Constant Mean Curvature Surfaces with Boundaryin Hyperbolic Space

 We study constant mean curvature compact surfaces immersed in hyperbolic space with non-empty boundary (=H-surfaces). We prove that the only H-surfaces with boundary circular and 0≤∣H∣≤1, are the umbilical examples. When the surface is embedded, conditions to be umbilical are given. Finally, we characterize umbilical surfaces bounded by a circle among all H-discs with small area.     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory.     

Mean-Value Theorems for Multiplicative Functions on Additive Arithmetic Semigroups via Halász’s Method

 We extend the mean-value theorems for multiplicative functions f on additive arithmetic semigroups, which satisfy the condition ∣f(a)∣≤1, to a wider class of multiplicative functions f for which ∣f(a)∣ is bounded in some average sense, via Halász’s method in classical probabilistic number theory. Received October 18, 2001; in final form April 11, 2002     

An algorithm for evolutionary surfaces

Summary An Algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a series of surfaces. This gives a new method to solve Plateau's problem forH-surfaces.     

Best linear methods for the approximation of functions of the Bergman class by algebraic polynomials

On concentric circles T _ϱ = {z ∈ ℂ: ∣z∣ = ϱ}, 0 ≤ ϱ < 1, we determine the exact values of the quantities of the best approximation of holomorphic functions of the Bergman class A _p , 2 ≤ p ≤ ∞, in the uniform metric by algebraic polynomials generated by linear methods of summation of Taylor series. For 1 ≤ p < 2, we establish exact order estimates for these quantities. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1674–1685, December, 2006.     

Constant mean curvature surfaces of annular type

We consider large solutions of annular type to the volume constrained Douglas problem. They are conformally immersed H-surfaces. By rescaling we set the volume functional at one while the boundary curves shrink to the origin. We show that the solutions become spherical in a precise manner. Spherical bubbling may fail if the conformality condition is dropped. We also discuss the rotationally symmetric annular solutions to the H-surface equation and consider some illustrative examples. Received: 2 May 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001     

Cyclic surfaces in E 5 generated by equiform motions

In this paper, we study cyclic surfaces in E⁵ generated by equiform motions of a circle. The properties of this cyclic surfaces up to the first order are discussed. We prove the following new result: A cyclic 2-surfaces in E⁵ in general are contained in canal hypersurfaces. Finally we give an example.     

Lower semicontinuity of quasiconvex integrals

Summary New lower semicontinuity results for quasiconvex integrals. In particular, under certain structure conditions and growth 0≤F(ξ)≤C(∣≤∣ ^q ) the functional ∫ _Ω F(∇u)dx is proved to be lower semicontinuous onW ^1,q with respect to the weak convergence inW ^1,p, p≥q−1. Research supported by the grant No. 201/93/2171 of Czech Grant Agency (GAČR) and by the grant No. 364 of Charles University (GAUK). This article was processed by the author using the Springer-VerlagTex P Jourlg marco package 1991.     

Affine umbilical surfaces inR 4

Summary A surface in ℝ⁴ is called affine umbilical if for each vector belonging to the affine normal plane the corresponding shape operator is a multiple of the identity. We will classify affine umbilical definite surfaces which either have constant curvature or which satisfy ∇^⊥ g ^⊥. Furthermore, it will be shown that for an affine umbilical definite surface, the affine mean curvature vector can not have constant non-zero length. The last author is a Senior Research Assistant of the National Fund for Scientific Research (Belgium) This article was processed by the author using the Springer-Verlag TEX PJour1g macro package 1991.     

Sui commutatori di alcuni operatori pseudo-differenziali

Riassunto Nel presente lavoro si considera una classe di simboli non-omogenei, più generale di quella introdotta nel [3]; per operatori pseudo-differenzialiEl(x,D) associati a tali simboli si considera il commutatore con l'operatore di Friedrichs Ψ(D) associato a una funzione Ψ(ξ) che gode di alcune speciali proprietà; prendendo inoltre una funzione ζ(ξ) nulla per ∣≤∣⩽1/2 e=1 per |ξ|≥1, si dimostra usando una tecnica dovuta a H?rmander [1] che l'operatore ζ(D)[Ψ(D),El(x,D)] è di ordine ≤-1/2.

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Résumé Dans ce travail on considère une classe de symboles non-homogènes plus générale de celle introduite dans [3]; pour les opérateurs pseudo-différentielsEl(x,D) associés à tels symboles on considère le commutateur [Ψ(D),El(x,D)] avec l'opérateur de Friedrichs Ψ(D) associé à une fonction Ψ(ξ) jouissant de certaines propriétés spéciales. En prenant ensuite une fonction ζ(ξ) nulle pour ∣≤∣⩽1/2 e=1 et =1 pour |ξ|≥1, on démontre avec une technique de H?rmander [1] que l'opérateur ζ(D)[Ψ(D),El(x,D)] est d'ordre ≤-1/2.

