In the following work we present a proof for the strong law of large numbers for pairwise negatively dependent random variables which relaxes the usual assumption of pairwise independence. Let be a double sequence of pairwise negatively dependent random variables. If for all non-negative real numbers t and , for 1 < p < 2, then we prove that (1). In addition, it also converges to 0 in ....

For simplicity, we adopt the following rules: n denotes an element of N, X, X1 denote sets, r, p denote real numbers, s, x0, x1, x2 denote real numbers, S, T denote real normed spaces, f , f1, f2 denote partial functions from R to the carrier of S, s1 denotes a sequence of real numbers, and Y denotes a subset of R. The following propositions are true: (1) Let s2 be a sequence of real numbers an...

In this work we prove that, given a simplicial graph Γ $\Gamma$ and family G $\mathcal {G}$ of linear groups over domain R $R$ , the product $\Gamma \mathcal is [ t ̲ ] $R[\underline{t}]$ where $\underline{t}$ tuple finitely many linearly independent variables. As consequence, obtain that any complex numbers again group numbers. This solves an open problem Hsu Wise (Michigan Math. J. 46 (1999), ...

We denote the real numbers by R and the positive real numbers by R+. Similarly we denote the rational numbers by Q and the positive rational numbers by Q+. We denote the natural numbers (including 0) by N. For all a and b in R with a < b, we denote the open interval between a and b by (a, b) and the closed interval by [a, b]. If S is a set, S∗ denotes the set of all finite sequences consisting ...

We give explicit evaluations of the linear and non-linear Euler sums hyperharmonic numbers $$h_{n}^{\left( r\right) }$$ with reciprocal binomial coefficients. These enable us to extend closed form formula an arbitrary integer r. Moreover, we reach at formulas for shifted Euler-type harmonic numbers. All are provided in terms Riemann zeta values, sums.

Text Classification is an important research field in information retrieval and text mining. The main task in text classification is to assign text documents in predefined categories based on documents’ contents and labeled-training samples. Since word detection is a difficult and time consuming task in Persian language, Bayesian text classifier is an appropriate approach to deal with different...