On the constellations of Weierstrass points

We prove that the constellation of Weierstrass points characterizes the isomorphism-class of double coverings of curves of genus large enough. The author was supported by a grant from the International Atomic Energy Agency and UNESCO.     

On the uniqueness of canonical points in the Hobby-Rice theorem

According to the Hobby-Rice theorem for anyn-dimensional subspaceU _n ofL ₁([a, b], ) ( positive, finite, nonatomic) there exist points =s ₀x ₁x _m+1=b, where 0mn, such that

The Gaussian primes contain arbitrarily shaped constellations

We show that the Gaussian primesP[i] ? ?[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, we show that given any distinct Gaussian integersv ₀,…,v _k?1, there are infinitely many sets {a+rv ₀,…,rv _k?1}, witha ∈?[i] andr ∈?{0}, all of whose elements are Gaussian primes. The proof is modeled on that in [9] and requires three ingredients. The first is a hypergraph removal lemma of Gowers and Rödl-Skokan or, more precisely, a slight strenghthening of this lemma which can be found in [22]; this hypergraph removal lemma can be thought of as a generalization of the Szemerédi-Furstenberg-Katznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument from [9], which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a Goldston-Yildirim type analysis for the Gaussian integers, similar to that in [9], which yields a pseudorandom measure. which is concentrated on Gaussian “almost primes”.     

D-semigroups and constellations

In a result generalising the Ehresmann–Schein–Nambooripad Theorem relating inverse semigroups to inductive groupoids, Lawson has shown that Ehresmann semigroups correspond to certain types of ordered (small) categories he calls Ehresmann categories. An important special case of this is the correspondence between two-sided restriction semigroups and what Lawson calls inductive categories. Gould and Hollings obtained a one-sided version of this last result, by establishing a similar correspondence between left restriction semigroups and certain ordered partial algebras they call inductive constellations (a general constellation is a one-sided generalisation of a category). We put this one-sided correspondence into a rather broader setting, at its most general involving left congruence D-semigroups (which need not satisfy any semiadequacy condition) and what we call co-restriction constellations, a finitely axiomatized class of partial algebras. There are ordered and unordered versions of our results. Two special cases have particular interest. One is that the class of left Ehresmann semigroups (the natural one-sided versions of Lawson’s Ehresmann semigroups) corresponds to the class of co-restriction constellations satisfying a suitable semiadequacy condition. The other is that the class of ordered left Ehresmann semigroups (which generalise left restriction semigroups and for which semigroups of binary relations equipped with domain operation and the inclusion order are important examples) corresponds to a class of ordered constellations defined by a straightforward weakening of the inductive constellation axioms.     

The canonical function game

The canonical function game is a game of length ω ₁ introduced by W. Hugh Woodin which falls inside a class of games known as Neeman games. Using large cardinals, we show that it is possible to force that the game is not determined. We also discuss the relationship between this result and Σ² ₂ absoluteness, cardinality spectra and Π₂ maximality for H(ω ₂) relative to the Continuum Hypothesis.     

On the canonical solution of indefinite problem with m turning points of even order

We consider the differential equation

(∗)     

On admissible constellations of consecutive primes

Admissible constellations of primes are patterns which, like the twin primes, no simple divisibility relation would prevent from being repeated indefinitely in the series of primes. All admissible constellations, formed ofconsecutive primes, beginning with a prime <1000, are established, and some properties of such constellations in general are conjectured.Dedicated to Peter Naur on the occasion of his 60th birthday     

The weighted log canonical threshold

In this note, we show how to apply the original L²$L^2$-extension theorem of Ohsawa and Takegoshi to the standard basis of a multiplier ideal sheaf associated with a plurisubharmonic function. In this way, we are able to reprove the strong openness conjecture and to obtain an effective version of the semicontinuity theorem for weighted log canonical thresholds.     

The canonical decomposition of bivariate distributions

The ordinary notion of a bivariate distribution has a natural generalisation. For this generalisation it is shown that a bivariate distribution can be characterised by a Hilbert space and a family _p, 0 ≤ p ≤ 1, of subspaces of . specifies the marginal distributions whilst _p is a summary of the dependence structure. This characterisation extends existing ideas on canonical correlation.     

Actions and partial actions of inductive constellations

Constellations were recently introduced by the authors as one-sided analogues of categories: a constellation is equipped with a partial multiplication for which ‘domains’ are defined but, in general, ‘ranges’ are not. Left restriction semigroups are the algebraic objects modelling semigroups of partial mappings, equipped with local identities in the domains of the mappings. Inductive constellations correspond to left restriction semigroups in a manner analogous to the correspondence between inverse semigroups and inductive groupoids.     

The canonical order and optimization problems

Using the partial order technique, we describe a subclass of objective functions taking their optimum at the greedy point of a given feasible polyhedron inR ⁿ. Supported by the Belarusian Foundatmental Research Found     

The canonical sheaf of Du Bois singularities

We prove that a Cohen-Macaulay normal variety X has Du Bois singularities if and only if π_∗ωX^′(G)?ωX for a log resolution π:X^′→X, where G is the reduced exceptional divisor of π. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen-Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.     

The asymptotic expansion of certain canonical integrals

Integrals of the form $\text{I(λ) = ∝}{}_{\mathrm{R}}{}^2\text{?(}\text{x}\text{?}\text{)}\text{exp}\text{{iλφ(}\text{x}\text{?}\text{)} d}\text{x}\text{?}$, common in diffraction theory, are usually evaluated asymptotically for large λ using the stationary phase technique. At caustic points the classical technique is not applicable. A computation-oriented algorithm is developed for determining the asymptotic expansion of such integrals with $\text{φ(}\text{x}\text{)}$ a Thom umbilic, the canonical form appropriate at caustic points for most cases of physical interest.     

The canonical model of a singular curve

We give refined statements and modern proofs of Rosenlicht’s results about the canonical model C′ of an arbitrary complete integral curve C. Notably, we prove that C and C′ are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C′ is equal to the blowup of C with respect to the canonical sheaf ω. We also prove some new results: we determine just when C′ is rational normal, arithmetically normal, projectively normal, and linearly normal.      

The canonical decomposition of once-punctured torus bundles

We determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defined by Floyd and Hatcher. Unlike previous work on this problem, the techniques we use are not geometric. Instead, they involve angled polyhedral decompositions, thin position and a version of almost normal surface theory.     

广义典型流形的实例构作

陈述了广义典型流形(或拓扑域流形)的概念,并构作了拓扑域流形的若干实例,丰富了这类流形的内容,也给今后研究和运用拓扑域流形提供一些参考.