Some remarks on the ring

Some properties like factoriality, seminormality and being a Krull domain, … are studied on power series rings , and over a commutative ring A. If \(\mathbb{X}\) is an uncountable set, there is an other sub-ring of that stands strictly between and , we denote it by . In this paper, we study properties mentioned before on the ring .     

Some remarks on the Clebsch's system

The almost Hamilton-Poisson realization, the stability problem, the existence of periodic solutions and the numerical integration via the Lie-Trotter integrator for the Clebsch system are discussed and some of their properties are pointed out.     

Some remarks on the Fefferman-Stein inequality

We investigate the Fefferman-Stein inequality related to a function f and the sharp maximal function f ^# on a Banach function space X. It is proved that this inequality is equivalent to a certain boundedness property of the Hardy-Littlewood maximal operator M. The latter property is shown to be self-improving. We apply our results in several directions. First, we show the existence of nontrivial spaces X for which the lower operator norm of M is equal to 1. Second, in the case when X is the weighted Lebesgue space L ^p (w), we obtain a new approach to the results of Sawyer and Yabuta concerning the C _p condition.     

Some remarks on the Steiner problem

Let Σ be a set of n points in the plane. The minimal network for Σ is the tree of shortest total length L_M(Σ) whose vertices are exactly the points of S. The Steiner minimal network for Σ is the tree of shortest possible total length L_S(Σ) when the vertices are allowed to be any set Σ′ ? Σ. Clearly L_S(Σ) ? L_M(Σ), since the minimization in L_S is over a larger set. It has long been conjectured that, conversely, $\text{L}{}_{\mathrm{<mtext>S</mtext>}}\text{(Σ) ? (}\text{3}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{2}\text{) L}{}_{\mathrm{<mtext>M</mtext>}}\text{(Σ)}$, but this has previously been proved only if n = 3. In this paper, among other results, this is proved for n = 4. Unfortunately the proof is sufficiently complicated that immediate generalization to arbitrary n, no matter how desirable, is unlikely.     

Some remarks on the Hook formula

Summary Without the Frame-Robinson-Thral hook formula we establish some divisibility properties of degrees of irreducible representations of the symmetric group.

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Riassunto Senza utilizzare una nota formula di Frame-Robinson-Thrall stabiliamo alcune proprietà di divisibilità di gradi di rappresentazioni irriducibili del grupposimmetrico.

     

Some remarks on the Lehmer conjecture

In 1933, Lehmer exhibited the polynomial

$$\begin{aligned} L(z)=z^{10} + z^9 - z^7 - z^6 - z^5 - z^4 - z^3 + z + 1 \end{aligned}$$

with Mahler measure \(\mu _0>1\). Then he asked if \(\mu _0\) is the smallest Mahler measure, not 1. This question became known as the Lehmer conjecture and it was apparently solved in the positive, while this paper was in preparation [19]. In this paper we consider those polynomials of the form \(\chi _A\), that is, Coxeter polynomials of a finite dimensional algebra A (for instance \(L(z)=\chi _{\mathbb {E}_{10}}\)). A polynomial in \(\mathbb {Z}[T]\) which is either cyclotomic or with Mahler measure \(\ge \mu _0\) is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that A being a tree algebra is a sufficient condition for \(\chi _A\) to be Lehmer.

     

Some remarks on the Chebyshev functional

Identities are provided for the Chebyshev functional defined on a real linear space equipped with a bilinear form. In addition, some inequalities for the functional are discussed.     

Some remarks on the Jacobian question

This revised version of Abhyankar's old lecture notes contains the original proof of the Galois case of then-variable Jacobian problem. They also contain proofs for some cases of the 2-variable Jacobian, including the two characteristic pairs case. In addition, proofs of some of the well-known formulas enunciated by Abhyankar are actually written down. These include the Taylor Resultant Formula and the Semigroup Conductor formula for plane curves. The notes are also meant to provide inspiration for applying the expansion theoretic techniques to the Jacobian problem.     

Some remarks on interval graphs

Using simplicial decompositions a new and simple proof of Lekkerkerker-Boland’s criterion for interval graphs is given. Also the infinite case is considered, and the problem is tackled to what extent the representation of a graph as an interval graph is unique. Dedicated to Tibor Gallai on his seventieth birthday     

Some remarks on greedy algorithms

Estimates are given for the rate of approximation of a function by means of greedy algorithms. The estimates apply to approximation from an arbitrary dictionary of functions. Three greedy algorithms are discussed: the Pure Greedy Algorithm, an Orthogonal Greedy Algorithm, and a Relaxed Greedy Algorithm.This research was supported by the Office of Naval Research Contract N0014-91-J1343.