The expected value of the number of real zeros of a random sum of Legendre polynomials

It is known that the expected number of zeros in the interval of the sum , in which is the normalized Legendre polynomial of degree and the coefficients are independent normally distributed random variables with mean 0 and variance 1, is asymptotic to for large . We improve this result and show that this expected number is for any positive .

     

Hyperspaces and open monotone maps of hereditarily indecomposable continua

We prove the following theorems:

Theorem 1. Let be an -dimensional hereditarily indecomposable continuum. Then there exist -dimensional hereditarily indecomposable continua and monotone maps such that is an embedding and the space of all subcontinua of is embeddable in by .

Theorem 2. For every open monotone map with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum with there exists a -dimensional subcontinuum such that and the restriction of to is also monotone and open.

The connection between these theorems and other results in Hyperspace theory is studied.

     

On Bernstein type theorems concerning the growth of derivatives of entire functions

A subspace of which is invariant under all left translation operators is called admissible if is a Banach space satisfying the following properties:

(i) If then there exists a subsequence such that almost everywhere.

(ii) The group is a bounded strongly continuous group. In this case, let

Typical admissible spaces are and all spaces for More generally, all of the Peetre interpolation spaces of two admissible spaces are also admissible.

A function is called subexponential if for every With these definitions our main result goes as follows: . If is an entire function of exponential type such that its restriction to the real axis, denoted by , is subexponential and belongs to some admissible space then the derivative is also in Moreover,
for each real

This result yields as consequences and in a systematic way many new and old Bernstein type inequalities.

     

Radial limit of lacunary Fourier series with coefficients in non-commutative symmetric spaces

Let be a rearrangement invariant space, an arbitrary set and a von Neumann algebra with a semifinite normal faithful trace. It is proved that the associated symmetric space of measurable operators has -RNP if and only if has -RNP extending in this way some previous results by Q. Xu.

     

An invariance of domain result for multi-valued maximal monotone operators whose domains do not necessarily contain any open sets

Let be a real, reflexive, locally uniformly convex Banach space with locally uniformly convex. Let be a maximal monotone operator and open and bounded. Assume that is pathwise connected and such that and Then If, moreover, is of type () on then may be replaced above by The significance of this result lies in the fact that it holds for multi-valued mappings which do not have to satisfy It has also been used in this paper in order to establish a general ``invariance of domain' result for maximal monotone operators, and may be applied to a greater variety of problems involving partial differential equations. No degree theory has been used. In addition to the above, necessary and sufficient conditions are given for the existence of a zero (in an open and bounded set ) of a completely continuous perturbation of a maximal monotone operator such that is locally monotone on

     

Smooth exhaustion functions in convex domains

We show that in every bounded convex domain in there exists a smooth convex exhaustion function such that the product of all eigenvalues of the matrix is . Moreover, if the domain is strictly convex, then can be chosen so that every eigenvalue is .

     

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 .

     

On the curves of contact on surfaces in a projective space. III

Suppose a smooth curve is a set-theoretic complete intersection of two surfaces and with the multiplicity of along less than or equal to the multiplicity of along . One obtains a relation between the degrees of , and , the genus of , and the multiplicity of along in case has only ordinary singularities. One obtains (in the characteristic zero case) that a nonsingular rational curve of degree 4 in is not set-theoretically an intersection of 2 surfaces, provided one of them has at most ordinary singularities. The same result holds for a general nonsingular rational curve of degree .

     

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.

     

A note on -hyponormal operators

Let be a -hyponormal operator on a Hilbert space with polar decomposition and let for and We study order and spectral properties of In particular we refine recent Furuta's result on -hyponormal operators.

     

On asymptotic estimates for arithmetic cost functions

Let (the cost of ) be the minimum number of arithmetic operations needed to obtain starting from 1. We prove that for almost all , and, given , for all sufficiently large. We prove analogous results for costs of polynomials with integer coefficients.

     

Krull dimension of modules over involution rings. II

Let be a ring with involution and invertible 2, and let be the subring of generated by the symmetric elements in . The following questions of Lanski are answered positively:

(i)

Must have Krull dimension when does?

(ii)

Is every Artinian -module Artinian as an -module?

     

Convolution of a measure with itself and a restriction theorem

Let and be the measure defined by . Let denote the measure obtained by restricting to the set . We prove estimates on . As a corollary we obtain results on the restriction to of the Fourier transform of functions on for , .

     

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.

     

On a class of subalgebras of and the intersection of their free maximal ideals

Let be a Tychonoff space and a subalgebra of containing . Suppose that is the set of all functions in with compact support. Kohls has shown that is precisely the intersection of all the free ideals in or in . In this paper we have proved the validity of this result for the algebra . Gillman and Jerison have proved that for a realcompact space , is the intersection of all the free maximal ideals in . In this paper we have proved that this result does not hold for the algebra , in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in . The paper terminates by showing that for any realcompact space , there exists in some sense a minimal algebra for which becomes -compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

On a theorem of Privalov and normal functions

A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

     

A joint spectral characterization of primeness for C-algebras

We prove that a C-algebra is prime iff for every where denotes Taylor spectrum and are the left and right multiplication operators acting on

     

Superrigid subgroups of solvable Lie groups

Let be a discrete subgroup of a simply connected, solvable Lie group , such that has the same Zariski closure as . If is any finite-dimensional representation of , we show that virtually extends to a continuous representation of . Furthermore, the image of is contained in the Zariski closure of the image of . When is not discrete, the same conclusions are true if we make the additional assumption that the closure of is a finite-index subgroup of (and is closed and is continuous).

     

Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

A classical result of W. Bade states that if is any complete Boolean algebra of projections in an arbitrary Banach space then, for every there exists an element (called a Bade functional for with respect to in the dual space , with the following two properties: (i) is non-negative on and, (ii) whenever satisfies It is shown that a Fréchet space has this property if and only if it does not contain an isomorphic copy of the sequence space