Superminimal surfaces in hyperquadric Q2

We study a superminimal surface M immersed into a hyperquadric Q₂ in several cases classified by two global defined functions {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y, which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : M\to {{\displaystyle Q}}_{\mathrm{2}}. In case both {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y are not identically zero, it is proved that f is superminimal if and only if f is totally real or i\circ f:M\to {ℂ\mathrm{P}}^{\mathrm{3}} is also minimal, where i:{{\displaystyle Q}}_{\mathrm{2}}\to {ℂ\mathrm{P}}^{\mathrm{3}} is the standard inclusion map. In the rest case that {{\displaystyle \tau }}_X\equiv \mathrm{0} or {{\displaystyle \tau }}_Y\equiv \mathrm{0}, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q₂ are completely described.     

An isometrical CPn-theorem

Let M^n(n\ge \mathrm{3}) be a complete Riemannian manifold with {{\displaystyle \sec ?}}_M\ge \mathrm{1}, and let {{\displaystyle M}}_i^{{{\displaystyle n}}_i}(i=\mathrm{1},\mathrm{2}) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n − 2 and if the distance \left|{{\displaystyle M}}_{\mathrm{1}}{{\displaystyle M}}_{\mathrm{2}}\right|\ge \pi /\mathrm{2}, then M_i is isometric to {\mathbb{S}}^{{{\displaystyle n}}_i}/{{\displaystyle \mathrm{?}}}_h,{\mathrm{?}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}}, or {\mathrm{?}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}} with the canonical metric when n_i>0, and thus, M is isometric to {\mathbb{S}}^n/{{\displaystyle \mathrm{?}}}_h,{\mathrm{?}\mathrm{P}}^{n/\mathrm{2}}, or {\mathrm{?}\mathrm{P}}^{n/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}} except possibly when n = 3 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}{\mathbb{S}}^{\mathrm{1}}/{{\displaystyle \mathrm{?}}}_h with h\ge \mathrm{2} or n = 4 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}\mathrm{?}{\mathrm{P}}^{\mathrm{2}}.     

Weighted product Hardy space associated with operators

Assuming that the operators L₁, L₂ are self-adjoint and {\mathrm{e}}^{-{{\displaystyle tL}}_i}(i=\mathrm{1},\mathrm{2}) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space {{\displaystyle H}}_{{{\displaystyle L}}_{\mathrm{1}},{{\displaystyle L}}_{\mathrm{2}},\omega }^{\mathrm{1}}({ℝ}^{{{\displaystyle n}}_{\mathrm{1}}}\times {ℝ}^{{{\displaystyle n}}_{\mathrm{2}}}) associated to operators L₁, L₂ on product domain, which is defined in terms of area function, has an atomic decomposition for some weight \omega.     

On multivariable Zassenhaus formula

We give a recursive algorithm to compute the multivariable Zassenhaus formula e^X1+X2+…+Xn=e^X1eX2…e^Xn∏∞k=2e^Wk and derive an effective recursion formula of Wk.     

Oscillatory hyper Hilbert transforms along variable curves

For n = 2 or 3 and x\in {ℝ}^n, we study the oscillatory hyper Hilbert transform{{\displaystyle T}}_{\alpha ,\beta }f(x)={{\displaystyle \int }}_{ℝ}f(x-\Gamma (t,x)){\mathrm{e}}^{{-i\left|t\right|}^{-\beta }}{\left|t\right|}^{-\mathrm{1}-\alpha }\mathrm{d}talong an appropriate variable curve \mathrm{\Gamma }(t,x) in {ℝ}^n (namely, \mathrm{\Gamma }(t,x) is a curve in {ℝ}^n for each fixed x), where \alpha >\beta >\mathrm{0}. We obtain some L^p boundedness theorems of {{\displaystyle T}}_{\alpha ,\beta }, under some suitable conditions on \alphaand \beta. These results are extensions of some earlier theorems. However, {{\displaystyle T}}_{\alpha ,\beta }f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.     

Derivative estimates of averaging opera tors and extension

We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp multiplier, we obtain the optimal range of exponents (γ,β,p)to ensure the H^p(R^n) boundedness of a^βS^γ1f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on H^p(R^n) spaces.     

Averages of shifted convolution sums for arithmetic functions

Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|≤H |ΣX相似文献   

Continuity of functors with respect to generalized inductive limits

Let(Ai,φi,i+1) be a generalized indue Live system of a sequeiiee (Ai) of unital separable C^*-algebras,with A =limi→∞(Ai,φi,i+1). Set φj,i=φi-1,i^0…0φj+1,j+2^0 φj,j+1 for all i>j. We prove that if φj,i are order zero completely positive contractions for all j and i>j, And L:=inf{λ|λ∈σ(φj,i(1Aj)) for all j uud i>j}>0, where σ(φj,i(1Aj)) is the speetrum of φj,i(1Aj),than limi→∞(Cu(Ai),Cu((φi,i+1))=Cu(A), where Cu(A) is a stable version of the Cuntz semigroup of C^*-algebra A. Let (An,φm,n) be a generalized inductive syfitem of C^*-algahrafl, with the ipmkn order zero completely positive contractions. We also prove that if the decomposition rank (nuclear dimension) of ,4n is no more t han some integer k for each n, then the decompostition rank (nuclear dimension) of A is also no more than k.     

