On solvable Dirac equation with polynomial potentials

One dimensional Dirac equation is analysed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.Comment: 3 pages, updated bibliograph

Injective convolution operators on ℓ∞(Γ){\ell}^{\infty}(\Gamma) are surjective

Let $\Gamma$ be a discrete group and let $f \in \ell^1(\Gamma)$. We observe that if the natural convolution operator $\rho_f:\ell^{\infty}(\Gamma)\to \ell^{\inf ty}(\Gamma)$ is injective, then f is invertible in $\ell^1(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt, by appealing to the direct finiteness of the algebra $\ell^1(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^p$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete G. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results.Comment: (v1) 3 pp. note, to be submitted (v2) Expanded version, now 7 pp. Extra material includes: more context/motivation: extra example for non-discrete case; new section on the weighted case. Some definitions also clarifie

The Social Objectives of Economic Development

NATURE ET ÉTENDUE DE L'ASSISTANCE TECHNIQUE DE L'O.I.T.L'Organisation internationale du travail a grandement changé dans les vingt dernières années. Alors qu'avant la guerre, elle recrutait ses effectifs parmi les pays fortement industrialisés d'Europe et d'Amérique, elle compte aujourd'hui plus de la moitié de ses membres parmi les pays en voie de développement. On comprend alors pourquoi son programme d'action a changé : en plus de vouloir protéger les travailleurs des inconvénients de l'industrialisation, l'O.I.T. se propose de favoriser le développement économique par la formation de travailleurs et de cadres compétents.Cette formation, l'O.I.T. cherche à la fournir non seulement au côté patronal, mais également au leadership syndical et aux fonctionnaires gouvernementaux des ministères du travail et des affaires sociales des pays en voie de développement.POURQUOI L'O.I.T. EST-ELLE IMPLIQUÉE DANS CE TRAVAIL ?« Le refus par un pays de favoriser l'établissement de conditions humaines de travail est un obstacle sérieux pour les autres pays qui cherchent à améliorer les conditions de travail à l'intérieur de leurs frontières ». Ceci est en fait le principe de base du Code international des normes de travail dont l'O.I.T. favorise l'adoption par les différentes autorités gouvernementales dans le but de promouvoir le changement social et économique. Comme corollaire de ce principe, l'O.I.T. croit fondamentalement qu'il ne peut y avoir de paix universelle sans qu'il y ait une justice sociale universelle et que la finalité du développement économique est l'épanouissement de l'homme tant sur le plan spirituel, culturel que matériel.UNE NOUVELLE ORIENTATIONLa stratégie de développement proposée pour les années 1970 reflète un certain changement d'attitudes et d'orientations. On abandonne l'hypothèse que le progrès économique est nécessairement suivi d'une amélioration des conditions de vie pour poursuivre des objectifs tels l'éducation, l'emploi et un niveau de mortalité plus bas.L'expérience a amené l'O.I.T. à croire qu'une approche internationale « concertée » est indispensable à la solution des problèmes de développement économique. C'est le défi que toutes les nations du monde auront à relever dans les dix prochaines années. Ce défi tient surtout au fait qu'il y a une tendance marquée depuis vingt ans à ce que le niveau d'emploi accuse un retard de plus en plus grand malgré la croissance économique. Ceci laisse donc entrevoir un effort encore plus grand de formation pour les années 70.La présente approche générale au problème du développement est, selon nous, trop traditionnelle. La stratégie employée ne vas pas assez loin. On ne peut rien faire en ce domaine sans la participation active des gens impliqués. C'est pourquoi on doit créer des motivations favorables au développement.L'APPROCHE TRADITIONNELLE EST-ELLE UTILE ?Historiquement, le développement des pays maintenant hautement industrialisés a pris la forme d'un glissement graduel des travailleurs du primaire vers le secondaire et le tertiaire, glissement accompagné d'une urbanisation toujours plus grande. On considérait alors le chômage comme la rançon des cycles de la croissance économique. Les modèles théoriques et les techniques de planification du développement reflètent encore aujourd'hui cette façon de penser.Les projections faites pour les dix prochaines années mettent sérieusement en doute l'utilité de cette approche surtout parce que l'emploi devient l'objectif de base sur lequel s'appuie le développement et qu'on ne peut pas s'attendre à des changements structurels majeurs à l'intérieur des pays en voie de développement.CONCLUSIONL'expérimentation de cette nouvelle approche du développement économiquedevra être appuyée, pour qu'elle réussisse, par les pays supporteurs d'une part etdevra être, d'autre part, l'occasion d'une participation active des pays récipiendaires.La coopération économique des pays en voie de développement offre un défi intéressant pour le Canada et les canadiens.The author summarizes the l.L.O.'s technical co-operation activities and their evolution from the early 1930's until the present time by highlighting some of the main topics and making a few supplementary observations

