scholarly journals Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Ana Portilla ◽  
Yamilet Quintana ◽  
José M. Rodríguez ◽  
Eva Tourís
Keyword(s):  
Asymptotic Behavior ◽  
Admissible Pair ◽  
Sobolev Norm ◽  
Diagonal Matrix ◽  
Extremal Polynomials ◽  
Sobolev Norms ◽  
The Matrix ◽  
Complex Coefficients ◽  
Zero Location

Let ℙ be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm∥ · ∥W1,p(Vμ), where the matrixVand the measureμconstitute ap-admissible pair for1≤p≤∞. In this paper we establish the zero location and asymptotic behavior of extremal polynomials associated to∥ · ∥W1,p(Vμ), stating hypothesis on the matrixVrather than on the diagonal matrix appearing in its unitary factorization.

scholarly journals Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms

2010 ◽  
Vol 162 (12) ◽  
pp. 2225-2242 ◽  
Author(s):  
Ana Portilla ◽  
Yamilet Quintana ◽  
José M. Rodríguez ◽  
Eva Tourís
Keyword(s):  
Asymptotic Behavior ◽  
Extremal Polynomials ◽  
Sobolev Norms ◽  
Zero Location

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Higher-order Darboux transformations for two-dimensional Dirac systems with diagonal matrix potential

2021 ◽  
Vol 2090 (1) ◽  
pp. 012038
Author(s):  
A Schulze-Halberg
Keyword(s):  
Dirac Equation ◽  
Explicit Form ◽  
Higher Order ◽  
Diagonal Matrix ◽  
Two Dimensional ◽  
Darboux Transformations ◽  
Dirac Systems ◽  
Matrix Potential ◽  
De Vries ◽  
The Matrix

Abstract We construct the explicit form of higher-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The matrix potential entries can depend arbitrarily on the two variables. Our construction is based on results for coupled Korteweg-de Vries equations [27].

scholarly journals New Bounds for the α-Indices of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1668
Author(s):  
Eber Lenes ◽  
Exequiel Mallea-Zepeda ◽  
Jonnathan Rodríguez
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Independence Number ◽  
Minimal Degree ◽  
Diagonal Matrix ◽  
The Matrix

Let G be a graph, for any real 0≤α≤1, Nikiforov defines the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal matrix of degrees of the vertices of G. This paper presents some extremal results about the spectral radius ρα(G) of the matrix Aα(G). In particular, we give a lower bound on the spectral radius ρα(G) in terms of order and independence number. In addition, we obtain an upper bound for the spectral radius ρα(G) in terms of order and minimal degree. Furthermore, for n>l>0 and 1≤p≤⌊n−l2⌋, let Gp≅Kl∨(Kp∪Kn−p−l) be the graph obtained from the graphs Kl and Kp∪Kn−p−l and edges connecting each vertex of Kl with every vertex of Kp∪Kn−p−l. We prove that ρα(Gp+1)<ρα(Gp) for 1≤p≤⌊n−l2⌋−1.

scholarly journals Structure of Personality Variables of Special Olympics Athletes and Unified Partners in Football

2015 ◽  
Vol 3 (1) ◽  
pp. 112
Author(s):  
Dragan Popović ◽  
Miloš Popović ◽  
Evagelia Boli ◽  
Hamidovoć Mensur ◽  
Marina Jovanović
Keyword(s):  
Covariance Matrix ◽  
Measurement Errors ◽  
Theory Of Measurement ◽  
Error Variance ◽  
Diagonal Matrix ◽  
Special Olympics ◽  
True Variance ◽  
The Matrix ◽  
Definition Of ◽  
Reliability Coefficients

Due to its simplicity and explicit algebraic and geometric meanings, latent dimensions, and identification structures associated with these dimensions, reliability of the latent dimensions obtained by orthoblique transformation of principal components can be determined in a clear and unambiguous manner. Let G = (gij); i = 1, ..., n; j = 1, ..., m is an acceptably unknown matrix of measurement errors in the description of a set E on a set V. Then the matrix of true results of entities from E on the variables from V will be Y = Z - G. Assume, in accordance with the classical theory of measurement (Gulliksen, 1950, Lord - Novick, 1968; Pfanzagl, 1968), that matrix G is such that YtG = 0 and GtGn-1 = E2 = (ejj2) where E2 is a diagonal matrix, the covariance matrix of true results will be H = YtYn-1 = R - E2 if R = ZtZn-1 is an intercorrelation matrix of variables from V defined on set E. Suppose that the reliability coefficients of variables from V are known; let P be a diagonal matrix whose elements j are these reliability coefficients. Then the variances of measurement errors for the standardized results on variables from V will be just elements of the matrix E2 = I - . Now the true values on the latent dimensions will be elements of the matrix  = (Z - G)Q with the covariance matrix  = tn-1 = QtHQ = QtRQ - QtE2Q = (pq). Therefore, the true variances of the latent dimensions will be the diagonal elements of matrix ; denote those elements with p2. Based on the formal definition of the reliability coefficient of some variable  = t2 /  where t2 is a true variance of the variable and  is the total variance of the variable, or the variance that also includes the error variance, the reliability coefficients of the latent dimensions, if the reliability coefficients of the variables from which these dimensions have been derived are known, will be p = p2 / sp2 = 1 - (qptE2qp )(qptRq )-1 p = 1,...,k

scholarly journals Laplacian integral graphs with a given degree sequence constraint

