A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on Two-Manifolds

We establish the peak point conjecture for uniform algebrasgenerated by smooth functions on two-manifolds: if A is a uniformalgebra generated by smooth functions on a compact smooth two-manifoldM, such that the maximal ideal space of A is M, and every pointof M is a peak point for A, then A = C(M). We also give an alternativeproof in the case when the algebra A is the uniform closureP(M) of the polynomials on a polynomially convex smooth two-manifoldM lying in a strictly pseudoconvex hypersurface in Cⁿ.     

Eva Kallin's Lemma On Polynomial Convexity

We give a survey on the use of Eva Kallin's lemma. This lemmagives a condition on two polynomially convex sets in Cⁿ underwhich their union is polynomially convex. This result has provedto be a useful tool in different areas of complex function theoryof several variables, for instance in the study of polynomialconvexity of the union of totally real surfaces, and in approximationproblems in function algebras.     

Riemann Boundary Value Problems on the Sphere in Clifford Analysis

We present and study a type of Riemann boundary value problems (for short RBVPs) for polynomially monogenic functions, i.e. null solutions to polynomially generalized Cauchy-Riemann equations, over the sphere of ${\mathbb{R}^{n+1}}$ . Making use of Fischer type decomposition and the Clifford calculus for polynomially monogenic functions, we obtain explicit expressions of solutions of this kind of boundary value problems over the sphere of ${\mathbb{R}^{n+1}}$ . As special cases the solutions of the corresponding boundary value problems for classical polyanalytic functions and metaanalytic functions are derived respectively.     

Projections of polynomial hulls

The following theorem is discussed. Let X be a compact subset of the unit sphere in $\text{C}$ⁿ whose polynomially convex hull, $\text{X}\text{?}$, contains the origin, then the sum of the areas of the n coordinate projections of $\text{X}\text{?}$ is bounded below by π. This applies, in particular, when $\text{X}\text{?}$ is a one-dimensional analytic subvariety V containing the origin, and in this case generalizes the fact that the “area” of V is at least π; in fact, the area of V is the sum of the areas of the n coordinate projections when these areas are counted with multiplicity. A convex analog of the theorem is obtained. Hartog's theorem that separate analyticity implies analyticity, usually proved with the use of subharmonic functions (Hartog's lemma), will be derived as a consequence of the theorem, the proof of which is based upon the elements of uniform algebras.     

Boundedness and compactness of Hankel operators on the sphere

We consider Hankel operators on the Hardy space of the unit sphere in Cn. We show that a large amount of information about the function f−Pf can be recovered from the Hankel operator Hf. For example, if Hf is compact, then the function f−Pf is necessarily in VMO.     

Ruled Special Lagrangian 3-Folds In C3

This is the fifth in a series of papers constructing explicitexamples of special Lagrangian submanifolds in C^m. A submanifoldof C^m is ruled if it is fibred by a family of real straightlines in C^m. This paper studies ruled special Lagrangian 3-foldsin C³, giving both general theory and families of examples.Our results are related to previous work of Harvey and Lawson,Borisenko, and Bryant. Special Lagrangian cones in C³ are automaticallyruled, and each ruled special Lagrangian 3-fold is asymptoticto a unique special Lagrangian cone. We study the family ofruled special Lagrangian 3-folds N asymptotic to a fixed specialLagrangian cone N0. We find that this depends on solving a linearequation, so that the family of such N has the structure ofa vector space. We also show that the intersection of N0 withthe unit sphere S⁵ in C³ is a Riemann surface, and constructa ruled special Lagrangian 3-fold N asymptotic to N0 for eachholomorphic vector field w on . As corollaries of this we writedown two large families of explicit special Lagrangian 3-foldsin C³ depending on a holomorphic function on C, which includemany new examples of singularities of special Lagrangian 3-folds.We also show that each special Lagrangian T2-cone N0 can beextended to a 2-parameter family of ruled special Lagrangian3-folds asymptotic to N0, and diffeomorphic to T²xR. 2000 Mathematical Subject Classification: 53C38, 53D12.     

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in ${\mathbb C^n}We compute the Szeg? kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb Cⁿ{\mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.     

Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve.     

From Trace States to States on the K0 Group of a Simple C*-Algebra

It is shown that any continuous affine surjection from a metrizableChoquet simplex onto a compact convex set occurs as the restrictionmap from the tracial state space onto the state space of theK₀ group of a separable unital simple C^*-algebra which is theinductive limit of a sequence of subhomogeneous C^*-algebras     

Uniform algebras on the sphere invariant under group actions

It is shown under certain conditions that a uniform algebra on the unit sphere S in C ² that is invariant under the action of the 2-torus must be C(S). Contrasting with this, an example is presented showing that the statement becomes false when 2 is replaced by n > 2. It is also shown that C(M) is the only uniform algebra on a smooth manifold M that is invariant under a transitive Lie group action on its maximal ideal space. The results presented answer a question raised by Ronald Douglas in connection with a conjecture of William Arveson.     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

A note on compact elements in the lattice of radicals

An example of radical compact ring A whose Dorroh extension A¹ is not radical compact is constructed.     

