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.     

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.     

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.     

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 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.     

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.     

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 .     

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.     

On weakly p-harmonic maps to a closed hemisphere

We consider weakly p-harmonic maps (p2) from a compact connected Riemannian manifold M^m(m2) to the the standard sphere Sⁿ with values in the closed hemisphere Sⁿ₊ = {x Sⁿ : x_n+1 0 } (n 2). We first prove that if u=(u₁,...,u_n+1):MSⁿ is a weakly p-harmonic map satisfying u_n+1(x)>0 a.e. on M, then it is a minimizing p-harmonic map. Next, we give a necessary and sufficient condition for the boundary data : M Sⁿ₊ to achieve uniqueness; and when this condition fails, we are able to describe the set of all minimizers. When M is without boundary, we obtain a Liouville type Theorem for weakly p-harmonic maps.Mathematics Subject Classification (2000): 58E20; 35J70     

New Gaussian estimates for enlarged balls

Consider a centered Gaussian measure on a separable Banach spaceX. Denote byK the unit ball of the reproducing kernel of , and consider a symmetric convex setC ofX. We provide two-sided estimates of (C+tK). We determine in a very general setting at which rate for the gauge ofC the variablesY _n (2 logn)^–1/2 cluster toK, when (Y _n ) is an i.i.d. sequence distributed like . The rate depends only on the behavior of the function (C) as 0.Partially supported by an NSF grant