A note on norm attaining functionals

We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space is reflexive if and only if it satisfies any of the following properties: (i) admits a norm with the Mazur Intersection Property and the set of all norm attaining functionals of contains an open set, (ii) the set of all norm one elements of contains a (relative) weak* open set of the unit sphere, (iii) has and contains a (relative) weak open set of the unit sphere, (iv) is , has and contains a (relative) weak open set of the unit sphere. Finally, if is separable, then is reflexive if and only if contains a (relative) weak open set of the unit sphere.

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

Relative to any nonrecursive set

There is a countable first order structure such that for any set of integers , is not recursive if and only if there is a presentation of which is recursive in .

     

A remark on Gelfand-Kirillov dimension

Let be a finitely generated non-PI Ore domain and the quotient division algebra of . If is the center of , then .

     

On scrambled sets and a theorem of Kuratowski on independent sets

The measure of scrambled sets of interval self-maps was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of ``-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map of the unit -cube is -chaotic on , then for any there is a map such that and are topologically conjugate, and has a scrambled set which has Lebesgue measure 1, and hence if , then there is a homeomorphism with a scrambled set satisfying that is an -set in and has Lebesgue measure 1.

     

Derived lengths and character degrees

Let be a finite solvable group. Assume that the degree graph of has exactly two connected components that do not contain . Suppose that one of these connected components contains the subset , where and are coprime when . Then the derived length of is less than or equal to .

     

Lie Incidence Systems from Projective Varieties

The homogeneous space , where is a simple algebraic group and a parabolic subgroup corresponding to a fundamental weight (with respect to a fixed Borel subgroup of in ), is known in at least two settings. On the one hand, it is a projective variety, embedded in the projective space corresponding to the representation with highest weight . On the other hand, in synthetic geometry, is furnished with certain subsets, called lines, of the form where is a preimage in of the fundamental reflection corresponding to and . The result is called the Lie incidence structure on . The lines are projective lines in the projective embedding. In this paper we investigate to what extent the projective variety data determines the Lie incidence structure.

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

Optimal estimation of shell thickness in Cutland's construction of Wiener measure

In Cutland's construction of Wiener measure, he used the product of Gaussian measures on , where is an infinite integer. It is mentioned by Cutland and Ng that for the product measure ,

where and with any positive infinite number. We prove here that may be replaced by with any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that . And for the *Lebesgue measure , is finite and not infinitesimal iff with finite, while for the *Lebesgue area of the sphere , should be .

     

On uniqueness of -adic meromorphic functions

Let be a complete ultrametric algebraically closed field of characteristic zero, and let be the field of meromorphic functions in . For all set in and for all we denote by the subset of : zero of order After studying unique range sets for entire functions in in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every , there exist sets of elements in such that, if have the same poles (counting multiplicities), and satisfy , then . We show how to construct such sets.

     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

Some properties of ordinary sense slice 1-links: some answers to problem (26) of Fox

We prove that, for any ordinary sense slice 1-link , we can define the Arf invariant, and Arf()=0. We prove that, for any -component 1-link , there exists a -component ordinary sense slice 1-link of which is a sublink.

     

Enveloping algebras of lie color algebras: primeness versus graded-primeness

Let be a finite abelian group and let be a, possibly restricted, -graded Lie color algebra. Then the enveloping algebra is also -graded, and we consider the question of whether being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field of characteristic . Specifically, we show that if and if has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an anti-automorphism of whose square is the automorphism of the enveloping algebra associated with its -grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if is graded-prime and if most homogeneous components of are infinite dimensional over , then is prime. Here we use -methods to study the grading on the extended centroid of . In particular, if is generated by the infinite support of , then we prove that is homogeneous.

     

Certain invariant subspace structure of

In this note, we study certain structure of an invariant subspace of . Considering the largest -invariant (resp. -invariant) subspace in the wandering subspace of with respect to the shift operator , we give an alternative characterization of Beurling-type invariant subspaces. Furthermore, we consider a certain class of invariant subspaces.

     

The moduli of substructures of a compact complex space

We construct a space of fine moduli for the substructures of an arbitrary compact complex space . A substructure of is given by a subalgebra of the structure sheaf with the additional feature that is also a complex space; and are called equivalent if and only if and are isomorphic as subalgebras of .

Since substructures are quotients, it is only natural to start with the fine moduli space of all complex-analytic quotients of . In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of , we realize that its open subset consisting of all substructures turns out to be a complex space which has the required universal property.

     

On certain character sums over

Let be the finite field with elements and let denote the ring of polynomials in one variable with coefficients in . Let be a monic polynomial irreducible in . We obtain a bound for the least degree of a monic polynomial irreducible in ( odd) which is a quadratic non-residue modulo . We also find a bound for the least degree of a monic polynomial irreducible in which is a primitive root modulo .

     

On complementary subspaces of Hilbert space

Every pair of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form on a Hilbert space . Here is possibly , is a positive injective contraction and denotes the graph of . For such a pair the following are equivalent: (i) is similar to a pair in generic position; (ii) and have a common algebraic complement; (iii) is similar to for some operators on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.

     

On the cohomology of regular differential forms and dualizing sheaves

If is a local Dedekind scheme and is a projective Cohen-Macaulay variety of relative dimension , then is torsionfree if and only if is arithmetically Cohen-Macaulay for a suitable embedding in . If is regular then is torsionfree whenever the multiplicity of the special fibre is not a multiple of the characteristic of the residue class field.

     

Bourgain's analytic projection revisited

For a positive function on the unit circle with , the following two statements are equivalent: (a) ; (b) there is an operator projecting onto for all at once and having weak type (1,1) with respect to .