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1
Content available remote

A method of approximate factorization of positive definite matrix functions

100%
Janashia G., Lagvilava E.
Studia Mathematica
|
1999
|
tom 137
|
nr 1
93-100
EN
An algorithm of factorization of positive definite matrix functions of second order is proposed.
2
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Théorèmes de factorisation dans les algèbres normées complètes non associatives

100%
Akkar M., Laayouni M.
Colloquium Mathematicum
|
1996
|
tom 70
|
nr 2
253-264
3
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Note on cyclic decompositions of complete bipartite graphs into cubes

80%
Fronček D.
Discussiones Mathematicae Graph Theory
|
1999
|
tom 19
|
nr 2
219-227
EN
So far, the smallest complete bipartite graph which was known to have a cyclic decomposition into cubes $Q_d$ of a given dimension d was $K_{d2^{d-1}, d2^{d-2}}$. We improve this result and show that also $K_{d2^{d-2}, d2^{d-2}}$ allows a cyclic decomposition into $Q_d$. We also present a cyclic factorization of $K_{8,8}$ into Q₄.
4
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On Twin Edge Colorings of Graphs

80%
Andrews E., Helenius L., Johnston D., VerWys J., Zhang P.
Discussiones Mathematicae Graph Theory
|
2014
|
tom 34
|
nr 3
613-627
EN
A twin edge k-coloring of a graph G is a proper edge coloring of G with the elements of Zk so that the induced vertex coloring in which the color of a vertex v in G is the sum (in Zk) of the colors of the edges incident with v is a proper vertex coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Among the results presented are formulas for the twin chromatic index of each complete graph and each complete bipartite graph
5
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Prime Factorization of Sums and Differences of Two Like Powers

80%
Ziobro R.
Formalized Mathematics
|
2016
|
tom 24
|
nr 3
187-198
EN
Representation of a non zero integer as a signed product of primes is unique similarly to its representations in various types of positional notations [4], [3]. The study focuses on counting the prime factors of integers in the form of sums or differences of two equal powers (thus being represented by 1 and a series of zeroes in respective digital bases). Although the introduced theorems are not particularly important, they provide a couple of shortcuts useful for integer factorization, which could serve in further development of Mizar projects [2]. This could be regarded as one of the important benefits of proof formalization [9].
6
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Elementary triangular matrices and inverses of k-Hessenberg and triangular matrices

80%
Verde-Star L.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict k-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenberg matrices. An n × n lower triangular matrix is called elementary if it is of the form I + C, where I is the identity matrix and C is lower triangular and has all of its nonzero entries in the k-th column,where 1 ≤ k ≤ n.
7
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Factorization through Hilbert space and the dilation of L(X,Y)-valued measures

80%
Mandrekar V., P.
Studia Mathematica
|
1993
|
tom 107
|
nr 2
101-113
EN
We present a general necessary and sufficient algebraic condition for the spectral dilation of a finitely additive L(X,Y)-valued measure of finite semivariation when X and Y are Banach spaces. Using our condition we derive the main results of Rosenberg, Makagon and Salehi, and Miamee without the assumption that X and/or Y are Hilbert spaces. In addition we relate the dilation problem to the problem of factoring a family of operators through a single Hilbert space.
8
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Fermat’s Little Theorem via Divisibility of Newton’s Binomial

80%
Ziobro R.
Formalized Mathematics
|
2015
|
tom 23
|
nr 3
215-229
EN
Solving equations in integers is an important part of the number theory [29]. In many cases it can be conducted by the factorization of equation’s elements, such as the Newton’s binomial. The article introduces several simple formulas, which may facilitate this process. Some of them are taken from relevant books [28], [14]. In the second section of the article, Fermat’s Little Theorem is proved in a classical way, on the basis of divisibility of Newton’s binomial. Although slightly redundant in its content (another proof of the theorem has earlier been included in [12]), the article provides a good example, how the application of registrations could shorten the length of Mizar proofs [9], [17].
9
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Factorizations of properties of graphs

70%
Broere I., Teboho Moagi S., Mihók P., Vasky R.
Discussiones Mathematicae Graph Theory
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1999
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tom 19
|
nr 2
167-174
EN
A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs 𝓟₁,𝓟₂,...,𝓟ₙ a vertex (𝓟₁, 𝓟₂, ...,𝓟ₙ)-partition of a graph G is a partition {V₁,V₂,...,Vₙ} of V(G) such that for each i = 1,2,...,n the induced subgraph $G[V_i]$ has property $𝓟_i$. The class of all graphs having a vertex (𝓟₁, 𝓟₂, ...,𝓟ₙ)-partition is denoted by 𝓟₁∘𝓟₂∘...∘𝓟ₙ. A property 𝓡 is said to be reducible with respect to a lattice of properties of graphs 𝕃 if there are n ≥ 2 properties 𝓟₁,𝓟₂,...,𝓟ₙ ∈ 𝕃 such that 𝓡 = 𝓟₁∘𝓟₂∘...∘𝓟ₙ; otherwise 𝓡 is irreducible in 𝕃. We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.
10
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On unique factorization semilattices

70%
Silva P.
Discussiones Mathematicae - General Algebra and Applications
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2000
|
tom 20
|
nr 1
97-120
EN
The class of unique factorization semilattices (UFSs) contains important examples of semilattices such as free semilattices and the semilattices of idempotents of free inverse monoids. Their structural properties allow an efficient study, among other things, of their principal ideals. A general construction of UFSs from arbitrary posets is presented and some categorical properties are derived. The problem of embedding arbitrary semilattices into UFSs is considered and complete characterizations are obtained for particular classes of semilattices. The study of the Munn semigroup for regular UFSs is developed and a complete characterization is accomplished with respect to being E-unitary.
11
Content available remote

Factorization of the Popov function of a multivariable linear distributed parameter system in the non-coercive case: a penalization approach

70%
Pandolfi L.
International Journal of Applied Mathematics and Computer Science
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2001
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tom 11
|
nr 6
1249-1260
EN
We study the construction of an outer factor to a positive definite Popov function of a distributed parameter system. We assume that is a non-negative definite matrix with non-zero determinant. Coercivity is not assumed. We present a penalization approach which gives an outer factor just in the case when there exists any outer factor.
12
Content available remote

Matrix quadratic equations column/row reduced factorizations and an inertia theorem for matrix polynomials

61%
Karelin I., Lerer L.
International Journal of Applied Mathematics and Computer Science
|
2001
|
tom 11
|
nr 6
1285-1310
EN
It is shown that a certain Bezout operator provides a bijective correspondence between the solutions of the matrix quadratic equation and factorizatons of a certain matrix polynomial (which is a specification of a Popov-type function) into a product of row and column reduced polynomials. Special attention is paid to the symmetric case, i.e. to the Algebraic Riccati Equation. In particular, it is shown that extremal solutions of such equations correspond to spectral factorizations of . The proof of these results depends heavily on a new inertia theorem for matrix polynomials which is also one of the main results in this paper.
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