     

Distribution of Primitive A-Roots of Composite Moduli Ⅱ

Abstract We improve estimates for the distribution of primitive λ-roots of a composite modulus q yielding an asymptotic formula for the number of primitive λ-roots in any interval I of length ∣I∣ ≫ q ^1/2+∈. Similar results are obtained for the distribution of ordered pairs (x, x ⁻¹) with x a primitive λ-root, and for the number of primitive λ-roots satisfying inequalities such as |x − x ⁻¹| ≤ B. (Dedicated to Professor Wang Yuan on the occasion of his 75th birthday) *Project supported by the National Natural Science Foundation of China (No.19625102) and the 973 Project of the Ministry of Science and Technology of China.     

Algebraic polynomials least deviating from zero in measure on a segment

We investigate the problem of algebraic polynomials with given leading coefficients that deviate least from zero on the segment [–1, 1] with respect to a measure, or, more precisely, with respect to the functional μ(f) = mes{x ∈ [–1, 1]: ∣f (x)∣ ≥ 1}. We also discuss an analogous problem with respect to the integral functionals ∫_–1¹ φ (∣f (x)∣) dx for functions φ that are defined, nonnegative, and nondecreasing on the semiaxis [0, +∞).     

Compositions of isometric immersions¶in higher codimension

Given a submanifold M ⁿ of Euclidean space ℝ ^{n + p} with codimension p≤6, under generic conditions on its second fundamental form, we show that any other isometric immersion of M ⁿ into ℝ ^{n + p + q} , 0≤q≤n− 2p−1 and 2q≤n+ 1 if q≥ 5, must be locally a composition of isometric immersions. This generalizes several previous results on rigidity and compositions of submanifolds. We also provide conditions under which our result is global. 14 March 2001     

Complete submanifolds in Euclidean spaces¶with parallel mean curvature vector

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E ^{n + p} are the totally geodesic Euclidean space E ⁿ , the totally umbilical sphere S ⁿ (c) or the generalized cylinder S ^{n − 1} (c) ×E ¹ if the second fundamental form h satisfies <h>²≤n ²|H|²/ (n− 1). Received: 28 November 2000 / Revised version: 7 May 2001     

The size of the shadow boundary projection

Among all embedded closed manifoldsM ^d ⊂ ℝ ^d+1 with positive exterior curvature ≤k the ratio between the (d − 1)-Hausdorff measure of the shadow boundary projection and the volume ofM ^d is maximized by the sphere of radius 1/k.     

Ruled surfaces of Weingarten type in Minkowski 3-space

In this paper, we study ruled Weingarten surfaces M : x (s, t) = α(s) + tβ (s) in Minkowski 3-space on which there is a nontrivial functional relation between a pair of elements of the set {K, K_II, H, H_II}, where K is the Gaussian curvature, K_II is the second Gaussian curvature, H is the mean curvature, and H_II is the second mean curvature. We also study ruled linear Weingarten surfaces in Minkowski 3-space such that the linear combination aK_II + bH + cH_II + dK is constant along each ruling for some constants a, b, c, d with a² + b² + c² ≠ 0.     

A remark on the unitary group of a tensor product of<Emphasis Type="Italic">n</Emphasis> finite-dimensional Hilbert spaces

LetH _i, 1 ≤ i ≤n be complex finite-dimensional Hilbert spaces of dimension d_i,1 ≤ i ≤n respectively withd _i ≥ 2 for everyi. By using the method of quantum circuits in the theory of quantum computing as outlined in Nielsen and Chuang [2] and using a key lemma of Jaikumar [1] we show that every unitary operator on the tensor productH =H ₁ ⊗H ₂ ⊗... ⊗H _n can be expressed as a composition of a finite number of unitary operators living on pair productsH _i ⊗H _j,1 ≤i,j ≤n. An estimate of the number of operators appearing in such a composition is obtained. Dedicated to Prof. A.K. Roy on his 62nd birthday.     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

Embedded Rotational Hypersurfaces with Constant Scalar Curvature in Sn: A Correction to a Statement in M. L. Leite, Manuscripta Math. 67 (1990), 285-304.

We prove that there exist (n − 1)-dimensional compact embedded rotational hypersurfaces with constant scalar curvature (n − 1)(n − 2)S (S > 1) of S ⁿ other than product of spheres for 4 ≤ n ≤ 6. As a result, we prove that Leite’s Assertion was incorrect.The project is supported by the grant No. 10531090 of NSFC.     

Complementary Sets of Finite Sets

Subsets 𝒜, 𝒮 of an additive group G are complementary if 𝒜 + 𝒮 = G. When 𝒜 is of finite cardinality ∣𝒜∣, and G is ℤ or ℝ, we give sufficient conditions for the existence of a complementary set 𝒮 with “density” not much larger than 1/∣𝒜∣.