On the Cohomology Sequence in a Semiabelian Category

In a semiabelian category, a strictly exact sequence 0ABC0 of cochain complexes gives rise to the cohomology sequence ...H ⁿ(A) H ⁿ(B) H ⁿ(C) H ⁿ⁺¹ (A) .... We study conditions for exactness of the homology sequence at a given term.     

Error bounds for continued fractionsK(1/b n)

Summary A priori truncation error bounds are obtained for continued fractions of the formK(1/b _n),b _n complex. The error bounds are easily applied to the case whenb _n0 asn. A numerical example involving the complex error function is given.     

Entire functions with univalent derivatives in a disc

It is proved that if the increasing sequence (n_p) of natural numbers satisfies the condition n_p+1/n_p 1 (p ) and all derivatives f^(np) of the analytic function f in D=¦¦< 1 are univalent in D, then f is an entire function. At the same time, for each increasing sequence (n_p) natural numbers such that n_p+1/n_p (p ) there exists an analytic function f in D all of whose derivatives f^(np) are univalent in D and D is the boundary for f. The growth of entire functions with derivatives univalent in the disc D is also studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 400–406, March, 1991.     

Codimensionm isometric immersions between pseudo-spheres

We describe all isometric immersionsf:S _p ⁿ S _p ^+1/n+m ,n –p 2m, whose first normal space is -parallel in the complement of totally geodesic points while the set of totally geodesic points does not disconnectS _p ⁿ .     

Complexes whose boundaries cannot be pushed around

LetK ^{n n} be a triangulatedn-ball. Examples are given to show that unlike in the two-dimensional case, the following hold for alln3: (1) there are nonconvexK ⁿ with no convex simplexwise linear embeddingsK ^{n n} , even though there are strictly convex simplexwise linear embeddings K ^{n n} ; (2) there are convexK ⁿ , with no spanning simplices, such that not every simplexwise linear embeddingf: K ^{n n} with convex image can be extended to a simplexwise linear embedding ofK ⁿ ; (3) there are convexK ⁿ such that the space of simplexwise linear homeomorphisms ofK ⁿ , fixed on K ⁿ , is not path connected.Partially supported by NSF Contract DMS-8503388.     

Lie Groupoids, Gerbes, and Non-Abelian Cohomology

We observe that any regular Lie groupoid G over a manifold M fits into an extension K G E of a foliation groupoid E by a bundle of connected Lie groups K. If F is the foliation on M given by the orbits of E and T is a complete transversal to F , this extension restricts to T, as an extension K _T G _T E _T of an étale groupoid E _T by a bundle of connected groups K _T . We break up the classification problem for regular Lie groupoids into two parts. On the one hand, we classify the latter extensions of étale groupoids by (non-Abelian) cohomology classes in a new ech cohomology of étale groupoids. On the other hand, given K and E and an extension K _T G _T E _T over T, we present a cohomological obstruction to the problem of whether this is the restriction of an extension K G E over M; if this obstruction vanishes, all extensions K G E over M which restrict to a given extension over the transversal together form a principal bundle over a group of bitorsors under K.     

Separation of hypersurfaces

We give a generalization of results obtained in [15]. LetK ⁿ denote the set of embedded hypersurfaces in ⁿ⁺¹; for all xSⁿ and MK ⁿ we denote by C _x ^M the apparent contour ofM in the directionx. Then we give a sufficient condition on WSⁿ such that the map _W ^{K n}:K ⁿ P(T Sⁿ) , defined by _W ^{K n} (M)={C _w ^M ¦ wW}, is injective.     

The Rank of a Direct Power of a Small-Cancellation Group

We construct an example of a finitely generated group G such that rank((G )ⁿ)=2 for all n1. For each n, we construct a finitely presented group G _n such that rank((G _n )ⁿ)=2. We conjecture that if G is a word-hyperbolic group then rank(G ⁿ ) as $ n. For each m we give an example of a residually finite group K _m such that K _m has exactly two relators, but K _m has no proper subgroups of index $ m. We construct a finitely generated group D such that there is an epimorphism DD×D.     

Resolvent of the Toeplitz operator may increase arbitrarily fast

It is proved that for every sequence of points _n from the unit circle, _n1, and for an arbitrary sequence of positive numbers A_n, A_n, there exists a continuous real function u, such that for the Toeplitz operator T (acting in the Hardy space H²) with the symbol =e ^iu we have the estimates (T–_nI)^–1>A_n, n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 157, pp. 175–177, 1987.     

