Global co-stationarity of the ground model from a new countable length sequence

Suppose are models of ZFC with the same ordinals, and that for all regular cardinals in , satisfies . If contains a sequence for some ordinal , then for all cardinals in with regular in and , is stationary in . That is, a new -sequence achieves global co-stationarity of the ground model.

     

The projective -character bounds the order of a -base

All spaces below are Tychonov. We define the projective - character of a space as the supremum of the values where ranges over all (Tychonov) continuous images of . Our main result says that every space has a -base whose order is ; that is, every point in is contained in at most -many members of the -base. Since for compact , this is a significant generalization of a celebrated result of Shapirovskii.

     

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Let and be compact Hausdorff spaces, and , be Banach lattices. Let denote the Banach lattice of all continuous -valued functions on equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism such that is non-vanishing on if and only if is non-vanishing on , then is homeomorphic to , and is Riesz isomorphic to . In this case, can be written as a weighted composition operator: , where is a homeomorphism from onto , and is a Riesz isomorphism from onto for every in . This generalizes some known results obtained recently.

     

On the normal bundle of submanifolds of

Let be a submanifold of dimension of the complex projective space . We prove results of the following type.i) If is irregular and , then the normal bundle is indecomposable. ii) If is irregular, and , then is not the direct sum of two vector bundles of rank . iii) If , and is decomposable, then the natural restriction map is an isomorphism (and, in particular, if is embedded Segre in , then is indecomposable). iv) Let and , and assume that is a direct sum of line bundles; if assume furthermore that is simply connected and is not divisible in . Then is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when this fact was proved by M. Schneider in 1990 in a completely different way.

     

Antiholomorphic involutions of spherical complex spaces

Let be a holomorphically separable irreducible reduced complex space, a connected compact Lie group acting on by holomorphic transformations, a Weyl involution, and an antiholomorphic map satisfying and for . We show that if is a multiplicity free -module, then maps every -orbit onto itself. For a spherical affine homogeneous space of the reductive group we construct an antiholomorphic map with these properties.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree

Let , , be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform of degree . It is proved that if is not in the Maaß space, then there exist infinitely many primes for which the sequence , , has infinitely many sign changes.

     

On the - boundedness of operators

We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that

then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.

     

-identities for the Grassmann algebra: The conjecture of Henke and Regev

Let be an algebraically closed field of characteristic 0, and let be the infinite dimensional Grassmann (or exterior) algebra over . Denote by the vector space of the multilinear polynomials of degree in , ..., in the free associative algebra . The symmetric group acts on the left-hand side on , thus turning it into an -module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The -modules and are canonically isomorphic. Letting be the alternating group in , one may study and its isomorphic copy in with the corresponding action of . Henke and Regev described the -codimensions of the Grassmann algebra , and conjectured a finite generating set of the -identities for . Here we answer their conjecture in the affirmative.

     

Isometries for the Carathéodory metric

Given two open unit balls and in complex Banach spaces, we consider a holomorphic mapping such that and is an isometry. Under some additional hypotheses on the Banach spaces involved, we prove that is a complex closed analytic submanifold of .

     

First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Let be an -dimensional compact hypersurface with constant scalar curvature , , in a unit sphere . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral of the mean curvature . In this paper, we first study the eigenvalue of the Jacobi operator of . We derive an optimal upper bound for the first eigenvalue of , and this bound is attained if and only if is a totally umbilical and non-totally geodesic hypersurface or is a Riemannian product , .

     

Topology of three-manifolds with positive -scalar curvature

Consider an -dimensional smooth Riemannian manifold with a given smooth measure on it. We call such a triple a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré's conjecture , where and is the scalar curvature of . In this note, we study the topological obstruction for the -stable minimal submanifold with positive -scalar curvature in dimension three under the setting of manifolds with density.

     

A topological reflection principle equivalent to Shelah's strong hypothesis

We notice that Shelah's Strong Hypothesis is equivalent to the following reflection principle:

Suppose is a first-countable space whose density is a regular cardinal, . If every separable subspace of is of cardinality at most , then the cardinality of is .

     

On isomorphisms between centers of integral group rings of finite groups

For finite nilpotent groups and , and a -adapted ring (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings and is monomial, i.e., maps class sums in to class sums in up to multiplication with roots of unity. As a consequence, and have identical character tables if and only if the centers of their integral group rings and are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

     

Infinite index subalgebras of depth two

An algebra extension is right depth two in this paper if its tensor-square is --isomorphic to a direct summand of any (not necessarily finite) direct sum of with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite-dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory (Kadison and Szlachányi, 2003) but extends the main theorem of depth two theory, as for example in (Kadison and Nikshych, 2001). That is, a right depth two extension has right bialgebroid over its centralizer . The main theorem: An extension is right depth two and right balanced if and only if is -Galois with respect to left projective, right -bialgebroid .

     

On exceptional eigenvalues of the Laplacian for

An explicit Dirichlet series is obtained, which represents an analytic function of in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on .

     

On the isolated points of the surjective spectrum of a bounded operator

For a bounded operator acting on a complex Banach space, we show that if is not surjective, then is an isolated point of the surjective spectrum of if and only if , where is the quasinilpotent part of and is the analytic core for . Moreover, we study the operators for which . We show that for each of these operators , there exists a finite set consisting of Riesz points for such that and is connected, and derive some consequences.

     

Spectra of operators with Bishop's property

Let be a Banach space and let be the class that consists of all operators such that for every , the range of has a finite-codimension when it is closed. For an integer , we define the class as an extension of . We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with .

     

On a fragment of the universal Baire property for sets

There is a well-known global equivalence between sets having the universal Baire property, two-step generic absoluteness, and the closure of the universe under the sharp operation. In this note, we determine the exact consistency strength of sets being -cc-universally Baire, which is below . In a model obtained, there is a set which is weakly -universally Baire but not -universally Baire.

     

Compact weighted composition operators on the Hardy space

A weighted composition operator takes an analytic map on the open unit disk of the complex plane to the analytic map , where is an analytic map of the open unit disk into itself and is an analytic map on the open unit disk. This paper studies how the compactness of depends on the interaction between the two maps and .