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[5] viXra:2308.0129 [pdf] submitted on 2023-08-19 06:34:27
Authors: Eckhard Hitzer
Comments: 19 Pages. accepted for D. DaSilva, D. Hildenbrand, E. Hitzer (eds.), Proceedings of ICACGA 2022, Springer Proceedings in Mathematics & Statistics, Springer, Heidelberg, 2023.
We study in full detail the inner product of oriented points in conformal geometric algebra and its geometric meaning. The notion of oriented point is introduced and the inner product of two general oriented points is computed, analyzed (including symmetry) and graphed in terms of point to point distance, and angles between the distance vector and the local orientation planes of the two points. Seven examples illustrate the results obtained. Finally, the results are extended from dimension three to arbitrary dimensions n.
Category: Algebra
[4] viXra:2308.0125 [pdf] submitted on 2023-08-19 23:25:38
Authors: Tai-Choon Yoon
Comments: 6 Pages.
This paper presents a solvable sextic equation under the condition that several coefficients of such polynomials are restricted to become dependent on the preceding or following coefficients. We can solve a sextic equation by restricting one or two in total seven coefficients available, and by solving a bisextic equation and a quintic equation. And we can also find the arbitrary coupling coefficients that generate a new solvable sextic equation as well.
Category: Algebra
[3] viXra:2308.0102 [pdf] submitted on 2023-08-14 23:48:11
Authors: Shao-Dan Lee
Comments: 8 Pages.
We may define a binary relation. Then a nonempty finite set equipped with the binary relation is called a circle set. And we define a bijective mapping of the circle set, and the mapping is called a shift. We may construct a pitch structure over a circle set. And we may define a tonic and step of a pitch structure. Then the ordered pair of the tonic and step is called the key of the pitch structure. Then we define a key transpose along a shift. And a key transpose is said to be regular if it consists of stretches, shrinks and a shift. A key transpose is regular if and only if it satisfies some hypotheses.
Category: Algebra
[2] viXra:2308.0028 [pdf] submitted on 2023-08-05 02:14:22
Authors: Yaroslav Shitov
Comments: 12 Pages.
Let S be a family of n x n matrices over a field such that, for some integer l, the products of the length l of the matrices in S span the full n x n matrix algebra. We show this for any positive integer l > n^2 + 2n − 5.
Category: Algebra
[1] viXra:2308.0001 [pdf] replaced on 2023-08-11 23:04:36
Authors: Deepak Ponvel Chermakani
Comments: 9 Pages. Explained more on the proofs
For a homogeneous linear-recurrence f(n) with integer coefficients and integer starting points, we derive a deterministic algorithm that finds the upper bound of the last non-periodic position n where f(n)=0, for a large family of special cases. First, when theta is a given irrational constant, then we show that, an eventual lower bound of minimum(absolute(cos(m PI theta)), over positive integers m less than n), for large positive integers n, is (2 theta / (sqrt(5) n)). Our deterministic algorithm is based on the key concept that this lower bound decreases at a lower rate than the nth power of the ratio of root-moduli since the ratio is lesser than 1. Our deterministic algorithm is developed for the special cases where G(x), the characteristic polynomial of f(n), has either equal absolute values of arguments or commensurable arguments of those complex roots whose moduli are equal. In an attempt to extend this algorithm as a general solution to Skolems problem, we obtain the lower bound of the distance between a zero and the next (2^(m+1))th zero, in the weighted sum of m continuous cosine functions, where the weights are given real-algebraic constants.
Category: Algebra
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