On sums of a prime and four prime squares in short intervals

In this paper, we are able to sharpen Hua's classical result by showing that each sufficiently large integer can be written as

     

Two, more readily computable, equivariant Nielsen numbers II

In this paper, we present and make computations of two equivariant Nielsen type numbers NG(f(H),k(H)) and NG(f(H),k(H)). The second one is new, while the first one extends and clarifies one given earlier by the author and Jianhan Guo. Both numbers were defined here in terms of Nielsen theory of M-ads introduced in the prequel to this work. The theory of M-ads is also used to give both upper and lower bounds on our numbers, and to make specific computations. Our numbers moreover, fit together in the same way that the two Nielsen type periodic point numbers NPn(f) and NΦn(f) fit together. In particular, we show that NG(f(H),k(H)) is greater than or equal to a sum of numbers of the form NG(f(K),k(K)), and give conditions for equality and Möbius inversion. The periodic point theory results are then seen to follow from what are actually generalizations of them.We work with both fixed point, and coincidence point classes, in the context of a category with essentiality which we introduced in the prequel on M-ads. It is intended that this paper be read in tandem with said prequel.     

Computation of Nielsen numbers for maps of compact surfaces with boundary

In this paper we study the Nielsen number of a self-map f:M→M of a compact connected surface with boundary. Let G=π₁(M) be the fundamental group of M which is a finitely generated free group. We introduce a new algebraic condition called “bounded solution length” on the induced endomorphism φ:G→G of f and show that many maps which have no remnant satisfy this condition. For a map f that has bounded solution length, we describe an algorithm for computing the Nielsen number N(f).     

On prime numbers of special kind on short intervals

Suppose that the Riemann hypothesis holds. Suppose that $$\psi _1 (x) = \mathop \sum \limits_{\begin{array}{*{20}c} {n \leqslant x} \\ {\{ (1/2)n^{1/c} \} < 1/2} \\ \end{array} } \Lambda (n)$$ where c is a real number, 1 < c ≤ 2. We prove that, for H>N ^1/2+10ε, ε > 0, the following asymptotic formula is valid: $$\psi _1 (N + H) - \psi _1 (N) = \frac{H}{2}\left( {1 + O\left( {\frac{1}{{N^\varepsilon }}} \right)} \right)$$ .     

Rotation numbers for curves on a torus

We define a regular homotopy invariant of closed curves on a surface, and give a formula for the rotation number of closed curves on torus, which is analogous to the Whitney formula for planar curves. As an application, we show a necessary condition for a Gauss word to be realized on torus.     

Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f^′ and k^′ which are homotopic as maps ofM-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X₁∪X₂).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.     

The Topological Tverberg Theorem and winding numbers

The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to R^d identifies points from q disjoint faces. (This has been proved for affine maps, for d?1, and if q is a prime power, but not yet in general.)The Topological Tverberg Theorem can be restricted to maps of the d-skeleton of the simplex. We further show that it is equivalent to a “Winding Number Conjecture” that concerns only maps of the (d-1)-skeleton of a (q-1)(d+1)-simplex to R^d. “Many Tverberg partitions” arise if and only if there are “many q-winding partitions.”The d=2 case of the Winding Number Conjecture is a problem about drawings of the complete graphs K_3q-2 in the plane. We investigate graphs that are minimal with respect to the winding number condition.     

The Betti numbers of some finite racks

We show that the lower bounds for Betti numbers given in (J. Pure Appl. Algebra 157 (2001) 135) are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.     

Onf-vectors and Betti numbers of multicomplexes

A multicomplexM is a collection of monomials closed under divisibility. For suchM we construct a cell complex M whosei-dimensional cells are in bijection with thef _i monomials ofM of degreei+1. The bijection is such that the inclusion relation of cells corresponds to divisibility of monomials. We then study relations between the numbersf _i and the Betti numbers of M. For squarefree monomials the construction specializes to the standard geometric realization of a simplicial complex.This work was supported by the Mittag-Leffler Institute during the Combinatorial Year program 1991–92. The second author also acknowledges support from the Serbian Science Foundation, Grant No. 0401D.     

On a property of functions on the sphere and its application

Given a continuous function f:S^m+n−2→R^m, and n points u₁,u₂,…,u_n∈S^m+n−2; does there exist a rotation r∈SO(m+n−1) such that f(ru₁)=f(ru₂)=?=f(ru_n)? In this paper, we study the property of a continuous map from a sphere to a Euclidean space by using the theory of Smith periodic transformation and Brouwer degree of map theorem. The conjecture is proved under the case of n=2 and m being even. Furthermore, this conjecture is proved for the case when u_j⋅u_j+1=λ and the dimension of the sphere is not less than m+n−2. This paper generalizes the Borsuk-Ulam theorem and then presents its application.     

On square-full numbers in short intervals

It is shown that the number of square-full numbers in the interval is asymptotically equal to for every in the range 1/6>0.14254, which extends P.Shiu's range 1/6>0.1526.     

On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals for all α ∈ [0,1] whenever . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.     

On sums of a prime and four prime squares in short intervals

In this paper, we prove that each sufficiently large integer N ≠1（mod 3） can be written as N=p＋p1^2＋p2^2＋p3^2＋p4^2, with
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes.     

On fixed points of stratified maps

Nielsen fixed point theory deals with the fixed point sets of self maps on compact polyhedra. In this note, we shall extend it to stratified maps, to consider fixed points on (noncompact) strata. The extension was motivated by our recent work on the braid forcing problem in which the deleted symmetric products are indispensable. The stratified viewpoint is theoretically as natural as the equivariant Nielsen fixed point theory, while it can be more tractable computationally and more flexible in applications. This work was partially supported by an NSFC grant and a BMEC grant.     

On the exceptional set in Goldbach’s problem in short intervals

Let 1/5 < θ ≤ 1. We prove that there exists a positive constant δ such that the number of even integers in the interval [X, X + X ^θ] which are not a sum of two primes is 《 X ^θ−δ. The proof uses the circle method, a sieve method, exponential sum estimates and zero-density estimates for L-functions. Current address: Department of Mathematics, 20014 University of Turku, Finland. Author’s address: Department of Mathematics, University of London, Royal Holloway, Egham, Surrey TW20 0EX, UK     

Distribution of rational numbers in short intervals

A simplified proof for a well-distribution property for rational numbers is given and a connection with Riemann’s Hypothesis is pointed out. More precisely, we consider rational numbers with denominators of a given order of magnitude and show that the number of such numbers lying in a short interval of given length is normally close to its expectation in a mean square sense. The proof is elementary, using only Fourier series and Ramanujan sums. At the end of the paper, a variant of the circle method is discussed as an application.