Almost periodic functions,Bohr compactification,and differential equations

The Bohr compactification is shown to be the natural setting for studying almost periodic functions. Applications to partial differential equations are also given.     

Rank one Eisenstein cohomology of local systems on the moduli space of abelian varieties

We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations.     

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.     

The Cross-ratio Compactification of the Configuration Space of Ordered Points on 

A natural compactification of the virtual configuration space of N points on the Riemann sphere  is constructed by using cross-ratios. We show that this compactification is homeomorphic to the Bers' compactification of the virtual moduli space of a punctured Riemann sphere of type N . In particular, the system of global and explicit coordinates of this standard compactification is given by cross-ratios.     

Compactifications determined by subsets of C1(X)

It is well known that every compactification of a completely regular space X can be generated, via a Tychonoff-type embedding, by some suitably chosen subset of C¹(X). Different subsets may give rise to equivalent compactifications, and we are concerned with the problem of finding all subsets of C¹(X) which yield a given compactification αX. The problem is easier if generalized: we say that a subset F of C¹(X) “determines” the compactification αX if αX is the smallest compactification to which every element of F extends, and give a simple necessary and sufficient condition for F to determine a given compactification αX. A number of sufficient conditions for two sets to determine the same compactification are given, and the relation between sets which determine αX and those which generate αX (via an embedding) is considered. Generally, a much smaller set of functions is required to determine αX than to generate it; the number needed to determine αX is never more than the weight of αX?X, while the number required to generate it is, if infinite, equal to the weight of αX.     

A new dependence ordering with applications

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed by Avérous et al. [J. Avérous, C. Genest, S.C. Kochar, On dependence structure of order statistics, Journal of Multivariate Analysis 94 (2005) 159-171]. Applications of this partial order to order statistics, k-record values and frailty models are given.     

The abelian monodromy extension property for families of curves

Necessary and sufficient conditions are given (in terms of monodromy) for extending a family of smooth curves over an open subset to a family of stable curves over S. More precisely, we introduce the abelian monodromy extension (AME) property and show that the standard Deligne–Mumford compactification is the unique, maximal AME compactification of the moduli space of curves. We also show that the Baily–Borel compactification is the unique, maximal projective AME compactification of the moduli space of abelian varieties.     

Singular points on compactifications of a Riemann surface

A notion of capacity on general compactifications of a Riemann surface is introduced. It is used to establish the connection between singular points on the Royden compactification and singular points on the Kuramochi compactification. The second named author was supported by Grant-in-Aid for Scientific Research, Japanese Ministry of Education, Science and Culture.     

Character sheaves on the semi-stable locus of a group compactification

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification.     

Low dimensional semiparametric estimation in a censored regression model

A new estimation procedure for a partial linear additive model with censored responses is proposed. To this aim, ideas of Lewbel and Linton [A. Lewbel, O. Linton, Nonparametric censored and truncated regression, Econometrica 70 (2002) 765-779] on censored model regression are combined with those of Kim et al. [W. Kim, O. Linton, N.W. Hengartner, A computationally efficient estimator for additive nonparametric regression with bootstrap confidence intervals, Journal of Computational and Graphical Statistics, 8 (1999) 278-297] on marginal integration and those on average derivatives. This allows for dimension reduction, interpretability and — depending on the context — for weights yielding computationally attractive estimates. Asymptotic behavior is provided for all proposed estimators.     

Diagonal structures on topological spaces

We investigate when a topological space admits a partial product operation satisfying some rather weak continuity restrictions and almost nothing else-the only algebraic requirement is that some element e of X is a left and a right identity with respect to this multiplication. The operation is called partial diagonalization of X at e. Several sufficient conditions for a space to be partially diagonalizable are established. On the other hand, it is shown that certain deep results about the topological structure of compact topological groups can be extended to partially diagonalizable compact spaces. We also discover that partial diagonalizability plays an important role in the theory of cardinal invariants, in the study of homogeneous spaces, and in such classical topics of general topology as the theory of Stone–Čech compactification and the theory of Hewitt–Nachbin compactification. The notions of a Moscow space and of a C-embedding are instrumental in our study.     

Constructing priors based on model size for nondecomposable Gaussian graphical models: A simulation based approach

A method for constructing priors is proposed that allows the off-diagonal elements of the concentration matrix of Gaussian data to be zero. The priors have the property that the marginal prior distribution of the number of nonzero off-diagonal elements of the concentration matrix (referred to below as model size) can be specified flexibly. The priors have normalizing constants for each model size, rather than for each model, giving a tractable number of normalizing constants that need to be estimated. The article shows how to estimate the normalizing constants using Markov chain Monte Carlo simulation and supersedes the method of Wong et al. (2003) [24] because it is more accurate and more general. The method is applied to two examples. The first is a mixture of constrained Wisharts. The second is from Wong et al. (2003) [24] and decomposes the concentration matrix into a function of partial correlations and conditional variances using a mixture distribution on the matrix of partial correlations. The approach detects structural zeros in the concentration matrix and estimates the covariance matrix parsimoniously if the concentration matrix is sparse.     

