Zeta functions of bielliptic surfaces over finite fields

Let S be a bielliptic surface over a finite field, and let the elliptic curve B be the image of the Albanese mapping S → B. In this case, the zeta function of the surface is equal to the zeta function of the direct product ?¹ × B. A classification of the possible zeta functions of bielliptic surfaces is also presented in the paper.     

Semipositivity of generic ordinary fibration over a curve

Let f : X → B be a generic ordinary proper fibration over a complete curve in positive characteristics, we prove that the dual of higher direct image sheaf R~1 f_*O_X is nef. As a corollary, we show that f_*ω_(S/B) is nef, if f : S → B is a fibration from a surface to a curve with generic ordinary fibres.     

Localization of the eigenvalues of a pencil of positive definite matrices

Let A and B be real square positive definite matrices close to each other. A domain S on the complex plane that contains all the eigenvalues λ of the problem Az = λBz is constructed analytically. The boundary ?S of S is a curve known as the limacon of Pascal. Using the standard conformal mapping of the exterior of this curve (or of the exterior of an enveloping circular lune) onto the exterior of the unit disc, new analytical bounds are obtained for the convergence rate of the minimal residual method (GMRES) as applied to solving the linear system Ax = b with the preconditioner B.     

A note on generating <Emphasis Type="Italic">P</Emphasis>-matrices

We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ^?1 B, AB ^?1, B ^?1 A and BA ^?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.     

On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

Assume that (N, ?) and (M, S) are two Riemann surfaces with conformal metrics ? and S. We prove that if there is a harmonic homeomorphism between an annulus A ? N with a conformal modulus Mod(A) and a geodesic annulus A _S (p, ρ₁, ρ₂)?M, then we have ρ₂/ρ₁ ≥ Ψ _S Mod(A)²+ 1, where Ψ _S is a certain positive constant depending on the upper bound of Gaussian curvature of the metric S. An application for the minimal surfaces is given.     

On the Blaschke isoparametric hypersurfaces in the unit sphere

Given an immersed submanifold x ： M^M → S^n in the unit sphere S^n without umbilics, a Blaschke eigenvalue of x is by definition an eigenvalue of the Blaschke tensor of x. x is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the MSbius geometry of submanifolds. When n = 4, the corresponding classification theorem was given by the authors. In this paper, we are able to complete the corresponding classification for n = 5. In particular, we shall prove that all the Blaschke isoparametric hypersurfaces in S^5 with more than two distinct Blaschke eigenvalues are necessarily Mobius isoparametric.     

Equivariant Classification of <Emphasis Type="Italic">b</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-symplectic Surfaces

Inspired by Arnold’s classification of local Poisson structures [1] in the plane using the hierarchy of singularities of smooth functions, we consider the problem of global classification of Poisson structures on surfaces. Among the wide class of Poisson structures, we consider the class of b^m-Poisson structures which can be also visualized using differential forms with singularities as b^m-symplectic structures. In this paper we extend the classification scheme in [24] for b^m-symplectic surfaces to the equivariant setting. When the compact group is the group of deck-transformations of an orientable covering, this yields the classification of these objects for nonorientable surfaces. The paper also includes recipes to construct b^m-symplectic structures on surfaces. The feasibility of such constructions depends on orientability and on the colorability of an associated graph. The desingularization technique in [10] is revisited for surfaces and the compatibility with this classification scheme is analyzed in detail.     

A characterization of the Ejiri torus in <Emphasis Type="Italic">S</Emphasis><Superscript>5</Superscript>

We conjecture that a Willmore torus having Willmore functional between 2π ² and 2π ² \(\sqrt 3 \) is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S ⁵ is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S ⁿ by reducing them into elastic curves in S ³, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S ⁵ attains the minimum 2π ² \(\sqrt 3 \), which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S ⁵. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S ³. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.     

On the geometry of Sasakian-Einstein 5-manifolds

On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [24]. We expand on the recent work of Demailly and Kollár [14] and Johnson and Kollár [20] who give methods for constructing Kähler-Einstein metrics on log del Pezzo surfaces. By a previous result of the first two authors [9], circle V-bundles over log del Pezzo surfaces with Kähler-Einstein metrics have Sasakian-Einstein metrics on the total space of the bundle. Here these simply connected 5-manifolds arise as links of isolated hypersurface singularities which by the well known work of Smale [36] together with [11] must be diffeomorphic to S ⁵ #l(S ² ×S ³ ). More precisely, using methods from Mori theory in algebraic geometry we prove the existence of 14 inequivalent Sasakian-Einstein structures on S ² ×S ³ and infinite families of such structures on #l(S ² ×S ³ ) with 2≤l≤7. We also discuss the moduli problem for these Sasakian-Einstein structures.     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

Steepest descent curves of convex functions on surfaces of constant curvature

Let S be a complete surface of constant curvature K = ±1, i.e., S ² or л ², and Ω ? S a bounded convex subset. If S = S ², assume also diameter(Ω) < π/2. It is proved that the length of any steepest descent curve of a quasi-convex function in Ω is less than or equal to the perimeter of Ω. This upper bound is actually proved for the class of G-curves, a family of curves that naturally includes all steepest descent curves. In case S = S ², the existence of G-curves, whose length is equal to the perimeter of their convex hull, is also proved, showing that the above estimate is indeed optimal. The results generalize theorems by Manselli and Pucci on steepest descent curves in the Euclidean plane.     

Pointwise multipliers for Besov spaces of dominating mixed smoothness-Ⅱ

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R~d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R~d).     

