There is no compactification theorem for the small inductive dimension

We give an example of a perfectly normal first countable space X¹ with ind X¹ = 1 such that if Z is a Lindelöf space containing X¹. then ind Z=dim Z=∞. Under CH, there is a perfectly normal, hereditarily separable and first countable such space.     

There is no Bogomolov type restriction theorem for strong semistability in positive characteristic

We give an example of a strongly semistable vector bundle of rank two on the projective plane such that there exist smooth curves of arbitrary high degree with the property that the restriction of the bundle to the curve is not strongly semistable anymore. This shows that a Bogomolov type restriction theorem does not hold for strong semistability in positive characteristic.

     

There is no plus-capping degree

Summary Answering a question of Per Lindström, we show that there is no plus-capping degree, i.e. that for any incomplete r.e. degreew, there is an incomplete r.e. degreea>w such that there is no r.e. degreev>w withav=w.Mathematics subject classification: 03D30The authors would like to thank Per Lindström for raising the question and Carl Jockusch for communicating it to them. The first author was partially supported by a U.S./New Zealand binational grant. The second author was partially supported by NSF grant DMS-9100114 and the U.S./New Zealand binational NSF grant INT-9020558     

There is no fat orbit

We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.     

There is no degree invariant half-jump

We prove that there is no degree invariant solution to Post's problem that always gives an intermediate degree. In fact, assuming definable determinacy, if is any definable operator on degrees such that on a cone then is low or high on a cone of degrees, i.e., there is a degree such that for every or for every .

     

There is no universal countable pentagon-free graph

No countable C_n-free graph contains every countable C_n-free graph as a subgraph, for n ≧ 4. For n = 4, this was proved earlier by J. Pach.     

Ohkawa's theorem: There is a set of Bousfield classes

We give a simple proof of Ohkawa's theorem, that there is a set of Bousfield classes. The proof leads us to consider the partially ordered set of Ohkawa classes, especially as it compares to the partially ordered set of Bousfield classes.

     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

There is no fast method for finding monochromatic complete subgraphs

A general theorem on the complexity of a class of recognition problems is proved. As a particular case the following result is given: There is no algorithm which, for any 2-coloration of the infinite complete graph, can produce a monochromatic subgraph of k vertices within $\text{2}{}^{\mathrm{<mtext>k</mtext><mtext>2</mtext>}}$ steps (at each step the color of an arbitrary edge is questioned).     

There is no safe pairing function over an arbitrary structure

In [1] the class of safe recursive functions over an arbitrary structure is defined. We prove that in this class, one cannot define a total pairing function independently of the structure. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

There is no Drake/Larson linear space on 30 points

A linear space is Drake/Larson if it contains at least two lines and there are no lines of size 2, 3 or 6. The existence or nonexistence of such linear spaces on v points is known except for v=30. The purpose of this article is to settle the remaining case on thirty points in the negative. This result relies on a combination of parameter calculation and exhaustive computer search. © 2009 Wiley Periodicals, Inc. J Combin Designs 28: 48–70, 2010     

There is no degree map for 0-cycles on Artin stacks

We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (1) the degree of an ordinary point is non-zero, and (1) degrees are compatible with closed immersions.     

A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

There is no upper bound of small transfinite compactness degree in metrizable spaces

In [J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993], Aarts and Nishiura investigated several types of dimensions modulo a class P of spaces. These dimension functions have natural transfinite extensions. The small transfinite compactness degree trcmp is such transfinite dimension function extending the small compactness degree cmp. We shall prove that there is no upper bound for trcmp in the class of metrizable spaces, i.e. for each ordinal number α there exists a metrizable space Xα such that trcmpXα=α. We also give a characterization of the dimension dim of a separable (compact) metrizable space in terms of the function cmp of the product of this space with a nowhere locally compact zero-dimensional factor.     

There is no (46, 6, 1) block design*

Abstact: In this paper we show that a (46, 6, 1) design does not exist. This result was obtained by a computer search. In the incidence matrix of such a design, there must exist a “c4” configuration—6 rows and 4 columns, in which each pair of columns intersect exactly once, in distinct rows. There can also exist a “c5” configuration with 10 rows and 5 columns, in which each pair of columns intersect exactly once, in distinct rows. Thus the search for (46, 6, 1) designs can be subdivided into two cases, the first of which assumes there is no “c5”, and the second of which assumes there is a “c5”. After completing the searches for both cases, we found no (46, 6, 1) design. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 60–71, 2001