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Register a new userThe General Form Of Cyclic Orthonormal Generators In R^N
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Kerry M. Soileau
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In this paper we give a definition of cyclic orthonormal generators (cogs) in R^N. We give a general canonical form for their expression. Further, we give an explicit formula for computing the canonical form of any given cog.
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In this paper we tried a different approach to work out the integrals of e^(x^n) and e^(-x^n). Integration by parts shows a nice pattern which can be reduced to a form of series. We have shown both the indefinite and definite integrals of the functions mentioned along with some essential properties e.g. conditions of convergence of the series. Further more, we used the integrals in form of series to find out series solution of differential equations of the form x[(d^2 y)/(dx^2)]-(n-1)(dy/dx)-n^2 x^(2n-1)y-nx^n=0 and x[(d^2 y)/(dx^2)] -(n-1)(dy/dx)-n^2x^(2n-1)y+(n-1)=0, using some non standard method. We introduced modified Normal distribution incorporating some properties derived from the above integrals and defined a generalized version of Skewness and Kurtosis. Finally we extended Starlings approximation to limit [n to infinity ] (2n)! ~ 2n * sqrt{(2pi)} [(2n/e)]^(2n).
Let $sum_{d|n}$ denote sum over divisors of a positive integer $n$, and $t_{r}(n)$ denote the number of representations of $n$ as a sum of $r$ triangular numbers. Then we prove that $$ sum_{d|n}frac{1+2,(-1)^{d}}{d}=sum_{r=1}^{n}frac{(-1)^{r}}{r}, binom{n}{r}, t_{r}(n) $$ using a result of Ono, Robbins and Wahl.
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In this paper, some sufficient conditions for the differentiability of the $n$-variable real-valued function are obtained, which are given based on the differentiability of the $n-1$-variable real-valued function and are weaker than classical conditions.
Continuous formal deformations of the Poisson superbracket defined on compactly supported smooth functions on R^2 taking values in a Grassmann algebra with N generating elements are described up to an equivalence transformation for N e 2.
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