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[3] viXra:2107.0091 [pdf] submitted on 2021-07-15 20:47:02
Authors: F. Maciala, A. Puindi
Comments: 15 Pages.
Mathematics is an area of knowledge whose learning is done in a phased manner. Each of these phases is filled with learning from mathematical entities or entities that in turn serve as a support for learning other new concepts, considered more complex in relation to those already seen. However, for the learning of the new concepts to occur without difficulties, it is necessary that the basic concepts are very well retained. And for the retention of these new concepts, this work presents a proposal in which one can work with modular equations, implementing heuristic instruction, supporting the literary G. Polya. The proposed approach is applied to the treatment of modular equations in the Second Cycle of Secondary Education.
Category: Algebra
[2] viXra:2107.0068 [pdf] submitted on 2021-07-11 07:27:36
Authors: Alex Kritov
Comments: 4 Pages. Text is in Russian
The article shows the correspondence of the seven principles of the esoteric doctrines, the Chinese trigrams Gua Fu Xi with Pauli matrices, which are the basis of the corresponding Lie algebras and groups, and widely used in modern fundamental physics. This article is intended for a very narrow category of readers, namely, for those who are familiar with the mathematical apparatus of the theory of groups and algebras
Clifford in their application to the physical world, with binary (bit) operations and, at the same time, with the esoteric teachings of Theosophy as presented by H.P. Blavatsky, Subba Row. For mathematicians, this work
may be interesting in connection with the proposed method for enumerating Pauli matrices using bitwise operations. [in Russian]
Category: Algebra
[1] viXra:2107.0049 [pdf] replaced on 2021-09-26 14:56:24
Authors: Yaroslav Shitov
Comments: 26 Pages. added results on matrix rigidity
We build a combinatorial technique to solve several long standing problems in linear algebra with a particular focus on algorithmic complexity of matrix completion and tensor decomposition problems. For all appropriate integral domains R, we show the polynomial time equivalence of the problem of the solvability of a system of polynomial equations over R to
• the minimum rank matrix completion problem (in particular, we answer a question asked by Buss, Frandsen, Shallit in 1999),
• the determination of matrix rigidity (we answer a question posed by Mahajan, Sarma in 2010 by showing the undecidability over Z, and we solve recent problems of Ramya corresponding to Q and R),
• the computation of tensor rank (we answer a question asked by Gonzalez, Ja'Ja' in 1980 on the undecidability over Z, and, additionally, the special case with R = Q solves a problem posed by Blaser in 2014),
• the computation of the symmetric rank of a symmetric tensor, whose algorithimic complexity remained open despite an extensive discussion in several foundational papers. In particular, we prove the NP-hardness conjecture proposed by Hillar, Lim in 2013.
In addition, we solve two problems on fractional minimal ranks of incomplete matrices recently raised by Grossmann, Woerdeman, and we answer, in a strong form, a recent question of Babai, Kivva on the dependence of the solution to the matrix rigidity problem on the choice of the target field.
Category: Algebra
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