Some remarks on the deterministic particle swarm optimization algorithm

In this paper, we give a simple proof for the convergence of the deterministic particle swarm optimization algorithm under the weak chaotic assumption and remark that the weak chaotic assumption does not relax the stagnation assumption in essence. Under the spectral radius assumption, we propose a convergence criterion for the deterministic particle swarm optimization algorithm in terms of the personal best and neighborhood best position of the particle that incorporates the stagnation assumption or the weak chaotic assumption as a special case.     

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 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 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 one-sided regularity

A ring is said to be right (resp., left) regular-duo if every right (resp., left) regular element is regular. The structure of one-sided regular elements is studied in various kinds of rings, especially, upper triangular matrix rings over one-sided Ore domains. We study the structure of (one-sided) regular-duo rings, and the relations between one-sided regular-duo rings and related ring theoretic properties.     

Some remarks on trigonometric interpolation

The object of this paper is to consider the problem of (0, 1, 2, 4) trigonometric interpolation when nodes are taken to bex _kn=(2kπ/n),k=0, 1 …,n−1. Here the interpolatory polynomials are explicitly constructed and the corresponding convergence theorem is proved, which is shown to be best possible in a certain sense. It is interesting to compare these results with those of Saxena [6], where the convergence theorem requires the existence off ^m (x). I take this opportunity to express my thanks to Professor P. Turán for some valuable conversation which led to this work. The author is at present a member of the faculty of the Dept. of Mathematics, University of Florida, Gainesville, U.S.A.