On Bernstein-Sato polynomials

We show that for fixed and the set of Bernstein-Sato polynomials of all the polynomials in at most variables of degrees at most is finite. As a corollary, we show that there exists an integer depending only on and such that generates as a module over the ring of the -linear differential operators of , where is an arbitrary field of characteristic 0, is the ring of polynomials in variables over and is an arbitrary non-zero polynomial of degree at most .

     

Inner derivations on ultraprime normed algebras

We show that, for every ultraprime Banach algebra , there exists a positive number satisfying for all in , where denotes the centre of and denotes the inner derivation on induced by . Moreover, the number depends only on the ``constant of ultraprimeness' of .

     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

Fixed points of the mapping class group in the moduli spaces

Let be a compact oriented surface with or without boundary components. In this note we prove that if then there exist infinitely many integers such that there is a point in the moduli space of irreducible flat connections on which is fixed by any orientation preserving diffeomorphism of . Secondly we prove that for each orientation preserving diffeomorphism of and each there is some such that has a fixed point in the moduli space of irreducible flat connections on . Thirdly we prove that for all there exists an integer such that the 'th power of any diffeomorphism fixes a certain point in the moduli space of irreducible flat connections on .

     

An infinite series of Kronecker conjugate polynomials

Let be a field of characteristic , a transcendental over , and be the absolute Galois group of . Then two non-constant polynomials are said to be Kronecker conjugate if an element of fixes a root of if and only if it fixes a root of . If is a number field, and where is the ring of integers of , then and are Kronecker conjugate if and only if the value set equals modulo all but finitely many non-zero prime ideals of . In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials and differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.

     

Volumes of hyperbolic Haken manifolds, II

We show that if is a closed hyperbolic 3-manifold and if has a non-abelian free quotient, then the volume of is greater than . If, in addition, contains no genus- surface groups, then the volume of is greater than . Using these results we show that if there are infinitely many primitive homology classes in which are not represented by fibroids, then the volume of is greater than .

     

Harmonic 2-spheres with pairs of extra eigenfunctions

In the present paper, deformations of harmonic 2-spheres in the unit -sphere respecting the degree are studied. The limit maps of such deformations are characterized as harmonic maps with extra eigenfunctions. The space of harmonic 2-spheres in with fixed degree is described in terms of such deformations and the limit maps.

     

Weighted weak-type inequalities for the maximal function of nonnegative integral transforms over approach regions

The relation between approach regions and singularities of nonnegative kernels is studied, where , , , and is a homogeneous space. For , a sufficient condition on approach regions () is given so that the maximal function

is weak-type with respect to a pair of measures and . It is shown that this condition is also necessary for operators of potential type in the sense of Sawyer and Wheedon (Amer. J. Math. 114 (1992), 813-874).

     

Noncommutative spaces

Let be a von Neumann algebra with a faithful, finite, normal tracial state , and let be a finite, maximal subdiagonal algebra of . Let be the closure of in the noncommutative Lebesgue space . Then possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an factorization theorem, Nehari's Theorem, and harmonic conjugates which are bounded.

     

Product of distinct simple integrally closed ideals in 2-dimensional regular local rings

Let be a two-dimensional regular local ring and an -primary integrally closed ideal in . In this paper, we give equivalent conditions for to be a product of distinct simple -primary integrally closed ideals (i.e., , where are distinct simple -primary integrally closed ideals of ) in terms of the regularity of for all and in terms of how to choose a minimal generating set for over its minimal reductions.

     

Open covers and the square bracket partition relation

An open cover of an infinite separable metric space is an -cover of if and for every finite subset of there is a such that . Let be the collection of -covers of . We show that the partition relation holds if, and only if, the partition relation holds.

     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

On -summable sequences in the range of a vector measure

Let . Among other results, we prove that a Banach space has the property that every sequence lies inside the range of an -valued measure if and only if, for all sequences in satisfying that the operator is 1-summing, the operator is nuclear, being the conjugate number for . We also prove that, if is an infinite-dimensional -space for , then can't have the above property for any .

     

Formulae and continuity for the index of subfactors

Let be an inclusion of -factors, the trace state of , and , the set of projections in and , respectively. We prove that the Jones index for the inclusion is

This formula is exploited to obtain continuity results for the index. In particular, we obtain a formula for the index which expresses in terms of the positions of and , in , when and are finite-dimensional -subalgebras with dense union in and , respectively.

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

Rudin's orthogonality problem and the Nevanlinna counting function

Let be a holomorphic function taking the open unit disk into itself. We show that the set of nonnegative powers of is orthogonal in if and only if the Nevanlinna counting function of , , is essentially radial. As a corollary, we obtain that the orthogonality of for a univalent implies for some constant . We also show that if is orthogonal, then the closure of must be a disk.

     

On normal solvability of the Riemann problem with singular coefficient

Suppose is a singular matrix function on a simple, closed, rectifiable contour . We present a necessary and sufficient condition for normal solvability of the Riemann problem with coefficient in the case where admits a spectral (or generalized Wiener-Hopf) factorization with essentially bounded. The boundedness of is not required when takes injective values a.e. on .

     

muni de l'arithmétique faible de Penzin est décidable

We prove the decidability of the additive ordered group equipped with a predicate for , the multiplication restricted to and the -adic valuation ranging in .