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

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

A one-sided Prime Ideal Principle for noncommutative rings

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how completely prime right ideals control the one-sided structure of a ring, and they recover earlier theorems stating that certain noncommutative rings are domains (namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian). In order to provide a deeper understanding of the set of completely prime right ideals in a general ring, we study the special subset of comonoform right ideals.Comment: 38 page

Experiment K-6-03. Gravity and skeletal growth, part 1. Part 2: Morphology and histochemistry of bone cells and vasculature of the tibia; Part 3: Nuclear volume analysis of osteoblast histogenesis in periodontal ligament cells; Part 4: Intervertebral disc swelling pressure associated with microgravity

Bone area, bone electrophysiology, bone vascularity, osteoblast morphology, and osteoblast histogenesis were studied in rats associated with Cosmos 1887. The results suggest that the synchronous animals were the only group with a significantly larger bone area than the basal group, that the bone electrical potential was more negative in flight than in the synchronous rats, that the endosteal osteoblasts from flight rats had greater numbers of transitional Golgi vesicles but no difference in the large Golgi saccules or the alkaline phosphatase activity, that the perioteal vasculature in the shaft of flight rats often showed very dense intraluminal deposits with adjacent degenerating osteocytes as well as lipid accumulations within the lumen of the vessels and sometimes degeneration of the vascular wall (this change was not present in the metaphyseal region of flight animals), and that the progenitor cells decreased in flight rats while the preosteoblasts increased compared to controls. Many of the results suggest that the animals were beginning to recover from the effects of spaceflight during the two day interval between landing and euthanasia; flight effects, such as the vascular changes, did not appear to recover

Type-Decomposition of a Pseudo-Effect Algebra

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.Comment: 18 page

Quasi-exactly solvable quartic: elementary integrals and asymptotics

We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we found an interesting identity which is used to describe the spectral locus. We also establish some asymptotic properties of the QES spectral locus.Comment: 20 pages, 1 figure. Added Introduction and several references, corrected misprint

The Hopf modules category and the Hopf equation

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules category. As an application, a five dimensional noncommutative noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres

Twisted K-Theory of Lie Groups

I determine the twisted K-theory of all compact simply connected simple Lie groups. The computation reduces via the Freed-Hopkins-Teleman theorem to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al.Comment: 16 page

Nonintegrability of the two-body problem in constant curvature spaces

We consider the reduced two-body problem with the Newton and the oscillator potentials on the sphere ${\bf S}^{2}$ and the hyperbolic plane ${\bf H}^{2}$. For both types of interaction we prove the nonexistence of an additional meromorphic integral for the complexified dynamic systems.Comment: 20 pages, typos correcte

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page