On compact splitting complex submanifolds of quotients of bounded symmetric domains

We study compact complex submanifolds S of quotient manifolds X = ?/Γ of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds S ? X, i.e., under the assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first type of results concerns S ? X which are characteristic complex submanifolds, i.e., embedding ? as an open subset of its compact dual manifold M by means of the Borel embedding, the non-zero(1, 0)-vectors tangent to S lift under a local inverse of the universal covering map π : ? → X to minimal rational tangents of M.We prove that a compact characteristic complex submanifold S ? X is necessarily totally geodesic whenever S is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic complex submanifolds of quotients of the complex unit ball Bnobtained by Mok(2005). The proof given here is however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify the holomorphic tangent bundle TSas a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle TXto S. The second type of results concerns characterization of total geodesic submanifolds among compact splitting complex submanifolds S ? X deduced from the results of Aubin(1978)and Yau(1978) which imply the existence of K¨ahler-Einstein metrics on S ? X. We prove that compact splitting complex submanifolds S ? X of sufficiently large dimension(depending on ?) are necessarily totally geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to TS,which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the sharp lower bound on dim(S) guaranteeing total geodesy of S ? X for the case of the type-I domains of rank2 and the case of type-IV domains, and examine a case which is critical for both conjectures, i.e., on compact complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual to the Grassmannian of 2-planes in C~4.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Estimation Problems for Periodically Correlated Isotropic Random Fields

Spectral theory of isotropic random fields in Euclidean space developed by M. I. Yadrenko is exploited to find a solution to the problem of optimal linear estimation of the functional

$$ A\zeta ={\sum\limits_{t=0}^{\infty}}\,\,\,{\int_{S_n}} \,\,a(t,x)\zeta (t,x)\,m_n(dx) $$

which depends on unknown values of a periodically correlated (cyclostationary with period T) with respect to time isotropic on the sphere S _n in Euclidean space E ⁿ random field ζ(t, x), t?∈?Z, x?∈?S _n . Estimates are based on observations of the field ζ(t, x)?+?θ(t, x) at points (t, x), t?=???1,???2, ..., x?∈?S _n , where θ(t, x) is an uncorrelated with ζ(t, x) periodically correlated with respect to time isotropic on the sphere S _n random field. Formulas for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional Aζ are obtained. The least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal estimates of the functional Aζ are determined for some special classes of spectral densities.

     

On the computation of <Emphasis Type="Italic">C</Emphasis>* certificates

The cone of completely positive matrices C* is the convex hull of all symmetric rank-1-matrices xx ^T with nonnegative entries. While there exist simple certificates proving that a given matrix \({B\in C^*}\) is completely positive it is a rather difficult problem to find such a certificate. We examine a simple algorithm which—for a given input B—either determines a certificate proving that \({B\in C^*}\) or converges to a matrix \({\bar S}\) in C* which in some sense is “close” to B. Numerical experiments on matrices B of dimension up to 200 conclude the presentation.     

Riemann Sums and Möbius

Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Möbius function has the property that it maps from \({ \cup _{0 < r < \infty }}{l^r}(s)\) to \({ \cap _{0 < r < \infty }}{l^r}(s)\), injectively. If 0 < r< 2 and ξ ∈ l^r (S), the series \({f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx} \) converges uniformly to an element of f_ξR₀, i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} {{f_\zeta }} (j/N) = 0,N = 1,2....\) If \({A_{\zeta \lambda }} = \lambda {\zeta _\lambda },{\zeta _\lambda }(1) = 1\), then ξ_λ is multiplicative. If \({f_{{\zeta _\lambda }}} \in {\Lambda _a}\), the space of α-Lipschitz continous functions, for some α > 0, and if χ is any Dirichlet character, then L(s, χ) ≠ 0, Res > 1 ? α. Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to f_ξ ∈ Λ_α, α < 1/2, ξ ∈ l^r (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies f_ξ ∈ Λ_α, α < 1/4, ξ ∈ l^r (S), 0 < r < 2. The question of whether R₀ ∩ Λ_α ≠ {0} for some positive α > 0 is open.     

Finite Sets as Complements of Finite Unions of Convex Sets

Suppose S?? ^d is a set of (finite) cardinality n, whose complement can be written as the union of k convex sets. It is perhaps intuitively appealing that when n is large k must also be large. This is true, as is shown here. First the case in which the convex sets must also be open is considered, and in this case a family of examples yields an upper bound, while a simple application of a theorem of Björner and Kalai yields a lower bound. Much cruder estimates are then obtained when the openness restriction is dropped. For a given set S the problem of determining the smallest number of convex sets whose union is ? ^d ?S is shown to be equivalent to the problem of finding the chromatic number of a certain (infinite) hypergraph ? _S . We consider the graph \(\mathcal {G}_{S}\) whose edges are the 2-element edges of ? _S , and we show that, when d=2, for any sufficiently large set S, the chromatic number of \(\mathcal{G}_{S}\) will be large, even though there exist arbitrarily large finite sets S for which \(\mathcal{G}_{S}\) does not contain large cliques.     

Block partitions: an extended view

Given a sequence \({S=(s_1, \ldots,s_m) \in [0, 1]^m}\), a block B of S is a subsequence \({B=(s_i,s_{i+1}, \ldots,s_j)}\). The size b of a block B is the sum of its elements. It is proved in [1] that for each positive integer n, there is a partition of S into n blocks B₁, \({\ldots,}\) B _n with \({|b_i - b_j| \le 1}\) for every i, j. In this paper, we consider a generalization of the problem in higher dimensions.     

Construction of hyperbolic Riemann surfaces with large systoles

Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface S_max a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.