Rational and transcendental growth series for the higher Heisenberg groups

This paper considers growth series of 2-step nilpotent groups with infinite cyclic derived subgroup. Every such group G has a subgroup of finite index of the form H _n ×ℤ ^m , where H _n is the discrete Heisenberg group of length 2n+1. We call n the Heisenberg rank of G. We show that every group of this type has some finite generating set such that the corresponding growth series is rational. On the other hand, we prove that if G has Heisenberg rank n ≧ 2, then G possesses a finite generating set such that the corresponding growth series is a transcendental power series. Oblatum 1-III-1995 & 28-XII-1995     

Distributional solutions of Burgers' equation and intrinsic regular graphs in Heisenberg groups

In the present paper we will characterize the continuous distributional solutions of Burgers' equation as those which induce intrinsic regular graphs in the first Heisenberg group H¹≡R³, endowed with a left-invariant metric d_∞ equivalent to its Carnot-Carathéodory metric. We will also extend the characterization to higher Heisenberg groups Hn≡R2n+1.     

Heisenberg群上无穷远处的集中列紧原理和具有Sobolev临界指数的p - 次Laplace方程多解的存在性

通过建立Heisenberg群上无穷远处的集中列紧原理, 研究了如下$p$ -次Laplace方程 -Δ_{H, p}u=λg(ξ)|u|^q-2u+f (ξ)|u|^p*-2u,在Hⁿ上, u∈ D^{1, p}(Hⁿ), 其中ξ∈Hⁿ,λ∈R,1j, 且m, j为整数.     

A theorem of the Wiener—Tauberian type forL 1(H n)

The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group Hⁿ and the unitary group U(n), acts on Hⁿ in a natural way. Here we prove a Wiener-Tauberian theorem for L¹ (Hⁿ) with this HM(n)-action on Hⁿ i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L¹ (Hⁿ) in order that the linear span of^gf : g∈HM(n) is dense in L¹(Hⁿ), where^gf(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈Hⁿ.     

Complete Hypersurfaces with Constant Mean Curvature in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with constant mean curvature and with two distinct principal curvatures, we give a characterization of the torus S ¹(a) × S^n-1(?{1-a²})S^{n-1}(\sqrt{1-a^2}) , where a²=\frac2+nH²±?{n²H⁴+4(n-1)H²}2n(1+H²)a^2=\frac{2+nH^2\pm\sqrt{n^2H^4+4(n-1)H^2}}{2n(1+H^2)} . We extend recent results of Hasanis et al. [5] and Otsuki [10].     

A class of Sasakian 5-manifolds

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H _{2n + 1}. Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H ₅ or a semidirect product ? ? (H ₃ × ?). In particular, the compact quotient is an S ¹-bundle over a four-dimensional Kähler solvmanifold.     

Submanifolds with parallel normalized mean curvature vector in a unit sphere

Let M ⁿ be an n-dimensional closed submanifold of a sphere with parallel normalized mean curvature vector. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M ⁿ , respectively. Assume that the fundamental group \({\pi_{1}(M^{n})}\) of M ⁿ is infinite and \({S\, \leqslant\, S(H)=n+\frac{n^{3}H^{2}}{2(n-1)}-\frac{n(n-2)H}{2(n-1)}\sqrt{n^{2}H^{2}+4(n-1)}}\), then S is constant, S = S(H), and M ⁿ is isometric to a Clifford torus \({S^{1}(\sqrt{1-r^{2}})\times S^{n-1}(r)}\) with \({r^{2}\leqslant \frac{n-1}{n}}\).     

Diameters of some classes of differentiable periodic functions in the space L

In this paper, diameters in the sense of A. N. Kolmogorov are found for the class of 2π-periodic functions W^(r)H_ω in the space L, that is, d_2n-1(W^(r)H_ω,L), where ω(t) is an upper-convex regular modulus of continuity (r, n=1, 2, ...). An estimate from below is found for diameters in the sense of I. M. Gel'fand, that is, d^2n-!(W^(r)H_ω,L).     