Estimation of means of multivariate normal populations with order restriction

Multivariate isotonic regression theory plays a key role in the field of statistical inference under order restriction for vector valued parameters. Two cases of estimating multivariate normal means under order restricted set are considered. One case is that covariance matrices are known, the other one is that covariance matrices are unknown but are restricted by partial order. This paper shows that when covariance matrices are known, the estimator given by this paper always dominates unrestricted maximum likelihood estimator uniformly, and when covariance matrices are unknown, the plug-in estimator dominates unrestricted maximum likelihood estimator under the order restricted set of covariance matrices. The isotonic regression estimators in this paper are the generalizations of plug-in estimators in unitary case.     

Tightness relative to some (co)reflections in topology

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given object B ∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the ?ech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the ?ech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice prin­ciples, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character.

At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflec­tion and for the Lindelöf reflection.     

Perfect compactifications of frames

Perfect compactifications of frames are introduced. It is shown that the Stone-Čech compactification is an example of such a compactification. We also introduce rim-compact frames and for such frames we define its Freudenthal compactification, another example of a perfect compactification. The remainder of a rim-compact frame in its Freudenthal compactification is shown to be zero-dimensional. It is shown that with the assumption of the Boolean Ultrafilter Theorem the Freudenthal compactification for spaces, as well as the Freudenthal-Morita Theorem for spaces, can be obtained from our frame constructions.     

Embedding in compact uniquely divisible semigroups

D. R. Brown and M. Friedberg have conjectured that each compact abelian semigroup can be embedded in a compact divisible semigroup. V. R. Hancock proved that each abelian algebraic semigroup can be embedded in a divisible abelian algebraic semigroup. In this paper we provide a partial solution to the conjecture of Brown and Friedberg by employing a topological version of Hancock's method as part of our construction. A theorem giving sufficient conditions for the Bohr compactification of weakly reductive semigroups to be injective is proved and used in the proof of our main result.     

Consistent variable selection in large panels when factors are observable

In this paper we develop an econometric method for consistent variable selection in the context of a linear factor model with observable factors for panels of large dimensions. The subset of factors that best fit the data is sequentially determined. Firstly, a partial R² rule is used to show the existence of an optimal ordering of the candidate variables. Secondly, We show that for a given order of the regressors, the number of factors can be consistently estimated using the Bayes information criterion. The Akaike will asymptotically lead to overfitting of the model. The theory is established under approximate factor structure which allows for limited cross-section and serial dependence in the idiosyncratic term. Simulations show that the proposed two-step selection technique has good finite sample properties. The likelihood of selecting the correct specification increases with the number of cross-sections both asymptotically and in small samples. Moreover, the proposed variable selection method is computationally attractive. For K potential candidate factors, the search requires only 2K regressions compared to 2^K for an exhaustive search.     

Compactification of the universal Picard over the moduli of stable curves

This article provides two different, but closely related, moduli problems, which in characteristic zero provide a type of compactification of the universal Picard over the moduli of stable curves. Although neither is of finite type, both are limits of a sequence of stacks, each of which is a separated algebraic stack of finite type. We discuss relations to previous compactifications and partial compactifications, give a number of examples related to this compactification, and work out the structure of its fibres over certain fixed curves. Some applications are also discussed. Received January 5, 1998; in final form April 1, 1999 / Published online July 3, 2000     

Remainders in compactifications of topological groups

In this paper, we consider the following question: when does a topological group G have a Hausdorff compactification bG with a remainder belonging to a given class of spaces? We extend the results of A.V. Arhangel'skii by showing that if a remainder of a non-locally compact topological group G has a countable open point-network or a locally Gδ-diagonal, then G and the compactification bG of G are separable and metrizable.     

Applications of nonstandard analysis to ideal boundaries in potential theory

A solution is given of the generalized Dirichlet problem for an arbitrary compactification of a Brelot harmonic space. A method of obtaining the Martin-Choquet integral representation of positive harmonic functions is given, and the existence is established of an ideal boundary Δ supporting the maximal representing measures for positive bounded and quasibounded harmonic functions with almost all points of Δ being regular for the Dirichlet problem. This work was supported by a grant from the U. S. National Science Foundation. The results in Sections 1–5 were presented at the 1974 Oberwolfach Conferences on Potential Theory and Nonstandard Analysis; Sections 1–6 were discussed at the Abraham Robinson Memorial Conference, Yale, University, May 1975.