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In this article, an attempt has been made to settle the question of existence of unbiased estimator of the key parameter p of the quasi-binomial distributions of Type I (QBD I) and of Type II (QBD II), with/without any knowledge of the other parameter φ appearing in the expressions for probability functions of the QBD's. This is studied with reference to a single observation, a random sample of finite size m as also with samples drawn by suitably defined sequential sampling rules. [ABSTRACT FROM AUTHOR]

We describe exercises in which students use PascGaloisJE to formulate conjectures about certain binomial identities which hold when the binomial coefficients are interpreted as elements in the cyclic group [image omitted] of integers modulo a prime integer p. In addition to having an appealing visual component, these exercises are open-ended and expose students to the investigative nature of mathematical discovery. [ABSTRACT FROM AUTHOR]

The two main classes of finite dimensional composition algebras, Hurwitz and symmetric, are characterized in terms of the flexible identity, and even of the third power associative identity, over fields of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

Let K be a field. A split basic finite-dimensional K-algebra with left quiver F is monomial if it can be represented as the path algebra KΓ modulo an ideal generated by paths in Γ, and binomial if it can be represented as KΓ modulo relations of the form p = λq, where λ ∈ c K and p and q are paths in Γ. Burgess et al. [Burgess, W. D., Fuller, K. R., Green, E. L., Zacharia, D. (1993). Left monomial rings-A generalization of monomial algebras. Osaka J. Math. 30(3):543-558] define monomial rings, generalizing monomial algebras; we here take this generalization one step further, introducing and studying the class of binomial rings, which includes the classes of binomial algebras, monomial rings, and square-free rings. We further show that every binomial ring has canonically associated with it a binomial algebra, with which the ring shares many properties. [ABSTRACT FROM AUTHOR]

A computatiorial algorithm is developed for generating the cumulative distribution of successes in a sequence of dependent trials, characterized by a Markov model, for which tables of the distribution may be compiled. Such tables may be used to assess the impact of such dependence on the calculation of confidence intervals and the testing of hypotheses. Examples are presented to show how such confidence intervals and hypothesis tests deteriorate with increasing dependence. [ABSTRACT FROM AUTHOR]

Shows that Green's theorem is sufficient in proving the fundamental theorem of algebra. Linearity of derivatives; Binomial theorem.

Provides a simple proof of Descartes' rule of signs in high school algebra. Proof of theorems; Variations in the signs of the sequence of coefficients; Number of positive zeros of p counting multiplicities by z; Nonnegative integers; Induction hypothesis; Arbitrary real numbers.