Structure of semisimple Hopf algebras of dimension p2q2p^2q^2

Let $p,q$ be prime numbers with $p^4<q$, and $k$ an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension $p^2q^2$ can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct R#kG, where $kG$ is the group algebra of group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$. As an application, the special case that the structure of semisimple Hopf algebras of dimension $4q^2$ is given.Comment: 11pages, to appear in Communications in Algebr

J. Sally's question and a conjecture of Y. Shimoda

In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most two. In this paper, having reduced the conjecture to the case of dimension three, if the ring is regular and local of dimension three, we explicitly describe a family of prime ideals of height two minimally generated by three elements. Weakening the hypothesis of regularity, we find that, to achieve the same end, we need to add extra hypotheses, such as completeness, infiniteness of the residue field and the multiplicity of the ring being at most three. In the second part of the paper we turn our attention to the category of standard graded algebras. A geometrical approach via a double use of a Bertini Theorem, together with a result of A. Simis, B. Ulrich and W.V. Vasconcelos, allows us to obtain a definitive answer in this setting. Finally, by adapting work of M. Miller on prime Bourbaki ideals in local rings, we detail some more technical results concerning the existence in standard graded algebras of homogeneous prime ideals with an "excessive" number of generators.Comment: 19 pages. Accepted in Nagoya Mathematical Journa

On the Content of Polynomials Over Semirings and Its Applications

In this paper, we prove that Dedekind-Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.Comment: Final version published at J. Algebra Appl., one reference added, three minor editorial change

Indecomposable modules and Gelfand rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings

Gauss composition over an arbitrary base

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the integers. However, such extensions have always included hypotheses on the rings, and the theorems involve only binary quadratic forms satisfying further hypotheses. We give a complete statement of the relationship between binary quadratic forms and modules for quadratic algebras over any base ring, or in fact base scheme. The result includes all binary quadratic forms, and commutes with base change. We give global geometric as well as local explicit descriptions of the relationship between forms and modules.Comment: submitte

Experiment K-6-04. Trace element balance in rats during spaceflight

Exposure to microgravity causes alterations in the skeletal and mineral homeostatic systems. Little is known about the effects of flight in an older skeleton; limited data suggest that bone resorption is increased after 5 days but no data are available about other metabolic effects. The response of a more slowly-growing skeleton to microgravity may be different than that of a younger animal, similar to the different responses seen in adolescents and adult humans to immobilization. This experiment was designed to investigate changes occurring in skeletal and mineral homeostatis in these older rats flown for two weeks in space. We may expect that the two portions of the rat vertebra, the vertebral body and the posterior elements, will show different responses to spaceflight. The results of the analyses from this study confirm major differences between portions of the vertebra. The posterior bone is more highly mineralized, evidenced by increased concentration (per unit weight of bone) of calcium (5 percent), phosphorus (6 percent) and osteocalcin (37 percent), similar to the differences seen between proximal and mid humerus in previous studies. The major increase in osteocalcin content indicates the presence of mature, low-turnover bone. The difference between flight and control animals were minimal in these older, slower-growing rats. Mass of whole vertebrae increased 6.2 percent in synchronous rats compared to less than 2 percent in flight rats over the 16 days when compared to basal controls, suggesting a decreased rate of bone growth in flight. Compared to young rats in which vertebral mass increased over 40 percent in 10 days in controls and 20 percent in flight rats, this may be a clear indication that even in the older skeleton bone growth will slow in microgravity

Adelic versions of the Weierstrass approximation theorem

Let $\underline{E}=\prod_{p\in\mathbb{P}}E_p$ be a compact subset of $\widehat{\mathbb{Z}}=\prod_{p\in\mathbb{P}}\mathbb{Z}_p$ and denote by $\mathcal C(\underline{E},\widehat{\mathbb{Z}})$ the ring of continuous functions from $\underline{E}$ into $\widehat{\mathbb{Z}}$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}}):=\{f(x)\in\mathbb{Q}[x]\mid \forall p\in\mathbb{P},\;\;f(E_p)\subseteq \mathbb{Z}_p\}$ is dense in the direct product $\prod_{p\in\mathbb{P}}\mathcal C(E_p,\mathbb{Z}_p)\,$ for the uniform convergence topology. Secondly, under the hypothesis that, for each $n\geq 0$, $\#(E_p\pmod{p})>n$ for all but finitely many $p$, we prove the existence of regular bases of the $\mathbb{Z}$-module ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}})$, and show that, for such a basis $\{f_n\}_{n\geq 0}$, every function $\underline{\varphi}$ in $\prod_{p\in\mathbb{P}}\mathcal{C}(E_p,\mathbb{Z}_p)$ may be uniquely written as a series $\sum_{n\geq 0}\underline{c}_n f_n$ where $\underline{c}_n\in\widehat{\mathbb{Z}}$ and $\lim_{n\to \infty}\underline{c}_n\to 0$.Comment: minor corrections the statement of Theorem 3.5, which covers the case of a general compact subset of the profinite completion of Z. to appear in Journal of Pure and Applied Algebra, comments are welcome