Lower deviations for supercritical branching processes with immigration

For a supercritical branching processes with immigration \{{{\displaystyle Z}}_n\}; it is known that under suitable conditions on the offspring and immigration distributions, Z_n/mⁿ converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P({{\displaystyle Z}}_n={{\displaystyle k}}_n) with {{\displaystyle k}}_n=o(m^n) as n\to \infty. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.     

The second moment of GL(3) × GL(2) L- functions at special points from GL(3) forms

For a fixed even SL(\mathrm{2},\mathbb{Z}) Hecke{Maass form f, we get an estimate for the second moment of L(s,{{\displaystyle \phi }}_j\times f) at special points, where {{\displaystyle \phi }}_j runs over an orthogonal basis of Hecke{Maass cusp forms for {{\displaystyle \mathrm{S}\mathrm{L}}}_{\mathrm{3}}(\mathbb{Z}).     

Fourier transform of anisotropic mixed-norm Hardy spaces

Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(Rⁿ)be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(Rⁿ).In this article,we show that the Fourier transform fcoincides with a continuous function g on?n in the sense of tempered distributions and,moreover,this continuous function g,multiplied by a step function associated with a,can be pointwisely controlled by a constant multiple of the Hardy space norm of f.These proofs are achieved via the known atomic characterization of Hpa(Rⁿ)and the establishment of two uniform estimates on anisotropic mixed-norm atoms.As applications,we also conclude a higher order convergence of the continuous function gat the origin.Finally,a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained.All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(Rⁿ)with p∈0,1],and are even new for isotropic mixed-norm Hardy spaces on∈n.     

Mean-square estimate of automorphic L-functions

Let f be a holomorphic Hecke cusp form with even integral weight k≥2 for the full modular group,and letχbe a primitive Dirichlet character modulo q.Let Lf(s,χ)be the automorphic L-function attached to f andχ-We study the mean-square estimate of Lf(s,χ)and establish an asymptotic formula.     

Convergence of complex martingale for a branching random walk in an independent and identically distributed environment

We consider anR^d-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_n(z)(z∈C^d)be the natural complex martingale of the process.We show necessary and sufficient conditions for the L^α-convergence of W_n(z)forα>1,as well as its uniform convergence region.     

Extremum of a time-inhomogeneous branching random walk

Consider a time-inhomogeneous branching random walk, generated by the point process L_n which composed by two independent parts: ‘branching’offspring X_n with the mean \mathrm{1}+{B(\mathrm{1}+n)}^{-\beta } for \beta \in (\mathrm{0},\mathrm{1}) and ‘displacement’ {{\displaystyle \xi }}_n with a drift {A(\mathrm{1}+n)}^{-\mathrm{2}\alpha } for \alpha \in (\mathrm{0},\mathrm{1}/\mathrm{2}), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ {{\displaystyle \xi }}_n is strictly positive or negative for \left|A\right|\rangle \mathrm{0} but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter \beta and \alpha.     

Forward and symmetric Wick-Itô integrals with respect to fractional Brownian motion

Let B^H=\{{{\displaystyle B}}_t^H,t\ge \mathrm{0}\} be a fractional Brownian motion with Hurst index H\in (\mathrm{0},\mathrm{1}). Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to B^H as follows: {\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{-}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_s^H)\mathrm{d}s,{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{°}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_{(s-\epsilon )\vee \mathrm{0}}^H)\mathrm{d}s,in probability, where ◊ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of B^H for all H\in (\mathrm{0},\mathrm{1}).     

Proper resolutions and Gorensteinness in extriangulated categories

Let(C,E,s)be an extriangulated category with a proper classξof E-triangles,and W an additive full subcategory of(C,E,s).We provide a method for constructing a proper W(ξ)-resolution(resp.,coproper W(ξ)-coresolution)of one term in an E-triangle inξfrom that of the other two terms.By using this way,we establish the stability of the Gorenstein category GW(ξ)in extriangulated categories.These results generalize the work of Z.Y.Huang[J.Algebra,2013,393:142–169]and X.Y.Yang and Z.C.Wang[Rocky Mountain J.Math.,2017,47:1013–1053],but the proof is not too far from their case.Finally,we give some applications about our main results.     

Generalized Vandermonde tensors

We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor A=(Ai1i2...im) a type-1 generalized Vandermonde (GV) tensor, or GV1 tensor, if there exists a vector v=(v1,v2...vn)T such that Ai1i2...im=vi1i2+i3+...+im-m+1, and call A a type-2 (mth order ndimensional) GV tensor, or GV2 tensor, if there exists an (m-1)th order tensor B=(Bi1i2...im-1) such that Ai1i2...im=Bi1i2...im-1im-1. In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.     

Upper bounds for signless Laplacian Z-spectral radius of uniform hypergraphs

Let ■ be a k-uniform hypergraph on n vertices with degree sequence △= d1≥…≥ dn =δ. In this paper, in terms of degree di , we give some upper bounds for the Z-spectral radius of the signless Laplacian tensor (Q(■)) of ■. Some examples are given to show the efficiency of these bounds.