Prüfer’s ideal numbers as Gelfand’s maximal ideals

Polyadic arithmetics is a branch of mathematics related to p-adic theory. The authors suggest two non-classical models for the Prüfer profinite completion Z of the ring Z. Firstly, letA be the algebra of all periodic functions on Z, then Z can be defined as the ring of all non-zero morphisms A → C with convolution ring operations equipped with some natural topology. Secondly, let (E, μ) be the maximal ideal space for the Banach algebra of almost periodic functions on Z with the Gelfand topology μ. One can define rings operations in (E, μ) which turn it into a topological ring isomorphic to Z.     

Weierstrass’s construction of the irrational numbers

We present an overview of the development of the irrational numbers due to Karl Weierstrass. This construction was first presented during lectures in the 1860s in Berlin. Weierstrass never published his construction. Several of his students (Kossak, Horwitz, von Dantscher and Pincherle, to name a few) gave accounts in lecture notes from the courses. However, these notes were not published under the direction of Weierstrass.     

Linearity defect of edge ideals and Fröberg’s theorem

Fröberg’s classical theorem about edge ideals with 2-linear resolution can be regarded as a classification of graphs whose edge ideals have linearity defect zero. Extending his theorem, we classify all graphs whose edge ideals have linearity defect at most 1. Our characterization is independent of the characteristic of the base field: The graphs in question are exactly weakly chordal graphs with induced matching number at most 2. The proof uses the theory of Betti splittings of monomial ideals due to Francisco, Hà, and Van Tuyl and the structure of weakly chordal graphs. Along the way, we compute the linearity defect of edge ideals of cycles and weakly chordal graphs. We are also able to recover and generalize previous results due to Dochtermann–Engström, Kimura and Woodroofe on the projective dimension and Castelnuovo–Mumford regularity of edge ideals.     

Jacobi and Kummer’s ideal numbers

In this article we give a modern interpretation of Kummer’s ideal numbers and show how they developed from Jacobi’s work on cyclotomy, in particular the methods for studying “Jacobi sums” which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.     

Waring’s problem in prime numbers

The existence of a constant V(n) such that any sufficiently large natural number can be represented as a sum of nth degrees of primes in total quantity not exceeding this constant is proved.     

Liouville numbers and Schanuel’s Conjecture

In this paper, using an argument of P. Erd?s, K. Alniaçik, and É. Saias, we extend earlier results on Liouville numbers, due to P. Erd?s, G.J. Rieger, W. Schwarz, K. Alniaçik, É. Saias, E.B. Burger. We also produce new results of algebraic independence related with Liouville numbers and Schanuel’s Conjecture, in the framework of ${G_\delta}$ -subsets.     

A quantitative program for Hadwiger’s covering conjecture

In 1957,Hadwiger made a conjecture that every n-dimensional convex body can be covered by 2n translates of its interior.Up to now,this conjecture is still open for all n 3.In 1933,Borsuk made a conjecture that every n-dimensional bounded set can be divided into n + 1 subsets of smaller diameters.Up to now,this conjecture is open for 4 n 297.In this article we encode the two conjectures into continuous functions defined on the spaces of convex bodies,propose a four-step program to attack them,and obtain some partial results.     

Local Łojasiewicz exponents,Milnor numbers and mixed multiplicities of ideals

Let be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I ₁,..., I _n . We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions and Newton non-degenerate functions. Work supported by DGICYT Grant MTM2006-06027.     

Geometric proof of R?dseth’s formula for Frobenius numbers

Using a geometric interpretation of continued fractions, we give a new proof of R?dseth’s formula for Frobenius numbers.     

Some of Melvin Henriksen?s contributions to spaces of ideals

The three of us have written this note to discuss Mel Henriksen?s joint paper with us, Joincompact spaces, continuous lattices andC^?-algebras. In this paper we learned that the space of closed primal ideals of a C^?-algebra is a continuous lattice, so it is joincompact when equipped with the lower and Scott topologies. In addition to its mathematics, we discuss how Mel brought us into this work, and his and its continuing influence on us.     

Branch values in Ahlfors’ theory of covering surfaces

In the study of the constant in Ahlfors' second fundamental theorem involving a set E_q consisting of q points,branch values of covering surfaces outside E_q bring a lot of troubles.To avoid this situation,for a given surface Σ_0,it is useful to construct a new surface Σ_1,such that L(■Σ_1)≤L(■Σ_0),and H(Σ_1,E_q)≥H(Σ_0,E_q).and all branch values of Σ_1 are contained in E_q.The goal of this paper is to prove the existence of such Σ_1,which generalizes a result found by Zhang(2013).     

A sufficient condition for Kim’s conjecture on the competition numbers of graphs

A hole of a graph $G$ is an induced cycle of length at least 4. Kim (2005) [3] conjectured that the competition number $k\left(G\right)$ is bounded by $h\left(G\right)+1$ for any graph $G$, where $h\left(G\right)$ is the number of holes of $G$. Li and Chang (2009) [5] proved that the conjecture is true for a graph whose holes all satisfy a property called ‘independence’. In this paper, by using similar proof techniques in Li and Chang (2009) [5], we prove the conjecture for graphs satisfying two conditions that allow the holes to overlap a lot.     

Correction to: On a variant of Pillai’s problem with transcendental numbers

Acta Mathematica Hungarica -     

The minimal and maximal operator ideals associated to $$$$-tensor norms of Michor’s type

We study an \((n+1)\)-tensor norm \(\alpha ^C_{\mathbf {r}}\) extending to \((n+1)\)-fold tensor products a tensor norm defined by Michor when \(n=1\) by convexification of a certain s-norm. We characterize the maps of the minimal and the maximal multilinear operator ideals related to \(\alpha ^C_{\mathbf {r}}\) in the sense of Defant, Floret and Hunfeld.     

Posner’s second theorem for Jordan ideals in rings with involution

Posner’s second theorem states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend this result to Jordan ideals of rings with involution. Moreover, some related results are also discussed.     

Optimal sequencing of a set of positive numbers with the variance of the sequence’s partial sums maximized

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a nice structure. It is interesting to note that the symmetric problem which aims at minimizing the variance of the same partial sums is proved to be NP-complete in the literature.