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On the determinant of a sparse 0-1 matrix
Linear Algebra and its Applications. 2018. Vol. 554. P. 49-50.
Shitov Y.
We prove that the determinant of an n x n 01-matrix with at most n+k non-zero entries does not exceed α^k with α=4^(1/3)≈1.316074.
Shitov Y., Linear Algebra and its Applications 2018 Vol. 554 P. 49-50
We prove that the determinant of an n-by-n 0-1 matrix with at most n + k non-zero entries does not exceed α^k with α ≈ 1.316074. ...
Added: September 26, 2018
Burman Y. M., Journal of Algebraic Combinatorics 2019 Vol. 50 No. 4 P. 427-446
The classical matrix-tree theorem discovered by G.Kirchhoff in 1847
expresses the principal minor of the (n x n) Laplace matrix as a
sum of monomials of matrix elements indexed by directed trees with n
vertices. We prove, for any k >= n, a three-parameter family of
identities between degree k polynomials of matrix elements of the
Laplace matrix. For k=n and special values of ...
Added: October 18, 2018
Баскаков А. Е., Новые информационные технологии в автоматизированных системах 2009 № 12 С. 279-280
Added: February 21, 2013
Shirokov D., Computational and Applied Mathematics 2021 Vol. 40 P. 1-29
In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only ...
Added: July 15, 2021
Ayzenberg A., Бухштабер В. М., Математический сборник 2021
An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space
$M_{\St_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove this space to be a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe ...
Added: November 6, 2020
Shitov Y., Communications in Algebra 2015 Vol. 43 No. 10 P. 4359-4366
We develop the technique useful for studying the problem of factoring nonnegative matrices. We illustrate our method, based on the tools from linear algebra over a semiring, by applying it to studying the problem of existence of a rank-three matrix with full nonnegative rank equal to n. ...
Added: July 7, 2015
Айзенберг А.А., Бухштабер В.М., Математический сборник 2021 Т. 212 № 5 С. 3-36
Матрицей-стрелкой называется матрица с нулями вне главной диагонали, первой строки и первого столбца. В работе исследуется пространство MStn,λ всех эрмитовых матриц-стрелок размера (n+1)×(n+1), имеющих заданный простой спектр λ. Доказано, что это пространство – гладкое 2n-мерное многообразие с локально стандартным действием тора, описана топология и комбинаторика его пространства орбит. При n⩾3 пространство орбит MStn,λ/Tn не является многогранником, а значит, MStn,λ не является квазиторическим многообразием. Тем не менее на MStn,λ имеется действие полупрямого ...
Added: June 18, 2021
Guterman A. E., Spiridonov I.A., Linear Algebra and its Applications 2020 Vol. 599 P. 140-155
Let $M_{n}(\mathbb{F})$ denote the set of square matrices of size $n$ over a field $\mathbb{F}$ with characteristics different from two. We say that the map $f: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ is additive if $f(A+B) = f(A) + f(B)$ for all $A, B \in M_{n}(\mathbb{F})$. The main goal of this paper is to prove that for $n>2$ ...
Added: November 9, 2020
Burman Y. M., Ploskonosov A., Trofimova A., / Cornell University. Series math "arxiv.org". 2011. No. 1109.6625.
We calculate determinants of weighted sums of reflections and of (nested) commutators of reflections. The results obtained generalize the matrix-tree theorem by Kirchhoff and the Pfaffian-hypertree theorem by Massbaum and Vaintrob. ...
Added: November 7, 2012
Beasley L., Guterman A., Shitov Y., Journal of Algebra 2015 Vol. 433 P. 168-182
Among different rank functions on tropical matrices, there is one known as tropical rank which is a lower bound for any other. Here we introduce a new concept (for being opposed to tropical rank, it is called arctic) which gives an upper bound for other ranks. Our definition is based on the perimeter notion previously ...
Added: April 14, 2015
Shitov Y., Sbornik Mathematics 2013 Vol. 204 No. 11 P. 1691-1699
It is demonstrated that every (0, 1)-matrix of size n×m having Boolean rank n contains a column with at least √n/2 − 1 zero entries. This bound is shown to be asymptotically optimal. As a corollary, it is established that the size of a full-rank Boolean matrix is bounded from above by a function of ...
Added: January 20, 2014
Rotmistrov A., Popova P., Социология: методология, методы, математическое моделирование 2016 № 43 С. 63-99
The focus of this article is the methodological aspect of political activism determinants identifying; specifically variants of handling with categorical predictors which hypothetically explain the level of activism. When using regression for explaining the issue, one may transform such predictors into dummy variables. Such a popular solution makes the model bulky and causes troubles with ...
