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Note on a conjecture for the sum of signless Laplacian eigenvalues

Xiaodan Chen, Guoliang Hao, Dequan Jin, Jingjian Li (2018)

Czechoslovak Mathematical Journal

For a simple graph G on n vertices and an integer k with 1 ≤ k ≤ n , denote by 𝒮 k + ( G ) the sum of k largest signless Laplacian eigenvalues of G . It was conjectured that 𝒮 k + ( G ) ≤ e ( G ) + k + 1 2 , where e ( G ) is the number of edges of G . This conjecture has been proved to be true for all graphs when k ∈ { 1 , 2 , n - 1 , n } , and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k ). In this note, this conjecture is proved to be true for all graphs when k = n - 2 , and for some new classes of graphs.

Note on the ANOVA of a completely confounded factorial experiment

Wiktor Oktaba (2005)

Applicationes Mathematicae

The purpose of this paper is to present a modern approach to the analysis of variance (ANOVA) of disconnected resolvable group divisible partially balanced incomplete block (GDPBIB) designs with factorial structure and with some interaction effects completely confounded. A characterization of a factorial experiment with completely confounded interaction is given. The treatment effect estimators and some relations between the matrix F of the reduced normal equations and the information matrix A are...

Note on the core matrix partial ordering

Jacek Mielniczuk (2011)

Discussiones Mathematicae Probability and Statistics

Complementing the work of Baksalary and Trenkler [2], we announce some results characterizing the core matrix partial ordering.

n-supercyclic and strongly n-supercyclic operators in finite dimensions

Romuald Ernst (2014)

Studia Mathematica

We prove that on ℝ N , there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if ℝ N has an n-dimensional subspace whose orbit under T ∈ ( ℝ N ) is dense in ℝ N , then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator T ∈ ( ℝ N ) is strongly n-supercyclic if ℝ N has an n-dimensional subspace whose orbit under T is dense in ℙ ₙ ( ℝ N ) , the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially in finite...

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