Uncountable equilateral sets in Banach spaces of the form <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)

The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin’s axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form C(K) has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form C(K) which has no uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε > 0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis and G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples of nonseparable Banach spaces which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.     

Constructive approximations of spherical functions

Approximation of functions on the sphere arises in almost all applications modeling data collected on the surface of the earth and for reconstruction of various processes in spherical coordinates. Constructive approximation of high dimensional spherical functions are useful for approximation of processes of several variables on a compact subset of a Euclidean space by mapping the data onto the unit sphere of a space having one higher dimension, avoiding the boundary effect of the set. Interpolation operators on the circle and periodic domains (based on a class of basis functions and data values) are essential for many high performance simulations. These operators are represented by analytical summation formulas that can be computed very efficiently using the fast Fourier transform. This work is concerned with construction of a similar class of interpolation and quasi-interpolation operators on the unit sphere in ℝ^q , for q = 3, 4, 5, · · ·, using a new class of basis functions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simplices of Maximal Volume or Minimal Total Edge Length in Hyperbolic Space

In this paper, we are mainly concerned with n-dimensional simplicesin hyperbolic space Hⁿ. We will also consider simplices withideal vertices, and we suggest that the reader keeps the Poincaréunit ball model of hyperbolic space in mind, in which the sphereat infinity Hⁿ() corresponds to the bounding sphere of radius1. It is known that all hyperbolic simplices (even the idealones) have finite volume. However, explicit calculation of theirvolume is generally a very difficult problem (see, for example,[1] or [16]). Our first theorem states that, amongst all simplicesin a closed geodesic ball, the simplex of maximal volume isregular. We call a simplex regular if every permutation of itsvertices can be realized by an isometry of Hⁿ. A correspondingresult for simplices in the sphere has been proved by Böröczky[4].     

An O(n2) algorithm for a controllable machine scheduling problem

A single-machine scheduling problem with controllable processingtimes is discussed in this paper. For some jobs, the processingtime can be crashed up to u units of time with the additionalcost c per unit of time crashed. The object is to find an optimalprocessing sequence as well as crash activities to minimizetotal costs of completion and crash. This problem is shown tobe polynomially solvable, and an O(n²) algorithm is given togetherwith the theoretical proof.     

The Webster scalar curvature problem on the three dimensional CR manifolds

In this paper, we prove some existence results for the Webster scalar curvature problem on the three dimensional CR compact manifolds locally conformally CR equivalent to the unit sphere S³ of C². Our methods are based on the techniques related to the theory of critical points at infinity.     

Traces of pluriharmonic functions on curves

We prove that, if γ is a simple smooth curve in the unit sphere inC ⁿ, the space o pluriharmonic functions in the unit ball, continuous up to the boundary, has a trace of finite cof dimension in the space of all continuous functions on the curve. First author partially supported by the Swedish Natural Science Research Council. Second author partially supported by CICYT grant PB85-0374.     

Integration of the equations of a rotational motion of a rigid body in quaternion algebra. The Euler case

A dynamical system is constructed in the multiplicative group of the quarternion algebra H that serves as the configuration space. A homomorphism H → SO(3) is used such that the unit sphere S³ ⊂ H, invariant under the system, is transformed into the rotation group SO(3). The homomorphic image of the system is identical with the dynamics of rotational motion of a rigid body. The equations of motion are completely integrated in the Euler case. To this end Weierstrass' elliptic functions are used. The following goals are achieved within the framework of the method: (a) when representing the algorithms for modelling the dynamics it suffices to use only one chart from the atlas of the phase space manifold, (b) the point in the configuration space of the actual motion lies on the unit sphere, which ensures the best accuracy in numerical procedures, and (c) in the majority of applications the right-hand sides of the equations of perturbed motion depend polynomially on the phase variables, which simplifies the use of computer algebra in analytic theories.     

First nonzero eigenvalue of a pseudo-umbilical hypersurface in the unit sphere

S. Deshmukh has obtained interesting results for first nonzero eigenvalue of a minimal hypersurface in the unit sphere. In the present article, we generalize these results to pseudoumbilical hypersurface and prove: What conditions are satisfied by the first nonzero eigenvalue λ ₁ of the Laplacian operator on a compact immersed pseudo-umbilical hypersurface M in the unit sphere S ⁿ⁺¹. We also show that a compact immersed pseudo-umbilical hypersurface of the unit sphere S ⁿ⁺¹ with λ ₁ = n is either isometric to the sphere S ⁿ or for this hypersurface an inequaluity is fulfilled in which sectional curvatures of the hypersuface M participate.