Cyclotomic factors of Coxeter polynomials

In this paper we show that the cyclotomic factors of the En Coxeter polynomials depend only on the value of nmod360, and come exclusively from spherical subdiagrams.     

Polynomial approximation using projections whose kernels contain the Chebyshev polynomials

Let P be any projection of f ε C^{n + 1}[−1, 1] onto _n such that P(T_{n + 1}) = 0. It is shown that, for a wide class of such projections, f − Pf may be expressed in terms of f^{n + 1} in the same manner as E_n(f), the error of minimax approximation. Furthermore, a general necessary condition is obtained for this phenomenon.     

Finite groups whose prime graphs do not contain triangles. I

Finite groups whose prime graphs do not contain triangles are investigated. In the present part of the study, the isomorphic types of prime graphs and estimates of the Fitting length of solvable groups are found and almost simple groups are determined.     

Class numbers with many prime factors

Here, we construct infinitely many number fields of any given degree d>1 whose class numbers have many prime factors.     

Cyclotomic polynomials and unions of groups

Using factorizations of the polynomial (1?xⁿ)/(1?x) we determine all monoids that are unions of groups, have a nontrivial group of units, and are syntactic monoids of languages generated by finite prefix codes.     

Critically prime interval orders

We prove that a prime interval order, not necessarily finite, has no prime upper cover if and only if it has width two. On the other hand, we prove that every finite prime N-free order has at least one prime upper cover.     

Families of non-congruent numbers with arbitrarily many prime factors

A method is given for generating families of non-congruent numbers with arbitrarily many prime factors. We then use this method to construct an infinite set of new families of these numbers with prime factors of the form $8k+3$.     

On rings whose prime ideals are completely prime

We investigate in this paper rings whose proper ideals are 2-primal. We concentrate on the connections between this condition and related concepts to this condition, and several kinds of π-regularities of rings which satisfy this condition. Moreover, we add counterexamples to the situations that occur naturally in the process of this note.     

Cut-and-stack simple weakly mixing map with countably many prime factors

Via the cut-and-stack construction we produce a 2-fold simple weakly mixing transformation which has countably many proper factors and all of them are 2-to-1 and prime.

     

On prime representing polynomials

A heuristic method is presented to determine the number of primesp ≤x, represented by an irreducible polynomialf(n), without non-trivial fixed factor (f(y)<∈Z[y]; n∈Z. The method is applied to two specific polynomials and the results are compared with those of the heuristic approach of Hardy and Littlewood.     

Integral points on quadrics in three variables whose coordinates have few prime factors

The main theorem states that if f(x ₁, x ₂, x ₃) is an indefinite anisotropic integral quadratic form with determinant d(f), and t a non-zero integer such that d(f)t is square-free, then as long as there is one integer solution to f(x ₁, x ₂, x ₃) = t there are infinitely many such solutions for which the product x ₁ x ₂ x ₃ has at most 26 prime factors. The proof relies on the affine linear sieve and in particular automorphic spectral methods to obtain a sharp level of distribution in the associated counting problem. The 26 comes from applying the sharpest known bounds towards Selberg’s eigenvalue conjecture. Assuming the latter the number 26 may be reduced to 22.     

The many faces of circle orders

A finite partially ordered set P is called a circle order if one can assign to each x P a circular disk C _x so that xy iff C _x C _y . It is interesting to observe that many other classes of posets, such as space-time orders, parabola orders, the Loewner order for 2×2 Hermitian matrices, etc. turn out to be exactly circle orders (or their higher dimensional analogues). We give a global proof for these equivalences.Research supported in part by the Office of Naval Research, the Air Force Office of Scientific Research and by DIMACS.