Generalized Burnside-Grothendieck ring functor and aperiodic ring functor associated with profinite groups

For every profinite group $G$, we construct two covariant functors $\Delta_G$ and ${\bf {\mathcal {AP}}}_G$ from the category of commutative rings with identity to itself, and show that indeed they are equivalent to the functor $W_G$ introduced in [A. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, {\it Adv. in Math.} {\bf{70}} (1988), 87-132]. We call $\Delta_G$ the generalized Burnside-Grothendieck ring functor and ${\bf {\mathcal {AP}}}_G$ the aperiodic ring functor (associated with $G$). In case $G$ is abelian, we also construct another functor ${\bf Ap}_G$ from the category of commutative rings with identity to itself as a generalization of the functor ${\bf Ap}$ introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, {\it Adv. in Math.} {\bf 81} (1990), 1-29]. Finally it is shown that there exist $q$-analogues of these functors (i.e, $W_G, \Delta_G, {\bf {\mathcal {AP}}}_G$, and ${\bf Ap}_G$) in case $G=\hat C$ the profinite completion of the multiplicative infinite cyclic group.Comment: minor corrections, 35 page

Necklace rings and logarithmic functions

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor $Nr$ from the category of special \ld-rings into the category of special \ld-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper (J. Algebra. 253 (2002); no.1, 167-188) to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincar\'e principle

Blowup of Solutions of the Hydrostatic Euler Equations

In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.Comment: 7 pages; added 1 reference in section 1, paraphrased lemma 2.2, but all mathematical details remain unchange

Impacts and Losses Caused By the Fraudulent and Manipulated Financial Information on Economic Decisions

Nowadays the effects of the fraudulent and manipulated financial information have been more controversial. We should take into consideration that the financial losses caused by fraudulent or manipulated financial information are remarkable. Preventing the fraud in the financial information has been an important issue by auditors all over the world. As the American economy is the dominant economy may cause and affect the capital market mostly all over the world. In the last decade we can see the financial losses caused by the fraudulent and manipulated financial information rather big. Today’s world has been affected by frauds and manipulation of the financial information. An investment decision based on false financial information causes the investors to suffer losses as was experienced in Enron and WorldCom cases. Financial information has, certainly, an important positive or negative effect in economic decisions. Positive or negative effects of financial information on economic decisions depend on reliability of the financial information. This paper aims to show the impacts of fraudulent on the financial information, effects on economic decision and what we should do for preventing the fraudulent or manipulation on the financial information.Financial information, fraudulent, manipulation, decision.