Added: March 15, 2017
Logvenkov S. A., Samovol V. S., М. : МЦНМО, 2017
Наряду с разделами линейной алгебры, традиционно включаемыми в подобные учебные издания, в данном учебнике представлены некоторые специальные темы, связанные прежде всего с теорией матриц, в том числе, многочленные матрицы и их преобразования, жордановы матрицы, методы приведения матриц к нормальной жордановой форме и др. В книге также нашли отражение такие вопросы, имеющие важное прикладное значение, как ...
Added: December 31, 2017
Ayzenberg A., Бухштабер В. М., / Arxiv Cornell University Library. Series 1803.10449 "1803.10449". 2018. No. 10449.
An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space MStn,λ of Hermitian arrow (n+1)×(n+1)-matrices with fixed simple spectrum λ. We prove that this space is a smooth 2n-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe ...
Added: October 15, 2018
Costara C., A. E. Guterman, A. M. Maksaev et al., Linear Algebra and its Applications 2023 Vol. 666 P. 129-143
In this paper, we characterize the automorphisms of the total graph for the ring M_n of matrices of order n≥2 over any field with at least 3 elements. To do this, we apply the technique of maps preserving matrix invariants; in particular, as an intermediate step, we characterize pairs of surjective maps φ_1,φ_2:M_n →M_n such that A+B is singular if and only if φ_1(A)+φ_2(B) is ...
Added: April 11, 2023
D. A. Spirin, E. N. Prokofeva, A. V. Vostrikov,, , in : Proceedings of 2022 IEEE Moscow Workshop on Electronic and Networking Technologies (MWENT). : M. : IEEE, 2022. P. 1-4.
Added: July 4, 2022
191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90
It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...
Added: September 23, 2016
Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620-624
Added: February 27, 2013
Ilyashenko Y., Яковенко С. Ю., М. : МЦНМО, 2013
Предлагаемая книга—первый том двухтомной монографии, посвящённой аналитической теории дифференциальных уравнений.
В первой части этого тома излагается формальная и аналитическая теория нормальных форм и теорема о разрешении особенностей для векторных полей на плоскости.
Вторая часть посвящена алгебраически разрешимым локальным задачам теории аналитических дифференциальных уравнений , квадратичным векторным полям и проблеме локальной классификации ростков векторных полей в комплексной области ...
Added: February 5, 2014
Kalyagin V.A., Koldanov A.P., Koldanov P.A. et al., Physica A: Statistical Mechanics and its Applications 2014 Vol. 413 No. 1 P. 59-70
A general approach to measure statistical uncertainty of different filtration techniques for market network analysis is proposed. Two measures of statistical uncertainty are introduced and discussed. One is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors. It is shown that for some important cases ...
Added: July 19, 2014
Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18
Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...
Added: November 16, 2020
Maslov V., Теоретическая и математическая физика 2019 Т. 201 № 1 С. 65-83
We study the process of a nucleon separating from an atomic nucleus from the mathematical standpoint
using experimental values of the binding energy for the nucleus of the given substance. A nucleon becomes
a boson at the instant of separating from a fermionic nucleus. We study the further transformations of
boson and fermion states of separation in a ...
Added: November 1, 2019
Pahomov F., Известия РАН. Серия математическая 2016 Т. 80 № 6 С. 173-216
Полимодальная логика доказуемости
GLP была введена Г. К. Джапаридзе в 1986 г. Она является логикой доказуемости для ряда цепочек предикатов доказуемости возрастающей силы. Всякой полимодальной логике соответствует многообразие полимодальных алгебр. Л. Д. Беклемишевым и А. Виссером был поставлен вопрос о разрешимости элементарной теории свободной GLP-алгебры, порожденной константами 0, 1 [1]. В этой статье для любого натурального n решается аналогичный вопрос для логик GLPn, являющихся ...
Added: December 4, 2017
Sinelshchikov D., Кудряшов Н. А., Theoretical and Mathematical Physics 2018 Vol. 196 No. 2 P. 1230-1240
We study a family of nonautonomous generalized Liénard-type equations. We consider the equivalence problem via the generalized Sundman transformations between this family of equations and type-I Painlevé–Gambier equations. As a result, we find four criteria of equivalence, which give four integrable families of Liénard-type equations. We demonstrate that these criteria can be used to construct ...
Added: February 9, 2019