The Loeb Space of Denumerable Infinite Dimensional Probability Product Measure Spaces

Let {(Xi, Si, μi) : i ℃ N} be a sequence of probability measure spaces and (*Xi, L(*Si), L(*μi)) be the Loeb measure space with respect to (Xi, Si, μi) for i ℃ N. Let X =× Xi, S = ×Si,μ = ×μi. We prove that × L(*Si) CL(*S) and in embedding meaning.

Generalized Series of Bernoulli Type

The problem of evaluating an infinite series whose successive terms are reciprocal squares of the natural numbers was posed without a solution being offered in the middle of the seventeenth century. In the modern era, it is part of the theory of the Riemann zeta-function, specifically ζ (2). Jakob Bernoulli attempted to solve it by considering other more tractable series which were superficially similar and which he hoped could be algebraically manipulated to yield a solution to the difficult series. This approach was eventually unsuccessful, however, Bernoulli did produce an early monograph on summation of series. It remained for Bernoulli’s student and countryman Leonhard Euler to ultimately determine the sum to be . We characterize a class of series based on generalizing Bernoulli’s original work by adding two additional parameters to the summations. We also develop a recursion formula that allows summation of any member of the class.

The Number of Representations of an Integer as a Sum of Eight Triangular Numbers

An infinite product is expanded to Laurent series by residue theorem.Applying this expansion, the formula for the number of representations of an integer as a sum of eight triangular numbers is easily obtained.

The Number of Representations of an Integer as a Sum of Eight Squares

Two identities are obtained by Jacobi's triple product identity and some basic operators. By applying these identities, Jacobi's theorem for the number of representations of an integer as a sum of eight squares is easily proved.

The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable-With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence

The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes—that is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and so on. The main conclusion revolves entirely around two points. First, on the one hand, it is shown that the prime sequence exhibits an extremely high level of organization. But second, on the other hand, it is also shown that the clearly detectable organization of the primes is ultimately beyond human comprehension. This conclusion runs radically counter and opposite—in regard to both points—to what may well be the default view held widely, if not universally, in current theoretical mathematics about the prime sequence, namely the following. First, on the one hand, the prime sequence is deemed by all appearance to be entirely random, not organized at all. Second, on the other hand, all hope has not been abandoned that the sequence may perhaps at some point be grasped by human cognition, even if no progress at all has been made in this regard. Current mathematical research seems to be entirely predicated on keeping this hope alive. In the present paper, it is proposed that there is no reason to hope, as it were. According to this point of view, theoretical mathematics needs to take a drastic 180-degree turn. The manner of demonstration that will be used is direct and empirical. Two key observations are adduced showing, 1), how the prime sequence is highly organized and, 2), how this organization transcends human intelligence because it plays out in the dimension of infinity and in relation to π. The present paper is part of a larger project whose design it is to present a complete and final mathematical and physical theory of rational human intelligence. Nothing seems more self-evident than that rational human intelligence is subject to absolute limitations. The brain is a material and physically finite tool. Everyone will therefore readily agree that, as far as reasoning is concerned, there are things that the brain can do and things that it cannot do. The search is therefore for the line that separates the two, or the limits beyond which rational human intelligence cannot go. It is proposed that the structure of the prime sequence lies beyond those limits. The contemplation of the prime sequence teaches us something deeply fundamental about the human condition. It is part of the quest to Know Thyself.

On Direct Limits of Finite Products of Fields as Subrings of a Commutative Ring

Let R be a commutative ring. In this paper, we develop the existence of direct limits of finite products of fields as subrings of R.

Ramanujan's cubic transformation and generalized modular equation

We study the quotient of hypergeometric functions in the theory of Ramanujan＇s generalized modular equation for a ∈ （0, 1/2], and find an infinite product for- mula for μ1/3（r） by use of the properties of μ＊a（r） and Ramanujan＇s cubic transformation. Besides, a new cubic transformation formula of hypergeometric function is given, which complements the Ramanujan＇s cubic transformation.

Factorizations that Involve Ramanujan＇s Function k（q） = r（q）r2（q2）

In the ＂lost notebook＂, Ramanujan recorded infinite product expansions for where r -= r（q） is the Rogers-Ramanujan continued fraction. We shall give analogues of these results that involve Ramanujan＇s function k = k（q） = r（q）r2 （q2）.

Two-level uncapacitated lot-sizing problem considering the financing cost of working capital requirement

During financial crisis,companies constantly need free cash flows to efficiently react to any uncertainty,thus ensuring solvency.Working capital requirement(WCR)has been recognized as a key factor for releasing tied up cash in companies.However,in literatures related to lot-sizing problem,WCR has only been studied in the single-level supply chain context.In this paper,we initially adopt WCR model for a multi-level case.A two-level(supplier–customer)model is established on the basis of the classic multi-level lot-sizing model integrated with WCR financing cost.To tackle this problem,we propose sequential and centralized approaches to solve the two-level case with a serial chain structure.The ZIO(Zero Inventory Ordering)property is further confirmed valid in both cases.This property allows us to establish a dynamic programming-based algorithm,which solves the problem in O(T).Finally,numerical tests show differences in optimal plans obtained by both approaches and the influence of varying delays in payment on the WCR of both actors.