Coarse Cohomology and lp-Cohomology

We prove that, for any exact category M, any element of K₁(M)can be described in terms of a pair of admissible monomorphisms A X, B Y and an isomorphism :A X/A Y B Y/B X.     

Generalizations of Lifting Modules

The notion of -supplemented modules is a proper generalization of lifting modules. As a proper generalization of -supplemented modules, we introduce (D ₁₂)-modules and obtain some properties, characterizations and decompositions of -supplemented modules and (D ₁₂)-modules.     

Iterated means of convex bodies

LetK be a convex body inR ⁿ with polarK . Let _p refer to Fireyp orp-dot means. If 0<<1,p1, andK _i+1 =K _{i p} (1–)K _i , fori1, then K _i is the unit ball inR ⁿ.     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

⊕-Cofinitely Supplemented Modules

Let R be a ring and M a right R-module. M is called -cofinitely supplemented if every submodule N of M with M/N finitely generated has a supplement that is a direct summand of M. In this paper various properties of the -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of -cofinitely supplemented modules is -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R-module is -cofinitely supplemented. In addition, if M has the summand sum property, then M is -cofinitely supplemented iff every maximal submodule has a supplement that is a direct summand of M.     

An abstract analogon of the de Branges-Rovnyak functional model

Given a Hilbert spaceK generated by two of its subspaces,K=K ₁K ₂, we investigate the representation ofK inK ₁K ₂ which assigns to eachk K the pairPk=P(K ₁)k P(K ₂)k consisting of its orthogonal projections ontoK ₁ andK ₂.A part of this work was done during the visit of the second author at the University of Sevilla, Spain in September-October 1989. The support of Junta Andalusia is gratefully aknowledged.     

A characterization of 2-trivial Banach spaces with unconditional basis

Let X be a Banach space with an unconditional basis such that each operator from X into ² is 2-absolutely summing. Then X is isomorphic either to c_o or to ¹ or to c_o¹.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 76–87, 1987.The author is grateful to I. A. Komarchev for a series of useful marks and for the permission to publish the proof of Lemma 1.     

Projections in minimax algebra

An axiomatic theory of linear operators can be constructed for abstract spaces defined over (R, , ), that is over the (extended) real numbersR with the binary operationsx y = max (x,y) andx y = x + y. Many of the features of conventional linear operator theory can be reproduced in this theory, although the proof techniques are quite different. Specialisation of the theory to spaces ofn-tuples provides techniques for analysing a number of well-known operational research problems, whilst specialisation to function spaces provides a natural formal framework for certain familiar problems of approximation, optimisation and duality.     

Iterated relative recursive enumerability

A result of Soare and Stob asserts that for any non-recursive r.e. setC, there exists a r.e.[C] setA such thatAC is not of r.e. degree. A setY is called [of]m-REA (m-REA[C] [degree] iff it is [Turing equivalent to] the result of applyingm-many iterated hops to the empty set (toC), where a hop is any function of the formXX W _e ^X . The cited result is the special casem=0,n=1 of our Theorem. Form=0,1, and any (m+1)-REA setC, ifC is not ofm-REA degree, then for alln there exists an-r.e.[C] setA such thatA C is not of (m+n)-REA degree. We conjecture that this holds also form2.German speakers should not be unduely influenced by the acronym for this titlePartially supported by an NSF Postdoctoral Fellowship and the US Army Research Office through the Mathematical Sciences Institute of Cornell University     

Isometric dilations in nest algebras

LetT be a contraction acting in a separable Hilbert space and leaving invariant a nest of subspaces of . We answer the question: when doesT have an isometric extension to which leaves invariant the nest = {N N :N ;}.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Weakly Associated Prime Filtration

Let R be a (not necessarily Noetherian) commutative ring and let M be a (not necessarily finitely generated) R-module. We characterize the modules with only finitely many weakly associated primes as those modules M admitting a chain 0 = M ₀ M ₁ ... M _n = M of submodules together with prime ideals p₁, p₂,...,p _n such that the set of weakly associated primes of M _i /M _i-1 is equal to {p _i } for all 1 i n. Let M = gr_a(M) = _n0a ⁿ M/a ⁿ⁺¹ M be the corresponding graded module over the graded ring R = gr_a(R) = _n0a ⁿ /a ⁿ⁺¹. It is shown that the union of the set of weakly associated primes of.....     

Projective Modules and Complete Intersections

Let A be a Noetherian ring of dimension n and P be a projective A module of rank n having trivial determinant. It is proved that if n is even and the image of a generic element g P^* is a complete intersection, then [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1. Further, it is proved that if n is odd, A is Cohen–Macaulay and [P] = [Q A] in K₀(A) for some projective A module Q of rank n – 1, then P has a unimodular element.     

Isomorphisms of Steinberg groups over commutative rings

LetA andR be commutative rings, andm andn be integers3. It is proved that, if :St _m (A)St _n (R) is an isomorphism, thenm=n. Whenn4, we have: (1) Every isomorphism :St _n(A)St _n(R) induces an isomorphism:E _n (A)E _n (R), and is uniquely determined by; (2) IfSt _n (A) St _n (R) thenK _2.n (A)K _2.n (R); (3) Every isomorphismE _n (A) E _n (R) can be lifted to an isomorphismSt _n(A)St _n(R); (4)St _n(A) St _n(R) if and only ifAR. For the casen=3, ifSt ₃(A) andSt ₃(R) are respectively central extensions ofE ₃(A) andE ₃ (R), then the above (1) and (2) hold.The Project supported by National Natural Science Foundation of China     

Legitimate colorings of projective planes

For a projective plane _n of ordern, let( _n ) denote the minimum numberk, so that there is a coloring of the points of _n ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 ( _n ) 8 for every projective plane _n of ordern. Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation     

The Translative Kissing Number of Cartesian Product of two Convex Bodies, one of which is Two-Dimensional

This article discusses the relation between the translative kissing numbers of convex bodies K ₁, K ₂ and K ₁ K ₂. As an application of the main theorem, we find that the translative kissing number of B Q, where B is a 24-dimensional ball and Q is a two-dimensional non-parallelogram convex domain, is 1375926.     

K-loops and Bruck loops on ℝ×ℝ

Using the function (x)=cosh x, K-loops on × are constructed. Since every K-loop is a Bruck loop, we have also examples for Bruck loops. Furthermore we investigate the group of the automorphisms _a,b of the K-loop which satisfy the equation a(bc)=(ab)_a,b(c).Dedicated to Professor Dr. Oswald Giering on the occasion of his 60 th birthday     

A survey of pseudo (v,k, λ)-designs

This work is an attempt to give a complete survey of all known results about pseudo (v, k, )-designs. In doing this, the author hopes to bring more attention to his conjecture given in Section 6; an affirmative answer to this conjecture would settle completely the existence and construction problem for a pseudo (v, k, )-design in terms of the existence of an appropriate (v, k, )-design.