Moduli of smoothness and approximation on the unit sphere and the unit ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a K-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding K-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.     

New Moduli of Smoothness on the Unit Ball and Other Domains,Introduction and Main Properties

A new set of moduli of smoothness on a large variety of Banach spaces of functions on the unit ball is introduced. These measures of smoothness utilize uniformly bounded holomorphic semigroups on the Banach space in question. The new moduli are “correct” in the sense that they satisfy direct (Jackson) and weak converse inequalities. The method used also applies to spaces of functions on the simplex and the unit sphere, and while the main goal is the investigation of properties and relations concerning the unit ball, many of the results will be given for other domains and situations. The classic properties, including equivalence with appropriate \(K\) -functionals or realization functionals, will be established. Bernstein- and Kolmogorov-type inequalities are proved.     

A compactification of the moduli variety of stable vector 2-bundles on a surface in the Hilbert scheme

A new compactification of the variety of moduli of stable vector 2-bundles with Chern classes c ₁ and c ₂ is constructed for the case in which the universal family of stable sheaves with given values of invariants is defined and there are no strictly semistable sheaves. The compactification is a subvariety in the Hilbert scheme of subschemes of a Grassmann manifold with fixed Hilbert polynomial; it is obtained from the variety of bundle moduli by adding points corresponding to locally free sheaves on surfaces which are modifications of the initial surface. Moreover, a morphism from the new compactification of the moduli space to its Gieseker-Maruyama compactification is constructed.     

Energy functions on moduli spaces of flat surfaces with erasing forest

This paper follows on from Nguyen (Geom Funct Anal 20(1):192–228, 2010), in which we study flat surfaces with erasing forest, these surfaces are obtained by deforming the metric structure of translation surfaces, and their moduli space can be viewed as a deformation of the moduli space of translation surfaces. We showed that the moduli spaces of such surfaces are complex orbifolds, and admit a natural volume form μ _Tr. The aim of this paper is to show that the volume of those moduli spaces with respect to μ _Tr, normalized by some energy function involving the area and the total length of the erasing forest, is finite. Note that translation surfaces and flat surfaces of genus zero can be viewed as special cases of flat surfaces with erasing forest, and on their moduli space, the volume form μ _Tr equals the usual ones up to a multiplicative constant. Using this result we obtain new proofs for some classical results due to Masur-Veech, and Thurston concerning the finiteness of the volume of the moduli space of translation sufaces, and of the moduli space of polyhedral flat surfaces.     

Short character sums for composite moduli

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square-free moduli and also on the result due to Gallagher and Iwaniec when the core q′ = Π _p|q p of the modulus q satisfies log q′ ～ log q. Some applications to zero free regions of Dirichlet L-functions and the Pólya and Vinogradov inequalities are indicated.     

The realizable extension problem and the weighted graph (K 3,3, l)

This note outlines the realizable extension problem for weighted graphs and provides a detailed analysis of this problem for the weighted graph (K _3,3, l). The main result of this analysis is that the moduli space of planar realizations of (K _3,3, l) can have one, two, four, six or eight connected components and explicit examples of each case are provided. The note culminates with two examples which show that in general, realizability and connectedness results relating to the moduli spaces of weighted cycles which are contained in a larger weighted graph cannot be extended to similar results regarding the moduli space of the larger weighted graph.     

A measure of symmetry for the moduli of spherical minimal immersions

A Grünbaum type of measure of symmetry is calculated and estimated for the DoCarmo-Wallach moduli spaces for eigenmaps and spherical minimal immersions. The DeTurck-Ziller classification of minimal imbeddings of 3-dimensional space forms is used to obtain exact determination of the measure for the SU(2)-equivariant moduli.     

Geršgorin variations III: On a theme of Brualdi and Varga

Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi’s conditions are sharp. Improving Varga’s results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular.     

Navigation in tree spaces

The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes, endowing the resolving space with an interesting canonical pseudometric. Indeed, the given map can be reinterpreted as relating the real and the tropical versions of the Deligne–Knudsen–Mumford compactification of the moduli space of Riemann spheres.     

On the 0-dimensional cusps of the Kähler moduli of a K3 surface

Let S be a complex algebraic K3 surface. It is proved that the 0-dimensional cusps of the Kähler moduli of S are in one-to-one correspondence with the twisted Fourier-Mukai partners of S. As a result, a counting formula for the 0-dimensional cusps of the Kähler moduli is obtained. Applications to rational maps between K3 surfaces are given. When the Picard number of S is 1, the bijective correspondence is calculated explicitly by using the Fricke modular curve.     

A generalization of the Rogers-Ramanujan identities for all moduli

The Rogers-Ramanujan identities have been extended to odd moduli by B. Gordon and to moduli of the form 4k + 2 by G. Andrews. We demonstrate and prove an extension to all even moduli, and provide a theorem which holds for all moduli.     

Injective resolutions of BG and derived moduli spaces of local systems

It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural “derived” moduli spaces which are always smooth in an appropriate sense and whose tangent spaces involve the entire cohomology of the sheaf of infinitesimal automorphisms, not just H¹. In this note we give an algebraic construction of such an extension for the simplest class of moduli spaces, namely for moduli of local systems (representations of the fundamental group).     

Universal moduli spaces of surfaces with flat bundles and cobordism theory

For a compact, connected Lie group G, we study the moduli of pairs (Σ,E), where Σ is a genus g Riemann surface and E→Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of is independent of g. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range, in terms of the homology of an explicit infinite loop space. Rationally, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H^∗(BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces.We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.     

Degenerations of the moduli spaces of vector bundles on curves I

LetY be a smooth projective curve degenerating to a reducible curveX with two components meeting transversally at one point. We show that the moduli space of vector bundles of rank two and odd determinant on Ydegenerates to a moduli space onX which has nice properties, in particular, it has normal crossings. We also show that a nice degeneration exists when we fix the determinant. We give some conjectures concerning the degeneration of moduli space of vector bundles onY with fixed determinant and arbitrary rank.     

具有非均匀界面相的颗粒和纤维增强复合材料弹性静力学问题的解析解

通过将以位移表示的平衡方程转化为黎卡提方程,得到了具有非均匀界面相的颗粒和纤维增强复合材料非均匀界面相内弹性场的解析解·　所得的解析解是弹性模量呈幂次方变化的非均匀界面相解的通用形式·　任意给定1个幂指数,可以得到具有非均匀界面相的颗粒和纤维增强复合材料体积模量的解析表达式·　通过改变幂指数及幂次方项的系数,此解析解可适用于具有多种不同性质的非均匀界面相·　结果表明:界面相模量和厚度对复合材料模量有很大的影响,当界面相存在时,粒子将出现一种"尺寸效应"·     

KP hierarchy for Hodge integrals

Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge integrals like Witten's conjecture, Virasoro constrains, Faber's λg-conjecture, etc. Among other results we show that a properly arranged generating function for Hodge integrals satisfies the equations of the KP hierarchy.     

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted L ^P spaces with weights satisfying the Muckenhoupt A _p condition. The proofs are based on an identity of Balázs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ω_r will be given. We are able to give values to the constants in the inequalities.