})}$. As applications of the simplicial complement, we also give methods to compute the cohomology of the moment-angle complex corresponding to a link and star linkKℙσ${\operatorname{link}_{K_\mathbb {P}}\sigma }$, starKℙσ${\operatorname{star}_{K_\mathbb {P}}\sigma }$ and the cohomology modules of the polyhedral products as well.

In the case G=ℤ2d$G=\mathbb {Z}_2^d$, we prove a multivariate Marcinkiewicz-type multiplier theorem for the Dunkl transform and a multivariate Marcinkiewicz-type multiplier theorem for the system of Dunkl harmonic oscillators.

We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving ( k +1)-point configurations in geometric measure theory, with k ≥ 2, including the distribution of simplices, volumes and angles determined by the points of fractal subsets E ⊂ ℝ d , d ≥ 2. If T k ( E ) denotes the set of noncongruent (k+1)${(k+1)}$-point configurations determined by E , we show that if the Hausdorff dimension of E is greater than d - ( d -1)/(2 k ), then the k+12${\binom{k+1}{2}}$-dimensional Lebesgue measure of T k ( E ) is positive. This complements previous work on the Falconer conjecture ([Int. Math. Res. Not. IMRN 23 (2005), 1411–1425] and the references there), as well as work on finite point configurations [Recent Advances in Harmonic Analysis and Applications, Springer-Verlag, New York (2013), 93–103; Anal. PDE 5 (2012), no. 2, 397–409]. We also give applications to Erdős-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in [Adv. Math. 228 (2011), 2385–2402].

We prove an algebraicity property for a certain ratio of Petersson norms associated to a Siegel cusp form of degree 2 (and arbitrary level) whose adelization generates a weak endoscopic lift. As a preparation for this, we explicate various features of the correspondence between scalar valued Siegel cusp forms of degree n and automorphic representations on GSp 2n${{\rm GSp}_{2n}}$.

For a local Cohen–Macaulay ring R of finite CM-type, Yoshino has applied methods of Auslander and Reiten to compute the Grothendieck group K 0 of the category mod R of finitely generated R -modules. For the same type of rings, we compute in this paper the first Quillen K-group K 1 (mod R ). We also describe the group homomorphism R * → K 1 (mod R ) induced by the inclusion functor proj R → mod R and illustrate our results with concrete examples.

We present some results on character degree sums in connection with several important characteristics of finite groups such as p -solvability, solvability, supersolvability, and nilpotency. Some of them strengthen known results in the literature.

We introduce a new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of affine Kac–Moody symmetric spaces; they have also important applications in the study of isoparametric submanifolds in tame Fréchet spaces.

Let L=-Δ+V${L=-\Delta +V}$ be a Schrödinger operator, where Δ is the Laplacian operator on ℝ n , while the nonnegative potential V belongs to the reverse Hölder class. In this paper, we establish the weighted norm inequalities for some Schrödinger type operators, which include Riesz transforms and fractional integrals and their commutators. These results generalize substantially some well-known results.