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Register a new userLinear Independence Of Some Irrational Numbers
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Authors
N. A. Carella
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This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples ${1, e, pi}$, ${1, e, pi^{-1}}$, and ${1, pi^r, pi^s}$, where $1leq r<s $ are fixed integers.
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The proofs that the real numbers are denumerable will be shown, i.e., that there exists one-to-one correspondence between the natural numbers $N$ and the real numbers $Re$. The general element of the sequence that contains all real numbers will be explicitly specified, and the first few elements of the sequence will be written. Remarks on the Cantors nondenumerability proofs of 1873 and 1891 that the real numbers are noncountable will be given.
From the more than two hundred partial orders for fuzzy numbers proposed in the literature, only a few are total. In this paper, we introduce the notion of admissible order for fuzzy numbers equipped with a partial order, i.e. a total order which refines the partial order. In particular, it is given special attention to the partial order proposed by Klir and Yuan in 1995. Moreover, we propose a method to construct admissible orders on fuzzy numbers in terms of linear orders defined for intervals considering a strictly increasing upper dense sequence, proving that this order is admissible for a given partial order. Finally, we use admissible orders to ranking the path costs in fuzzy weighted graphs.
Let $alpha=0.a_1a_2a_3ldots$ be an irrational number in base $b>1$, where $0leq a_i<b$. The number $alpha in (0,1)$ is a $textit{normal number}$ if every block $(a_{n+1}a_{n+2}ldots a_{n+k})$ of $k$ digits occurs with probability $1/b^k$. A conditional proof of the normality of the real number $pi$ in base $10$ is presented in this note.
We present in this work a heuristic expression for the density of prime numbers. Our expression leads to results which possesses approximately the same precision of the Riemanns function in the domain that goes from 2 to 1010 at least. Instead of using a constant as was done by Legendre and others in the formula of Gauss, we try to adjust the data through a function. This function has the remarkable property: its points of discontinuity are the prime numbers.
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Xiao-Jun Yang (School of Mathematics
, China University of Mining andn Technology
, Xuzhou
2021
This article proves the products, behaviors and simple zeros for the classes of the entire functions associated with the Weierstrass-Hadamard product and the Taylor series.
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