Displaying similar documents to “A method for determining constants in the linear combination of exponentials”

Some properties complementary to Brualdi-Li matrices

Chuanlong Wang, Xuerong Yong (2015)

Czechoslovak Mathematical Journal

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In this paper we derive new properties complementary to an 2 n × 2 n Brualdi-Li tournament matrix B 2 n . We show that B 2 n has exactly one positive real eigenvalue and one negative real eigenvalue and, as a by-product, reprove that every Brualdi-Li matrix has distinct eigenvalues. We then bound the partial sums of the real parts and the imaginary parts of its eigenvalues. The inverse of B 2 n is also determined. Related results obtained in previous articles are proven to be corollaries.

Sufficient Conditions for Integrability of Distortion Function Kf 1

Costantino Capozzoli (2009)

Bollettino dell'Unione Matematica Italiana

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Assume that Ω , Ω are planar domains and f : Ω onto Ω is a homeomorphism belonging to Sobolev space W loc 1 , 1 ( Ω ; 2 ) with finite distortion. We prove that if the distortion function K f of f satisfies the condition dist EXP ( K f , L ) < 1 , then the distortion function K f - 1 of f - 1 belongs to L loc 1 ( Ω ) . We show that this result is sharp in sense that the conclusion fails if dist EXP ( K f , L ) = 1 . Moreover, we prove that if the distortion function K f satisfies the condition dist EXP ( K f , L ) = λ for some λ > 0 , then K f - 1 belongs to L loc p ( Ω ) for every p ( 0 , 1 2 λ ) . As special case of this result we show that if...

Investigating generalized quaternions with dual-generalized complex numbers

Nurten Gürses, Gülsüm Yeliz Şentürk, Salim Yüce (2023)

Mathematica Bohemica

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We aim to introduce generalized quaternions with dual-generalized complex number coefficients for all real values α , β and 𝔭 . Furthermore, the algebraic structures, properties and matrix forms are expressed as generalized quaternions and dual-generalized complex numbers. Finally, based on their matrix representations, the multiplication of these quaternions is restated and numerical examples are given.

Nonlinear mappings preserving at least one eigenvalue

Constantin Costara, Dušan Repovš (2010)

Studia Mathematica

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We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either F ( x ) = u x u - 1 or F ( x ) = u x t u - 1 for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying...

Persistence of Coron’s solution in nearly critical problems

Monica Musso, Angela Pistoia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the problem - Δ u = u N + 2 N - 2 + λ in Ω ε ω , u &gt; 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε &gt; 0 and λ . We prove that if the size of the hole ε goes to zero and if, simultaneously, the parameter λ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

On the first sign change in Mertens' theorem

Jan Büthe (2015)

Acta Arithmetica

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The function p x 1 / p - l o g l o g ( x ) - M is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).

G-matrices, J -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D 1 and D 2 such that A - T = D 1 A D 2 , where A - T denotes the transpose of the inverse of A . Denote by J = diag ( ± 1 ) a diagonal (signature) matrix, each of whose diagonal entries is + 1 or - 1 . A nonsingular real matrix Q is called J -orthogonal if Q T J Q = J . Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation...

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

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Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) i = 1 n X i - d i = 1 n Y ² i = - I with d ∈ N for nonsingular X i , Y i M ( Z ) , i=1,...,n.

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

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Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product k R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y , then k R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

On a sequence formed by iterating a divisor operator

Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer (2019)

Czechoslovak Mathematical Journal

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Let be the set of positive integers and let s . We denote by d s the arithmetic function given by d s ( n ) = ( d ( n ) ) s , where d ( n ) is the number of positive divisors of n . Moreover, for every , m we denote by δ s , , m ( n ) the sequence d s ( d s ( ... d s ( d s ( n ) + ) + ... ) + ) m -times = d s ( n ) for m = 1 , d s ( d s ( n ) + ) for m = 2 , d s ( d s ( d s ( n ) + ) + ) for m = 3 , We present classical and nonclassical notes on the sequence ( δ s , , m ( n ) ) m 1 , where , n , s are understood as parameters.

A Fiedler-like theory for the perturbed Laplacian

Israel Rocha, Vilmar Trevisan (2016)

Czechoslovak Mathematical Journal

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The perturbed Laplacian matrix of a graph G is defined as L D = D - A , where D is any diagonal matrix and A is a weighted adjacency matrix of G . We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore,...

Asymptotic properties of a ϕ -Laplacian and Rayleigh quotient

Waldo Arriagada, Jorge Huentutripay (2020)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider the ϕ -Laplacian problem with Dirichlet boundary condition, - div ϕ ( | u | ) u | u | = λ g ( · ) ϕ ( u ) in Ω , λ and u | Ω = 0 . The term ϕ is a real odd and increasing homeomorphism, g is a nonnegative function in L ( Ω ) and Ω N is a bounded domain. In these notes an analysis of the asymptotic behavior of sequences of eigenvalues of the differential equation is provided. We assume conditions which guarantee the existence of stationary solutions of the system. Under these rather stringent hypotheses we prove that any extremal is both...