四阶非线性抛物方程解之存在性

四阶椭圆方程解之极值原理最先由Dunninger,D.R.提出.Goyal,V.B.和Singl,K.P.推广到半线性方程的情形,文献[5]、[6]作进一步的推广,都对文献[3]的某些结论作了修正,并且都建立四阶椭圆方程边值问题解的存在性定理。关于四阶抛物方程解的极值原理及唯一性定理作者作过讨论。这篇短文研究四阶非线性抛物方程的初值问题和混合问题解的存在性,其前提是所有解的最大模有一致先验的上界。     

Large time behavior of solutions for critical and subcritical complex Ginzburg-Landau equations in H<Superscript>1</Superscript>

Considering the Cauchy problem for the critical complex Ginzburg-Landau equation in H¹(Rⁿ), we shall show the asymptotic behavior for its solutions in C(0, ∖;H¹(Rⁿ)) ∩ L²(0, ∖;H^1,2n/(n-2)(R²)), n≥3. Analogous results also hold in the case that the nonlinearity has the subcritical power in H¹(Rⁿ), n≥1. Dedicated to Professor Zhou Yulin for his 80th birthday.     

Estimates for spectral projection operators of the sub-laplacian on the heisenberg group

In this paper, we use Laguerre calculus to find theLP spectrum (λ, Μ) of the pair (L, iT). Here md T = ∂/∂t with a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray λ > 0 in the Heisenberg brush and show that they are Calderón-Zygmund-Mikhlin operators. Estimates for these operators in L _k ^p (H_n), H^P(H_n), and S _k ^pv (H_n) spaces can therefore be deduced.     

Conformal paracontact curvature and the local flatness theorem

A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.     

Two versions of the Nikodym maximal function on the Heisenberg group

The classical Nikodym maximal function on the Euclidean plane R² is defined as the supremum over averages over rectangles of eccentricity N; its operator norm in L²(R²) is known to be O(logN). We consider two variants, one on the standard Heisenberg group H¹ and the other on the polarized Heisenberg group . The latter has logarithmic L² operator norm, while the former has the L² operator norm which grows essentially of order O(N1/4). We shall imbed these two maximal operators in the family of operators associated to the hypersurfaces {(x₁,x₂,αx₁x₂)} in the Heisenberg group H¹ where the exceptional blow up in N occurs when α=0.     

The Number of Triangulations of the Cyclic Polytope C (n , n -4)

We show that the exact number of triangulations of the standard cyclic polytope C(n,n-4) is (n+4)2 ^(n-4)/2 -n if n is even and \left((3n+11)/2\right)2 ^(n-5)/2 -n if n is odd. These formulas were previously conjectured by the second author. Our techniques are based on Gale duality and the concept of virtual chamber. They further provide formulas for the number of triangulations which use a specific simplex. We also compute the maximum number of regular triangulations among all the realizations of the oriented matroid of C(n,n-4) . Received October 24, 2000, and in revised form July 8, 2001. Online publication November 7, 2001.     

Divergence symmetries of critical Kohn-Laplace equations on Heisenberg groups

We show that every Lie point symmetry of semilinear Kohn-Laplace equations with a power-law nonlinearity on the Heisenberg group H ¹ is a divergence symmetry if and only if the corresponding exponent takes a critical value.     

Spherical means and CR functions on the Heisenberg group

The injectivity of the spherical mean value operator on the Heisenberg group is studied. Whenf ∈L ^P (Hⁿ), 1 ≤p < ∞ it is proved that the spherical mean value operator is injective. When 1 ≤p ≤ 2,f(z, ·) ∈L ^P (ℝ) the same is proved under much weaker conditions in the z-variable. Some extensions of recent results of Agranovskyet al. regardingCR functions on the Heisenberg group are also obtained.     

On Cauchy's bound for zeros of a polynomial

In this paper we have improved Cauchy's bound for zeros of a polynomialp(z)=z ⁿ+a _n+1 z ^n-1+a _n-2 z ^n-2+......+a ₁ z+a ₀. Our result is best possible and sharpens some other results also.     

Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the $2n + 1$ -dimensional Heisenberg group H _n and the 4-manifolds $Nil^4 $ and $Nil^3 \times \mathbb{R}$ endowed with an arbitrary left-invariant metric admit no C ³-regular immersions into Euclidean spaces $\mathbb{R}^{2n + 2} $ and $\mathbb{R}^5 $ , respectively.     

Group classification of semilinear Kohn–Laplace equations

We study the Lie point symmetries of semilinear Kohn–Laplace equations on the Heisenberg group H¹$H^1$ and obtain a complete group classification of these equations.     

TheT-ideal generated by the standard identitys 3[x 1,x 2,x 3]

LetK = T_o(s₃), {c_n} its codimensions, {l_n} its colengths and {Χ_n} its sequence of co-characters. For 9≦n, c_n =2n - 1 or c_n =n(n + l)/2- 1, 3≦l_n ≦4 and χ_n =[n] + 2[n-1,1] + α[n-2,2] + β[2²,1^n?4] where α + β≦l.