Transformations conjugate to their inverses have even essential values

Let be an ergodic automorphism defined on a standard Borel probability space for which and are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of . It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97--124) that if is ergodic having simple spectrum and isomorphic to its inverse, and if is a conjugation between and (i.e. satisfies ), then , the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation , the unitary operator induced by on must have a multiplicity function whose essential values on the ortho-complement of the subspace are always even. In particular, we see that can be weakly mixing, so the corresponding must have even maximal spectral multiplicity (regarding as an even number).

     

The local cohomology modules of Matlis reflexive modules are almost cofinite

We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

     

Oscillatory singular integrals on and Hardy spaces

We consider boundedness properties of oscillatory singular integrals on and Hardy spaces. By constructing a phase function, we prove that boundedness may fail while boundedness holds for all . This shows that the theory and theory for such operators are fundamentally different.

     

On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

A Wolff-Denjoy theorem for infinitely connected Riemann surfaces

We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let be a non-parabolic Riemann surface with Martin boundary . Suppose each Martin function , , extends continuously to and vanishes there. We show that if is an endomorphism of and the iterates of converge to the point at infinity, then the iterates converge locally uniformly to a point in . As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on but none on itself.

     

The statistics of continued fractions for polynomials over a finite field

Given a finite field of order and polynomials of degrees respectively, there is the continued fraction representation . Let denote the number of such pairs for which and for . We give both an exact recurrence relation, and an asymptotic analysis, for . The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of . Averaged over all and as above, this presents no difficulties. The average value of is , and there is full information about the distribution. When is fixed and only is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of that differs from this average by more than

     

On fully invariant ideals of the free group algebra

Let be the group algebra of a free noncyclic group over an integral domain . It is proved that if is not a field, then there exists a fully invariant ideal of such is torsion-free but not projective as an -module. In other words, there exists a pure nonprojective variety of group representations over .

     

Operator versions of the Kantorovich inequality

The Operator Kantorovich Inequality

holds for a wide class of operators on a Hilbert space and all operators for which is a partial isometry, being the range projection of

     

Derivations with Engel conditions on multilinear polynomials

Let be a prime algebra over a commutative ring with unity and let be a multilinear polynomial over . Suppose that is a nonzero derivation on such that for all in some nonzero ideal of , with fixed. Then is central--valued on except when char and satisfies the standard identity in 4 variables.

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

and the associated line bundles

Let be a compact semi-simple Lie group, and let be a maximal unipotent subgroup of the complexified group . In this paper, we classify all the -invariant Kaehler structures on . For each Kaehler structure , let be the line bundle with connection whose curvature is . We then study the holomorphic sections of , which constitute a -representation space.

     

On contravariant finiteness of subcategories of modules of projective dimension

Let be an artin algebra. This paper presents a sufficient condition for the subcategory of to be contravariantly finite in , where is the subcategory of consisting of --modules of projective dimension less than or equal to . As an application of this condition it is shown that is contravariantly finite in for each when is stably equivalent to a hereditary algebra.

     

Integer-valued polynomials on Krull rings

If is a subring of a Krull ring such that is a valuation ring for every finite index , in Spec, we construct polynomials that map into the maximal possible (for a monic polynomial of fixed degree) power of , for all in Spec simultaneously. This gives a direct sum decomposition of Int, the -module of polynomials with coefficients in the quotient field of that map into , and a criterion when Int has a regular basis (one consisting of 1 polynomial of each non-negative degree).

     

Countable network weight and multiplication of Borel sets

A space Borel multiplies with a space if each Borel set of is a member of the -algebra in generated by Borel rectangles. We show that a regular space Borel multiplies with every regular space if and only if has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multiply with any non-separable metric space. In passing, we obtain a characterization of those spaces which Borel multiply with the space of countable ordinals, and an internal necessary and sufficient condition for to Borel multiply with every metric space.

     

On the higher delta invariants of a Gorenstein local ring

Let be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by for each module and for each integer . We propose a conjecture asking if for any positive integers and . We prove that this is true provided the associated graded ring of has depth not less than . Furthermore we show that there are only finitely many possibilities for a pair of positive integers for which .

     

The local zeta function for the non-trivial characters associated with the singular Jordan algebras

This paper investigates the local integrals

where represents the integers of a composition algebra over a non-archimedean local field and is a non-trivial character on the units in the ring of integers of extended to by setting . The local zeta function for the trivial character is known for all composition algebras . In this paper, we show in the quaternion case that for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for equal to the quadratic character. In this latter case, for any character of order greater than .

     

Polynomial rings over Goldie-Kerr commutative rings II

An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. 120 (1994), 989--993) is that any Goldie ring of Goldie dimension 1 has Artinian classical quotient ring , hence is a Kerr ring in the sense that the polynomial ring satisfies the on annihilators . More generally, we show that a Goldie ring has Artinian when every zero divisor of has essential annihilator (in this case is a local ring; see Theorem ). A corollary to the proof is Theorem 2: A commutative ring has Artinian iff is a Goldie ring in which each element of the Jacobson radical of has essential annihilator. Applying a theorem of Beck we show that any ring that has Noetherian local ring for each associated prime is a Kerr ring and has Kerr polynomial ring (Theorem 5).