The MoravaK-theory of algebraicK-theory spectra

Ifn2 the MoravaK-theoryK(n) _* of an algebraicK-theory spectrumKX vanishes for any ring or schemeX. This is proved using thev _n -complexes of Hopkins and Smith, together with the following theorem. The natural mapf:Q ₀S⁰BGL⁺ factors through the space ImJ. In particularf _*: _* ^s K _* annihilates CokerJ. These results are closely related to the Lichtenbaum-Quillen conjectures.Partially supported by an NSF grant.     

Almost everywhere convergence of the spherical partial sums for radial functions

LetfL ^p( ⁿ ),n2, be a radial function and letS _Rf be the spherical partial sums operator. We prove that if thenS _Rf(x)f(x) a.e. asR. The result is false for and \frac{{2n}}{{n + 1}}$$ " align="middle" border="0"> .Partially supported by M.P.I.     

The convergence to a Gaussian process of the number of empty cells in the classical problem of distributing particles among cells

We consider a case in which n particles are distributed independently of one another in N cells. We examine the behavior of the number of empty cells, ₀ (n), as a random function of the parameter n when n, N. We prove that for suitable variation of the time parameter, ₀ (n) will converge to a Gaussian process in the following cases: a) n/N, n/N^– l _nN; b) n/N0, n²/N.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp, 97–104, July, 1968.     

On the shape of the convex hull of random points

Summary Denote by E _n the convex hull of n points chosen uniformly and independently from the d-dimensional ball. Let Prob(d, n) denote the probability that E _n has exactly n vertices. It is proved here that Prob(d, 2 ^d/2 d ^-)1 and Prob(d, 2 ^d/2 d ^(3/4)+)0 for every fixed >0 when d. The question whether E _n is a k-neighbourly polytope is also investigated.     

On random minimum length spanning trees

We extend and strengthen the result that, in the complete graphK _n with independent random edge-lengths uniformly distributed on [0, 1], the expected length of the minimum spanning tree tends to(3) asn. In particular, ifK _n is replaced by the complete bipartite graphK _{n, n} then there is a corresponding limit of 2 (3).     

Homotopies for computation of fixed points on unbounded regions

Givenf: R ⁿ R ^n* with some conditions, our aim is to compute a fixed pointx f(x) off; hereR ⁿ isn-dimensional Euclidean space andR ^n* is the collection of nonempty subsets ofR ⁿ . A typical application of the algorithm can be motivated as follows: Beginning with the constant mapf ₀:R ⁿ {0} R ⁿ and its fixed pointx ₀ = 0, we deformf _t ast tof f and follow the pathx _t of fixed points off _t . Cluster points of thex _t 's ast are fixed points off. This research was supported in part by Army Research Office-Durham Contract DAHC-04-71-C-0041 and by National Science Foundation Grant GK-5695.     

Divergence of the (C, 1) Means of d-dimensional Walsh-Fourier Series

In 1992, Móricz, Schipp and Wade [MSW] proved for functions in L log⁺ L(I ²) (I ² is the unit square) the a.e. convergence of the double (C, 1) means of the Walsh-Fourier series _n f f as min(n ₁, n ₂) , n = (n ₁, n ₂ N ²). In the same paper, they also proved the restricted convergence of the (C, 1) means of functions in L(I ²): (2 ⁿ 1,2 ⁿ 2)f f a.e. as min (n ₁, n ₂) provided |n ₁ – n ₂| < C. The aim of this paper is to demonstrate the sharpness of these results of Móricz, Schipp and Wade with respect to both the space L log⁺ L(I ²) and the restrictedness |n ₁ – n ₂| < C.     

Random Walks Crossing High Level Curved Boundaries

Let {S _n} be a random walk, generated by i.i.d. increments X _i which drifts weakly to in the sense that as n . Suppose k0, k1, and E|X ₁|^1\k = if k>1. Then we show that the probability that S. crosses the curve nan ^K before it crosses the curve n –an ^k tends to 1 as a . This intuitively plausible result is not true for k = 1, however, and for 1/2 <k<1, the converse results are not true in general, either. More general boundaries g(n) than g(n) = n ^k are also considered, and we also prove similar results for first passages out of regions like { (n, y): n1, |y| (a + n) ^k } as a .     

Random Lazy Random Walks on Arbitrary Finite Groups

This paper considers lazy random walks supported on a random subset of k elements of a finite group G with order n. If k=a log₂ n where a>1 is constant, then most such walks take no more than a multiple of log₂ n steps to get close to uniformly distributed on G. If k=log₂ n+f(n) where f(n) and f(n)/log₂ n0 as n, then most such walks take no more than a multiple of (log₂ n) ln(log₂ n) steps to get close to uniformly distributed. To get these results, this paper extends techniques of Erdös and Rényi and of Pak.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Rates of convergence and optimal spectral bandwidth for long range dependence

Summary For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like ^1–2H as 0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2j/n,j=1,..., m, wherem andm/n0 asn. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.