Abstract:
We give an extension result of Watanabe’s
characterization for 2-dimensional Poisson processes. By using this result, the
equivalence of uniqueness in law and joint uniqueness in law is proved for
one-dimensional stochastic differential equations driven by Poisson processes.
After that, we give a simplified Engelbert theorem for the stochastic
differential equations of this type.

Abstract:
The traveller Engelbert Kaempfer, a Christian German in Swedish diplomatic service, here in Schamachi, comes into real contact with a Muslim culture for the first time. As an academically educated physician he makes profitable use of his profession, as with Nawi Jusbascy, probably a born Azerbaijani, Mohammed Hossein and probably Martiros, perhaps Hafi Saburjan, too. As a Christian living in Schamachi within the Armenian quarter he meets other Christians, the Priest Arakhel, the already mentioned Martiros and maybe Hafi Saburjan. Besides the Armenian priest Arakhel he meets a cleric of another religion, Maheb Aali, the Muslim Molla, probably a born Azerbaijani and having lived there in Schamachi. As a diplomat he makes good contacts with diplomats of other nations, like Mohammed Hossein, a Persian and envoy to Poland, and esspecially with his friend Christophorov, the Greek in Russian service. And there are contacts with a military background, too, with the already-mentioned captain Nawi, and with . if correctly identified . the ghulam Ali Kuli Chan, possibly a born Georgian and all his life in Persian service, as commander-in-chief under two Shahs, one of the most powerful men in Persia and now, 1684, governor of the province of Shirvan.On the one hand all these Schamachi contacts were surely a good start for Kaempfer.s later cultural experiences and his attempts to understand foreign cultures, such as India, Siam and Japan. On the other hand this shows above else that, during those days, Schamachi was a meeting point between great empires and an important place of intercultural contacts at the European-Asian border.

Abstract:
English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat’s Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time—such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles’s proof employed very advanced mathematical tools and methods that were not at all available in the known World during Fermat’s days. Given that Fermat claimed to have had the “truly marvellous” proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat’s time, has led many to doubt that Fermat actually did possess the “truly marvellous” proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat’s Last Theorem actually yields to our efforts to prove it.

Abstract:
In this work, we study the following problem. , where ？is the
fractional Laplacian and Ω？is a bounded
domain in R^{N}？with Lipschitz
boundary. g: R→R？is an
increasing locally Lipschitz continuous function. and f∈L^{m}(Ω), . We use Stampacchia’s theorem to study existence of
the solution u

Abstract:
First, the numerous claims that the theory of natural selection would be a tautology, just empty circular reasoning, are shown to be erroneous, and that they follow from an essentialistic and deterministic way of thinking, which is not consistent with the dynamic theory of evolution. Secondly, it is proposed that a careful analysis applying Fisher’s Fundamental Theorem of Natural Selection of the seemingly tautologous sentence in question: “those who reproduce most, reproduce most” shows that in actual fact it is a predictive statement. Consequently, the analysis presented reduces the essence of the theory of natural selection to that one single statement.

The virial theorem is written by using the canonical equations of motion in classical mechanics. A moving particle with an initial speed in an n-particle system is considered. The distance of the moving particle from the origin of the system to the final position is derived as a function of the kinetic energy of the particle. It is thought that the considered particle would not collide with other particles in the system. The relation between the final and initial distance of the particle from the origin of the system is given by a single equation.

In this paper, an approach to
Pythagoras’ Theorem is presented within the historical context in which it was
developed and from the underlying intellectual outline of the Pythagorean
School. This was analyzed from a rationalism standpoint. An experiment is
presented to the reader so that they, through direct observation, can analyze
Pythagoras’ Theorem and its relation to the creation of knowledge. The theory
of knowledge conceptualization is used.

Abstract:
In this paper we show that, under some conditions, if M is a manifold with Bakry-émery Ricci curvature bounded below and with bounded potential function then M is compact. We also establish a volume comparison theorem for manifolds with nonnegative Bakry-émery Ricci curvature which allows us to prove a topolological rigidity theorem for such manifolds.

Abstract:
This paper shows that, if firms borrow at an interest rate that is greater than the rate at which they can lend, the value of a firm declines with the amount borrowed. The model assumes the possibility that a firm may go bankrupt, which introduces the need for financial intermediation. A modified version of the homemade lev-erage examples introduced by Modigliani and Miller [2] is used to introduce the concept. A state-preference model is used for a more formal proof.

Abstract:
In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality instead of Maximum Principle.