scholarly journals Nonderogatory Unicyclic Digraphs

2007 ◽  
Vol 2007 ◽  
pp. 1-8 ◽  
Author(s):  
Diego Bravo ◽  
Juan Rada
Keyword(s):  
Characteristic Polynomial ◽  
Minimal Polynomial ◽  
Complete Product

A digraph is nonderogatory if its characteristic polynomial and minimal polynomial are equal. We find a characterization of nonderogatory unicyclic digraphs in terms of Hamiltonicity conditions. An immediate consequence of this characterization ia that the complete product of difans and diwheels is nonderogatory.

ART STUDIOS – A WAY TO DEVELOPE CHILDREN’S IMAGINATION

2020 ◽  
Vol 11 (1) ◽  
pp. 144-148
Author(s):  
Liuba Zlatkova ◽  
Keyword(s):  
Fairy Tale ◽  
Psychological Effects ◽  
Complete Product ◽  
The Arts ◽  
The Creation ◽  
Role Of Parents

The report describes the steps for creating a musical tale by children in the art studios of „Art Workshop“, Shumen. These studios are led by students volunteers related to the arts from pedagogical department of Shumen University, and are realized in time for optional activities in the school where the child studies. The stages of creating a complete product with the help of different arts are traced – from the birth of the idea; the creation of a fairy tale plot by the children; the characterization of the fairy-tale characters; dressing them in movement, song and speech; creating sets and costumes and creating a finished product to present on stage. The role of parents as a link and a necessary helper for children and leaders is also considered, as well as the positive psychological effects that this cooperation creates.

An extension of the Cobham-Semënov Theorem

2000 ◽  
Vol 65 (1) ◽  
pp. 201-211 ◽  
Author(s):  
Alexis Bès
Keyword(s):  
Characteristic Polynomial ◽  
Minimal Polynomial ◽  
Pisot Numbers ◽  
Numeration Systems

AbstractLet θ, θ′ be two multiplicatively independent Pisot numbers, and letU,U′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For everyn≥ 1, ifA⊆ ℕnisU-andU′ -recognizable thenAis definable in 〈ℕ: + 〉.

scholarly journals From Fibonacci Sequence to the Golden Ratio

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Alberto Fiorenza ◽  
Giovanni Vincenzi
Keyword(s):  
Characteristic Polynomial ◽  
Golden Ratio ◽  
Fibonacci Sequence ◽  
Maximum Modulus ◽  
Point Of View ◽  
General Point ◽  
Constant Coefficients ◽  
The Difference ◽  
New Perspective

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

scholarly journals SIMETRISASI BENTUK KANONIK JORDAN

2021 ◽  
Vol 15 (1) ◽  
pp. 015-028
Author(s):  
Darlena Darlena ◽  
Ari Suparwanto
Keyword(s):  
Scalar Field ◽  
Linear Operator ◽  
Characteristic Polynomial ◽  
Canonical Form ◽  
Minimal Polynomial ◽  
Splitting Field ◽  
Jordan Decomposition ◽  
Jordan Canonical Form ◽  
Canonical Matrix

If the characteristic polynomial of a linear operator  is completely factored in scalar field of  then Jordan canonical form  of  can be converted to its rational canonical form  of , and vice versa. If the characteristic polynomial of linear operator  is not completely factored in the scalar field of  ,then the rational canonical form  of  can still be obtained but not its Jordan canonical form matrix . In this case, the rational canonical form  of  can be converted to its Jordan canonical form by extending the scalar field of  to Splitting Field of minimal polynomial   of , thus forming the Jordan canonical form of  over Splitting Field of  . Conversely, converting the Jordan canonical form  of  over Splitting Field of  to its rational canonical form uses symmetrization on the Jordan decomposition basis of  so as to form a cyclic decomposition basis of  which is then used to form the rational canonical matrix of

scholarly journals Extending the Characteristic Polynomial for Characterization of C20 Fullerene Congeners

Mathematics ◽  
2017 ◽  
Vol 5 (4) ◽  
pp. 84 ◽  
Author(s):  
Dan-Marian Joiţa ◽  
Lorentz Jäntschi
Keyword(s):  
Characteristic Polynomial ◽  
C20 Fullerene

