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.

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 Dynamics of f(R) gravity models and asymmetry of time

2018 ◽  
Vol 27 (02) ◽  
pp. 1850002 ◽  
Author(s):  
Murli Manohar Verma ◽  
Bal Krishna Yadav
Keyword(s):  
Fixed Points ◽  
Quadratic Forms ◽  
Scale Factor ◽  
Gravity Models ◽  
Field Equations ◽  
Classical Level ◽  
The Matrix ◽  
The Stability ◽  
Effects Of Radiation ◽  
Linear And Quadratic Forms

We solve the field equations of modified gravity for [Formula: see text] model in metric formalism. Further, we obtain the fixed points of the dynamical system in phase-space analysis of [Formula: see text] models, both with and without the effects of radiation. The stability of these points is studied against the perturbations in a smooth spatial background by applying the conditions on the eigenvalues of the matrix obtained in the linearized first-order differential equations. Following this, these fixed points are used for analyzing the dynamics of the system during the radiation, matter and acceleration-dominated phases of the universe. Certain linear and quadratic forms of [Formula: see text] are determined from the geometrical and physical considerations and the behavior of the scale factor is found for those forms. Further, we also determine the Hubble parameter [Formula: see text], the Ricci scalar [Formula: see text] and the scale factor [Formula: see text] for these cosmic phases. We show the emergence of an asymmetry of time from the dynamics of the scalar field exclusively owing to the [Formula: see text] gravity in the Einstein frame that may lead to an arrow of time at a classical level.

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.

scholarly journals Theorems Relating to Quadratic Forms and their Discriminant Matrices

1953 ◽  
Vol 10 (1) ◽  
pp. 13-15
Author(s):  
S. Vajda
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Normal Population ◽  
Royal Statistical Society ◽  
Unit Variance ◽  
The Matrix ◽  
Chi Squared ◽  
Image Position ◽  
Latent Roots

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:Let the expression be given; introduce, for c, a linear form in and obtainIf the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.

Sign in / Sign up

Export Citation Format

Share Document