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Displaying similar documents to “Estimates of the principal eigenvalue of the p -Laplacian and the p -biharmonic operator”

Inequalities for real number sequences with applications in spectral graph theory

Emina Milovanović, Şerife Burcu Bozkurt Altındağ, Marjan Matejić, Igor Milovanović (2022)

Czechoslovak Mathematical Journal

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Let a = ( a 1 , a 2 , ... , a n ) be a nonincreasing sequence of positive real numbers. Denote by S = { 1 , 2 , ... , n } the index set and by J k = { I = { r 1 , r 2 , ... , r k } , 1 r 1 < r 2 < < r k n } the set of all subsets of S of cardinality k , 1 k n - 1 . In addition, denote by a I = a r 1 + a r 2 + + a r k , 1 k n - 1 , 1 r 1 < r 2 < < r k n , the sum of k arbitrary elements of sequence a , where a I 1 = a 1 + a 2 + + a k and a I n = a n - k + 1 + a n - k + 2 + + a n . We consider bounds of the quantities R S k ( a ) = a I 1 / a I n , L S k ( a ) = a I 1 - a I n and S k , α ( a ) = I J k a I α in terms of A = i = 1 n a i and B = i = 1 n a i 2 . Then we use the obtained results to generalize some results regarding Laplacian and normalized Laplacian eigenvalues of graphs.

Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Catherine Bandle, Joachim von Below, Wolfgang Reichel (2008)

Journal of the European Mathematical Society

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We consider linear elliptic equations - Δ u + q ( x ) u = λ u + f in bounded Lipschitz domains D N with mixed boundary conditions u / n = σ ( x ) λ u + g on D . The main feature of this boundary value problem is the appearance of λ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient σ ( x ) . We study positivity principles and anti-maximum principles. One of our main results states that if σ is somewhere negative, q 0 and D q ( x ) d x > 0 then there exist two eigenvalues λ - 1 , λ 1 such the positivity...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...

Lower bounds for the largest eigenvalue of the gcd matrix on { 1 , 2 , , n }

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

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Consider the n × n matrix with ( i , j ) ’th entry gcd ( i , j ) . Its largest eigenvalue λ n and sum of entries s n satisfy λ n > s n / n . Because s n cannot be expressed algebraically as a function of n , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that λ n > 6 π - 2 n log n for all n . If n is large enough, this follows from F. Balatoni (1969).

Computing the greatest 𝐗 -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector x is said to be an eigenvector of a square max-min matrix A if A x = x . An eigenvector x of A is called the greatest 𝐗 -eigenvector of A if x 𝐗 = { x ; x ̲ x x ¯ } and y x for each eigenvector y 𝐗 . A max-min matrix A is called strongly 𝐗 -robust if the orbit x , A x , A 2 x , reaches the greatest 𝐗 -eigenvector with any starting vector of 𝐗 . We suggest an O ( n 3 ) algorithm for computing the greatest 𝐗 -eigenvector of A and study the strong 𝐗 -robustness. The necessary and sufficient conditions for strong 𝐗 -robustness are introduced...