Linear isometries between subspaces of continuous functions

We say that a linear subspace of is strongly separating if given any pair of distinct points of the locally compact space , then there exists such that . In this paper we prove that a linear isometry of onto such a subspace of induces a homeomorphism between two certain singular subspaces of the Shilov boundaries of and , sending the Choquet boundary of onto the Choquet boundary of . We also provide an example which shows that the above result is no longer true if we do not assume to be strongly separating. Furthermore we obtain the following multiplicative representation of : for all and all , where is a unimodular scalar-valued continuous function on . These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

     

Randomness and semigenericity

Let contain only the equality symbol and let be an arbitrary finite symmetric relational language containing . Suppose probabilities are defined on finite structures with `edge probability' . By , the almost sure theory of random -structures we mean the collection of -sentences which have limit probability 1. denotes the theory of the generic structures for (the collection of finite graphs with hereditarily nonnegative). . , the almost sure theory of random -structures, is the same as the theory of the -generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

     

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Let be a domain in , a nonnegative and a positive function on such that is locally bounded, the space of all holomorphic functions on square-integrable with respect to the measure , where is the -dimensional Lebesgue measure, and the reproducing kernel for . It has been known for a long time that in some special situations (such as on bounded symmetric domains with and the Bergman kernel function) the formula

holds true. [This fact even plays a crucial role in Berezin's theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function can be approximated by certain pluriharmonic functions lying below it. For instance, () holds if is convex (and, hence, can be approximated from below by linear functions), for any function . Counterexamples are also given to show that in general () may fail drastically, or even be true for some and fail for the remaining ones. Finally, we also consider the question of convergence of for , which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of : for instance, if is not real-analytic at some point, then must have zeroes for all sufficiently large.

     

Coherent functors, with application to torsion in the Picard group

Let be a commutative noetherian ring. We investigate a class of functors from commutative -algebras to sets, which we call coherent. When such a functor in fact takes its values in abelian groups, we show that there are only finitely many prime numbers such that is infinite, and that none of these primes are invertible in . This (and related statements) yield information about torsion in . For example, if is of finite type over , we prove that the torsion in is supported at a finite set of primes, and if is infinite, then the prime is not invertible in . These results use the (already known) fact that if such an is normal, then is finitely generated. We obtain a parallel result for a reduced scheme of finite type over . We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.

     

A bound on the geometric genus of projective varieties verifying certain flag conditions

Fix integers and let be the set of all integral, projective and nondegenerate varieties of degree and dimension in the projective space , such that, for all , does not lie on any variety of dimension and degree . We say that a variety satisfies a flag condition of type if belongs to . In this paper, under the hypotheses , we determine an upper bound , depending only on , for the number , where denotes the geometric genus of . In case and , the study of an upper bound for the geometric genus has a quite long history and, for , and , it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data . For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension in as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in . Next we discuss how far is from and show a sort of lifting theorem which states that, at least in certain cases, the varieties of maximal geometric genus must in fact lie on a flag such as , where denotes a subvariety of of degree and dimension . We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.

     

The Mizohata complex

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ at of a -valued, formally integrable (), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for -forms , of the equation

where is a given -form satisfying is estimated in terms of the De Rham cohomology relative to

     

Generalized Weil's reciprocity law and multiplicativity theorems

Fix a one-dimensional group variety with Euler-characteristic , and a quasi-projective variety , both defined over . For any and constructible sheaf on , we construct an invariant , which provides substantial information about the topology of the fiber-structure of and the structure of along the fibers of . Moreover, is a group homomorphism.

     

Restriction of stable bundles in characteristic

Let be an algebraically closed field of characteristic . Let be a nonsingular projective variety defined over and an ample line bundle on . We shall prove that there exists an explicit number such that if is a -stable vector bundle of rank at most three, then the restriction is -stable for all and all smooth irreducible divisors . This result has implications to the geometry of the moduli space of -stable bundles on a surface or a projective space.

     

Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of -Functions

Let be a congruence subgroup of type and of level . We study congruences between weight 2 normalized newforms and Eisenstein series on modulo a prime above a rational prime . Assume that , is a common eigenfunction for all Hecke operators and is ordinary at . We show that the abelian variety associated to and the cuspidal subgroup associated to intersect non-trivially in their -torsion points. Let be a modular elliptic curve over with good ordinary reduction at . We apply the above result to show that an isogeny of degree divisible by from the optimal curve in the -isogeny class of elliptic curves containing to extends to an étale morphism of Néron models over if . We use this to show that -adic distributions associated to the -adic -functions of are -valued.

     

Partial regularity of solutions to a class of degenerate systems

We consider the system , in coupled with suitable initial-boundary conditions, where is a bounded domain in with smooth boundary and is a continuous and positive function of . Our main result is that under some conditions on there exists a relatively open subset of such that is locally Hölder continuous on , the interior of is empty, and is essentially bounded on .

     

Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Liouville type theorems for fourth order elliptic equations in a half plane

Consider an elliptic equation in the half plane with boundary conditions if and if where are second and third order differential operators. It is proved that if and, for some , if if where for some nonnegative integer , then . Results of this type are also established in case under different conditions on and ; furthermore, in one case has a lower order term which depends nonlocally on . Such Liouville type theorems arise in the study of coating flow; in fact, they play a crucial role in the analysis of the linearized version of this problem. The methods developed in this paper are entirely different for the two cases (i) and (ii) ; both methods can be extended to other linear elliptic boundary value problems in a half plane.

     

On the strong equality between supercompactness and strong compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension ZFC + GCH in which, (a) (preservation) for regular, if is supercompact', then is supercompact' and so that, (b) (equivalence) for regular, is strongly compact' iff is supercompact', except possibly if is a measurable limit of cardinals which are supercompact.

     

On the second adjunction mapping. The case of a 1-dimensional image

Let be a very ample line bundle on an -dimensional projective manifold , i.e., assume that for some embedding . In this article, a study is made of the meromorphic map, , associated to in the case when the Kodaira dimension of is , and has a -dimensional image. Assume for simplicity that . The first main result of the paper shows that is a morphism if either or . The second main result of this paper shows that if , then the genus, , of a fiber, , of the map induced by on hyperplane sections is . Moreover, if then , a connected component of a general fiber of is either a surface or the blowing up at one point of a surface, and . Finally the structure of the finite to one part of the Remmert-Stein factorization of is worked out.

     

The local dimensions of the Bernoulli convolution associated with the golden number

Let be a sequence of i.i.d. random variables each taking values of 1 and with equal probability. For satisfying the equation , let be the probability measure induced by . For any in the range of , let

be the local dimension of at whenever the limit exists. We prove that

where , are respectively the maximum and minimum values of the local dimensions. If , then is the golden number, and the approximate numerical values are and .

     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

The class number one problem for some non-abelian normal CM-fields

Let be a non-abelian normal CM-field of degree any odd prime. Note that the Galois group of is either the dicyclic group of order or the dihedral group of order We prove that the (relative) class number of a dicyclic CM-field of degree is always greater then one. Then, we determine all the dihedral CM-fields of degree with class number one: there are exactly nine such CM-fields.