2021 ◽  
Vol 40 (6) ◽  
pp. 1431-1448
Author(s):  
Ansderson Fernandes Novanta ◽  
Carla Silva Oliveira ◽  
Leonardo de Lima
Keyword(s):  
Adjacency Matrix ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Bipartite Graphs ◽  
Diagonal Matrix ◽  
Connected Graphs ◽  
Sequence Constraint ◽  
The Matrix ◽  
Vertex Degrees ◽  
Integral Graphs

Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) −A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be L-integral if all eigenvalues of the matrix L(G) are integers. In this paper, we characterize all Lintegral non-bipartite graphs among all connected graphs with at most two vertices of degree larger than or equal to three.

Best Polynomial Approximation with Linear Constraints

1992 ◽  
Vol 44 (6) ◽  
pp. 1289-1302
Author(s):  
K. Pan ◽  
E. B. Saff
Keyword(s):  
Asymptotic Behavior ◽  
Polynomial Approximation ◽  
Asymptotic Relation ◽  
Sufficient Conditions ◽  
Linear Constraints ◽  
Necessary And Sufficient Conditions ◽  
Compact Set ◽  
Approximation By Polynomials ◽  
The Matrix ◽  
Necessary And Sufficient

AbstractLet A be a (k + 1) × (k + 1) nonzero matrix. For polynomials p ∈ Pn, set and . Let E ⊂ C be a compact set that does not separate the plane and f be a function continuous on E and analytic in the interior of E. Set and . Our goal is to study approximation to f on E by polynomials from Bn(A). We obtain necessary and sufficient conditions on the matrix A for the convergence En(A,f) → 0 to take place. These results depend on whether zero lies inside, on the boundary or outside E and yield generalizations of theorems of Clunie, Hasson and Saff for approximation by polynomials that omit a power of z. Let be such that . We also study the asymptotic behavior of the zeros of and the asymptotic relation between En(f) and En(A,f).

A Method of Identification of Analyser Characteristics That is Free of Probability Summation Effect

Perception ◽  
1997 ◽  
Vol 26 (1_suppl) ◽  
pp. 245-245
Author(s):  
A D Logvinenko
Keyword(s):  
Decision Rule ◽  
Quadratic Forms ◽  
Diagonal Matrix ◽  
Weighted Sum ◽  
General Belief ◽  
Probability Summation ◽  
Detection Model ◽  
Similarity Transform ◽  
Summation Effect ◽  
The Matrix

A detection model (originally proposed by Quick) comprising, in a sequence of linear analysers, varphi1, …, varphi n, nonlinear transducer functions, and the Minkowski decision rule, is widely used, especially when it is necessary to take into account the effect of probability summation. However, there is a general belief that the analyser characteristics cannot be determined in detection experiments since there is a trade-off between these characteristics and the decision rule. Here we show how to overcome this problem, ie how to identify the analysers varphi1, …, varphi n despite the probability summation between them. The observer's performance is assumed to be quantitatively defined in terms of an equidetection surface (EDS). Each analyser varphi i is expressed as a weighted sum of linear (coordinate) analysers {phi j}: varphi i=sum j=1 n a ijphi j, so that an identification of the analysers {phi i} is then reduced to evaluating the weight matrix A={ a ij}. It has been proven that A can be uniquely recovered from an ellipsoidal approximation of EDS in the neighbourhood of at least two points. More specifically, the following equation holds true: A−1 DA= H1−1 H2, where D is a diagonal matrix, H1 and H2 are the matrices of the quadratic forms determining the n-dimensional ellipsoids approximating EDS. Thus, the matrix H1−1 H2 known from experiment is a similarity transform of the diagonal matrix, the columns of A being the eigenvectors of H1−1 H2. Hence, any eigensystem routine can be used to derive A from H1−1 H2.

DOMINANCE OF LONG DISTANCE EFFECTS IN THE KL-KS MASS DIFFERENCE

1990 ◽  
Vol 05 (19) ◽  
pp. 1477-1483 ◽  
Author(s):  
K. TERASAKI ◽  
S. ONEDA
Keyword(s):  
Asymptotic Behavior ◽  
Vector Meson ◽  
Ground State ◽  
Mass Shell ◽  
Mass Difference ◽  
Matrix Elements ◽  
Long Distance ◽  
Distance Effects ◽  
Intermediate States ◽  
The Matrix

It is argued that the asymptotic behavior of the matrix elements of Hw involving on-mass-shell ground-state mesons with infinite momenta, which were instrumental in explaining the |ΔI|=1/2 rule in the K→ππ decays, actually implies [Formula: see text]. The long distance effects (contributions of PS and vector meson poles and ππ intermediate states) may reproduce the observed KL−Ks mass difference, consistent with the |ΔI|=1/2 rule. Its estimate is however sensitive to the η-η′-… mixing.

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