The Spectral Pictures of Completely Irreducible Operators and Decomposition Theorem Of Hilbert Space Operators

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Equivalent Separating Curves on Closed Surface of Genus 2

Let {A,B} be a complete system of the closed orientable surface F of genus 2. A simple closed curve C on F is separating with respect to (A, B) if it is disjoint from A∪B and it cuts F into two once-punctured tori X, Y with A(?)X, B(?)Y. Letγbe a simple closed curve on F which is disjoint from A∪B and intersects C essentially in two points. In this paper, we show that up to isotopy, {hnγ(C):n∈Z} is the set containing all the simple closed curves on F which is separating with respect to (A,B), where hγis the Dehn twist alongγon F. This also shows how two simple closed curves on F which are separating with respect to (A,B) are related. The result can be applied to yield all Haken spheres of a Heegaard splitting V∪F W which are weakly equivalent to a given Heken sphere of the splitting.     

The Most Symmetric Nonsingular Plane Curves of Degree n≤20, I*

If n is an odd prime less than 20, then the most symmetric nonsingular plane curves in P ² of degree n are projectively equivalent to the Fermat curve x ⁿ +y ⁿ +z ⁿ .     

The Degree of Unsolvability of Models

The purpose of this paper is to describe the complexity of models by theirdegrees of unsolvability,J.Richter defined the degree of a structure to be deg ()=sup{deg(),deg(R_i),i=1,…,n},Where is a model for afinite language L={R_i,i=1,…,n}and the universe of is a subset of ω. Shepointed out that, according to her definition, there can be models which areisomorphic but their degrees are different.Also,her discussions are restricted tofinite languages and models whose universes are subsets of ω.     

The Moduli Space of Curves, Double Hurwitz Numbers, and Faber��s Intersection Number Conjecture

We define the dimension 2g − 1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of \mathbbP₁{{\mathbb{P}_1}} with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by “top intersections” of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a “combinatorialization” of top intersections of Y{\Psi} -classes. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ are well understood, we obtain Faber’s Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga’s theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of κ _g–2.     

Applications of Automata and Graphs: Labeling-Operators in Hilbert Space I

We show that certain representations of graphs by operators on Hilbert space have uses in signal processing and in symbolic dynamics. Our main result is that graphs built on automata have fractal characteristics. We make this precise with the use of Representation Theory and of Spectral Theory of a certain family of Hecke operators. Let G be a directed graph. We begin by building the graph groupoid $\Bbb{G}$ induced by G, and representations of  $\Bbb{G}$ . Our main application is to the groupoids defined from automata. By assigning weights to the edges of a fixed graph G, we give conditions for $\Bbb{G}$ to acquire fractal-like properties, and hence we can have fractaloids or G-fractals. Our standing assumption on G is that it is locally finite and connected, and our labeling of G is determined by the “out-degrees of vertices”. From our labeling, we arrive at a family of Hecke-type operators whose spectrum is computed. As applications, we are able to build representations by operators on Hilbert spaces (including the Hecke operators); and we further show that automata built on a finite alphabet generate fractaloids. Our Hecke-type operators, or labeling operators, come from an amalgamated free probability construction, and we compute the corresponding amalgamated free moments. We show that the free moments are completely determined by certain scalar-valued functions.     

Doob–Meyer-Decomposition of Hilbert Space Valued Functions

(1) We give here a new proof of the Doob–Meyer-decomposition which is rather quick and elementary. It is more general in some aspects, and weaker in other aspects if compared to other approaches. The functions are defined on a totally ordered set with image in a Hilbert space. (2) We also give a second variant of the Doob–Meyer-decomposition; it is more specialized. (3) We apply (1) by reproving the Doléans-measure in a special setting and demonstrate that the stochastic integral could be defined on more general totally ordered time scales then .     

Multiple Degree Reduction and Elevation of Bézier Curves Using Jacobi–Bernstein Basis Transformations

In this article, we find the optimal r times degree reduction of Bézier curves with respect to the Jacobi-weighted L ₂-norm on the interval [0, 1]. This method describes a simple and efficient algorithm based on matrix computations. Also, our method includes many previous results for the best approximation with L ₁, L ₂, and L _∞-norms. We give some examples and figures to demonstrate these methods.     

The Hilbert function of generic plane sections of curves of ℙ3

Summary A characteristic condition is given on a zero-dimensional differentiable 0-sequenceH={h _i}_i0,h ₁3, in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of ³, hence of points of ² with the uniform position property. In this way an answer is given to some question stated by Harris in [Ha₂].The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in ³ whose generic plane section has an assigned Hilbert function satisfying that condition.A.M.S. (1980) Subject classification. Primary 14C20, 14H99. Secondary 14C05     

The Degree Theory of a New Operator

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The Equivalence of Ishikawa-Mann and Multistep Iterations in Banach Space

Let E be a real Banach space and T be a continuous Φ-strongly accretive operator. By using a new analytical method, it is proved that the convergence of Mann, Ishikawa and three-step iterations are equivalent to the convergence of multistep iteration. The results of this paper extend the results of Rhoades and Soltuz in some aspects.     

Properties for a Class of Holomorphic Mapson Hilbert Space

Itiswell-knownthattheanalyticfunctionwithpositiverealpartintheunitdiscofthecomplexplaneplaysaveryimportantroleinthegeometrictheoryofcomplexvariablefunc-tions.Ithasbeenremarkablehowtoextendtheclassicalresultofthetheoryoffunctionstoabstractspacesinrecentyears[2].Inthisnotewedefineageneralizedhalf-planeinHilbertspaceandfindthebiholomorphicmapoftheunitballonHilbertspaceontothishalf-plane.Asacorollary,wegiveashortproofofaresultin[5].UsingtheaboveholomooorphicmapweobtainthedistortiontheoremandPick…     

Sur le schéma de Hilbert des courbes de degré d et genre (d−3)(d−4)2 de ℙk3

In this Note we study the Hilbert scheme of space curves of degree d and genus (d − 3)(d − 4)/2.     

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.     

Hilbert Space of Probability Density Functions Based on Aitchison Geometry

The set of probability functions is a convex subset of L1 and it does not have a linear space structure when using ordinary sum and multiplication by real constants. Moreover, difficulties arise when dealing with distances between densities. The crucial point is that usual distances are not invariant under relevant transformations of densities. To overcome these limitations, Aitchison＇s ideas on compositional data analysis are used, generalizing perturbation and power transformation, as well as the Aitchison inner product, to operations on probability density functions with support on a finite interval. With these operations at hand, it is shown that the set of bounded probability density functions on finite intervals is a pre-Hilbert space. A Hilbert space of densities, whose logarithm is square-integrable, is obtained as the natural completion of the pre-Hilbert space.     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2×V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

The Moduli Space of Points in the Boundary of Complex Hyperbolic Space

We consider the space M(n,m)\mathcal{M}(n,m) of ordered m-tuples of distinct points in the boundary of complex hyperbolic n-space, H_\mathbbCⁿ\mathbf{H}_{\mathbb{C}}^{n}, up to its holomorphic isometry group PU(n,1). An important problem in complex hyperbolic geometry is to construct and describe the moduli space for M(n,m)\mathcal{M}(n,m). In particular, this is motivated by the study of the deformation space of complex hyperbolic groups generated by loxodromic elements. In the present paper, we give the complete solution to this problem.