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Register a new userA new analytical formula for the inverse of a square matrix
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A concise analytical formula is developed for the inverse of an invertible 3 x 3 matrix using a telescoping method, and is generalized to larger square matrices. The formula is confirmed using randomly generated matrices in Matlab
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The inversion of nabla Laplace transform, corresponding to a causal sequence, is considered. Two classical methods, i.e., residual calculation method and partial fraction method are developed to perform the inverse nabla Laplace transform. For the first method, two alternative formulae are proposed when adopting the poles inside or outside of the contour, respectively. For the second method, a table on the transform pairs of those popular functions is carefully established. Besides illustrating the effectiveness of the developed methods with two illustrative examples, the applicability are further discussed in the fractional order case.
New analytical expressions are found for the Levi-Civita symbol using the Kronecker delta symbol. The expressions are derived up to 3 dimensions, extended to higher dimensions, and confirmed in Matlab for 5 dimensions. The expressions can be re-cast in terms of elementary and/or special functions, which lead to the conclusion that the Levi-Civita Symbol can be treated as a generalized, discrete function
We propose a new algorithm to obtain max flow for the multicommodity flow. This algorithm utilizes the max-flow min-cut theorem and the well known labeling algorithm due to Ford and Fulkerson [1]. We proceed as follows: We select one source/sink pair among the n distinguished source/sink pairs at a time and treat the given multicommodity network as a single commodity network for such chosen source/sink pair. Then applying standard labeling algorithm, separately for each sink/source pair, the feasible flow which is max flow and the corresponding minimum cut corresponding to each source/sink pair is obtained. A record is made of these cuts and the paths flowing through the edges of these cuts. This record is then utilized to develop our algorithm to obtain max flow for multicommodity flow. In this paper we have pinpointed the difficulty behind not getting a max flow min cut type theorem for multicommodity flow and found out a remedy.
We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{mathrm{min}}(n)$ of integers $ngeq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $mathbb{Q}$. Then we prove that $$-lim_{Xrightarrowinfty}sum_{substack{2leq nleq X[1pt]left[frac{K/mathbb{Q}}{p_{mathrm{min}}(n)}right]=C}}frac{mu(n)}{n}=frac{#C}{#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{mathrm{max}}(n)$ are equidistributed in arithmetic progressions modulo an integer $k$, which occurs when $K$ is a cyclotomic field $mathbb{Q}(zeta_k)$.
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Xiao-Jun Yang (School of Mathematics
, China University of Mining andn Technology
, Xuzhou
2021
In this article an alternative infinite product for a special class of the entire functions are studied by using some results of the Laguerre-P{o}lya entire functions. The zeros for a class of the special even entire functions are discussed in detail. It is proved that the infinite product and series representations for the hyperbolic and trigonometric cosine functions, which are coming from Euler, are our special cases.
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