A cohesive set which is not high

We study the degrees of unsolvability of sets which are cohesive (or have weaker recursion-theoretic “smallness” properties). We answer a question raised by the first author in 1972 by showing that there is a cohesive set A whose degree a satisfies a' = 0″ and hence is not high. We characterize the jumps of the degrees of r-cohesive sets, and we show that the degrees of r-cohesive sets coincide with those of the cohesive sets. We obtain analogous results for strongly hyperimmune and strongly hyperhyperimmune sets in place of r-cohesive and cohesive sets, respectively. We show that every strongly hyperimmune set whose degree contains either a Boolean combination of ∑2 sets or a 1-generic set is of high degree. We also study primitive recursive analogues of these notions and in this case we characterize the corresponding degrees exactly. MSC: 03D30, 03D55.     

On a construction of critical graphs which are not sensitive

Let ?(G) be theclosed-set lattice of a graphG. G issensitive if the following implication is always true for any graphG′: ?(G)??(G′)?(G)?G′G iscritical if ?(G)??(G-e) for anye inE(G) and ?(G)??(G+e) for anye in \(\left( {\bar G} \right)\) where \(\bar G\) is the complement ofG. Every sensitive graph is, a fortiori, critical. Is every critical graph sensitive? A negative answer to this question is given in this note.     

A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Let be a sequence of positive real numbers. We define as the space of functions which are analytic in the unit disc , continuous on and such that

where is the Fourier coefficient of the restriction of to the unit circle . Let be a closed subset of . We say that is a Beurling-Carleson set if

where denotes the distance between and . In 1980, A. Atzmon asked whether there exists a sequence of positive real numbers such that for all and that has the following property: for every Beurling-Carleson set , there exists a non-zero function in that vanishes on . In this note, we give a negative answer to this question.

     

The first example of a recursive function which is not primitive recursive

The first example of a recursive function which is not primitive recursive is usually attributed to W. Ackermann. The authors of the present paper show that such an example can also be found in a paper by G. Sudan, published concomitantly with Ackermann's paper.     

Estimating the direction in which a data set is most interesting

Summary Estimation of orientation is a key operation at each step in projection pursuit. Since projection pursuit is a nonparametric algorithm, and since even low-dimensional approximations to the target function must converge to their limits at rates considerably slower than n _-1 ² (where n is sample size), then it might be thought that the same is true of orientation estimates. It is shown in the present paper that this is not the case, and that estimation of orientation is a parametric operation, in the sense that, under mild nonparametric assumptions, correctly-chosen kernel-type orientation estimates converge to their limits at rate n _-1 ² . This property is not enjoyed by standard projection pursuit orientation estimates, which converge at a slower rate than n _-1 ² . Most attention in the present paper is focussed on the case of projection pursuit density approximation, but it is pointed out that our arguments hold generally. An important practical conclusion is that data should be smoothed less when estimating orientation than when constructing the final projection pursuit approximation.     

A Lie grading which is not a semigroup grading

This note is devoted to the construction of a graded Lie algebra, whose grading is not given by a semigroup; thus providing a counterexample to an assertion by Patera and Zassenhaus.     

A relation algebra which is not a cylindric reduct

There is an incomplete atomic relation algebra which is not the reduct of any 4-dimensional cylindric algebra. This completes the answer to a problem in [Mo61].Presented by Bjarni Jónsson.     

The union of a Riesz set and a Lust-Piquard set is a Riesz set

We prove that the union of a Riesz set and a Lust-Piquard set is a Riesz set. This gives as corollaries known results of Y. Katznelson, R.E. Dressler-L. Pigno, and D. Li. Moreover, we give an example of a Rosenthal set which is dense in Z for the Bohr topology.     

A contractible Levi-flat hypersurface which is a determining set for pluriharmonic functions

We find a real analytic Levi-flat hypersurface in C² containing a bounded contractible domain which is a determining set for pluriharmonic functions.     

The class of groups which have a subgroup of index 2 is not elementary

F. Oger proved that if A is a finite group, then the class of groups which are abelian-by-A can be axiomatized by a single first order sentence. It is established here that, in Oger's result, the word abelian cannot be replaced by group. Received: 15 March 1996 / Published online: 18 July 2001     

Banach spaces in which a theorem of orlicz is not true

Let the Banach space X be such that for every numerical sequencet _n ↘0 there exists in X an unconditionally convergent series σx_n, the terms of which are subject to the condition ∥x_n∥=t_n (n=1,2,...). Then $$\mathop {sup}\limits_n \mathop {inf}\limits_{X_n } d(X_n ,l_\infty ^n )< \infty ,$$ where X_n ranges over all the n-dimensional subspaces of X.     

A cooperative system which does not satisfy the limit set dichotomy

The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x<y, then either ω(x)<ω(y), or ω(x)=ω(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.     

A universal deformation ring which is not a complete intersection ring

Bleher and Chinburg recently used modular representation theory to produce an example of a linear representation of a finite group whose universal deformation ring is not a complete intersection ring. We prove this by using only elementary cohomological obstruction calculus. To cite this article: J. Byszewski, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On a function algebra which is not generated by its idempotents

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 3–7, January, 1992.     

An indecomposable direct summand of a serial module which is not uniserial

Abstract

We present a construction of indecomposable direct summands of serial modules which are not uniserial.     

An example of a commutative basic algebra which is not an MV-algebra

Many algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in [CHK1] that these lattices are in a one-to-one correspondence with so-called basic algebras. In the recent papers [BOTUR, M.—HALAŠ, R.: Finite commutative basic algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. and [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] whether every commutative basic algebra on the interval [0, 1] of the reals has to be an MV-algebra.