ON THE MINIMAL POLYNOMIAL OF A MATRIX

2004 ◽  
Vol 15 (01) ◽  
pp. 89-105
Author(s):  
THANH MINH HOANG ◽  
THOMAS THIERAUF
Keyword(s):  
Characteristic Polynomial ◽  
Minimal Polynomial ◽  
Constant Term ◽  
Open Problems ◽  
Matrix Rank ◽  
The Matrix ◽  
Exact Counting ◽  
Class C

We investigate the complexity of the degree and the constant term of the minimal polynomial of a matrix. We show that the degree of the minimal polynomial is computationally equivalent to the matrix rank. We compare the constant term of the minimal polynomial with the constant term of the characteristic polynomial. The latter is known to be computable in the logspace counting class GapL. We show that if this holds for the minimal polynomial as well, then the exact counting in logspace class C=L is closed under complement. Whether C=L is closed under complement is one of the main open problems in this area. As an application of our techniques we show that the problem of deciding whether a matrix is diagonalizable is complete for AC0(C=L), the AC0-closure ofC=L.

Signless Laplacian spectral characterization of some joins

2015 ◽  
Vol 30 ◽  
Author(s):  
Xiaogang Liu ◽  
Pengli Lu
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Regular Graph ◽  
Spectral Characterization ◽  
Laplacian Eigenvalue ◽  
Laplacian Spectra ◽  
Signless Laplacian

The join of two disjoint graphs G and H, denoted by G ∨ H, is the graph obtained by joining each vertex of G to each vertex of H. In this paper, the signless Laplacian characteristic polynomial of the join of two graphs is first formulated. And then, a lower bound for the i-th largest signless Laplacian eigenvalue of a graph is given. Finally, it is proved that G ∨ K_m, where G is an (n − 2)-regular graph on n vertices, and K_n ∨ K_2 except for n = 3, are determined by their signless Laplacian spectra.

scholarly journals Closed sets of finitary functions between finite fields of coprime order

2020 ◽  
Vol 81 (4) ◽  
Author(s):  
Stefano Fioravanti
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Linear Transformation ◽  
Minimal Polynomial ◽  
Invariant Subspaces ◽  
Closed Sets ◽  
The Right ◽  
Coprime Order

AbstractWe investigate the finitary functions from a finite field $$\mathbb {F}_q$$ F q to the finite field $$\mathbb {F}_p$$ F p , where p and q are powers of different primes. An $$(\mathbb {F}_p,\mathbb {F}_q)$$ ( F p , F q ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the invariant subspaces of the vector space $$\mathbb {F}_p^{\mathbb {F}_q\backslash \{0\}}$$ F p F q \ { 0 } with respect to a certain linear transformation with minimal polynomial $$x^{q-1} - 1$$ x q - 1 - 1 . Furthermore we prove that each of these subsets of functions is generated by one unary function.

scholarly journals The rank of Pseudo walk matrices: controllable and recalcitrant pairs

2020 ◽  
Vol 3 (3) ◽  
pp. 41-52
Author(s):  
Alexander Farrugia ◽  
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Minimal Polynomial ◽  
Full Rank ◽  
Gram Matrix ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

A pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) of a graph \(G\) having adjacency matrix \(\mathbf{A}\) is an \(n\times n\) matrix with columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{n-1}\mathbf{v}\) whose Gram matrix has constant skew diagonals, each containing walk enumerations in \(G\). We consider the factorization over \(\mathbb{Q}\) of the minimal polynomial \(m(G,x)\) of \(\mathbf{A}\). We prove that the rank of \(\mathbf{W}_\mathbf{v}\), for any walk vector \(\mathbf{v}\), is equal to the sum of the degrees of some, or all, of the polynomial factors of \(m(G,x)\). For some adjacency matrix \(\mathbf{A}\) and a walk vector \(\mathbf{v}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable if \(\mathbf{W}_\mathbf{v}\) has full rank. We show that for graphs having an irreducible characteristic polynomial over \(\mathbb{Q}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\). We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors \(\mathbf{v}\), the degree of the minimal polynomial of the largest eigenvalue of \(\mathbf{A}\) is a lower bound for the rank of \(\mathbf{W}_\mathbf{v}\). If the rank of \(\mathbf{W}_\mathbf{v}\) attains this lower bound, then \((\mathbf{A},\mathbf{v})\) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that \((\mathbf{A},\mathbf{v})\) is neither controllable nor recalcitrant for any walk vector \(\mathbf{v}\).

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