Matrix Maps Research Articles

Glaucoma diagnostic performance of humphrey matrix and standard automated perimetry

To evaluate and compare the performance of Humphrey Matrix perimetry (Matrix) and standard automated perimetry (SAP) for glaucoma discrimination. Forty-seven healthy and 68 glaucomatous subjects were included in this study. Glaucoma was defined as having a glaucomatous optic disc and a nerve fiber indicator (NFI) GDx VCC scanning laser polarimetry score of >40. Glaucomatous subjects were classified into two groups, early (40 < NFI < or = 50) and moderate to advanced (NFI > 50). The number of clusters on Matrix and SAP pattern deviation maps were estimated and scored. Areas under the receiver operating characteristic curves (AUCs) of the cluster scores were calculated for early and moderate to advanced stages of glaucoma and compared between Matrix and SAP. Among 68 glaucomatous subjects, 24 eyes were classified as having early and 44 eyes as having moderate to advanced glaucoma. The overall AUC of the Matrix cluster score was comparable to that of SAP (0.857, 0.881, P = 0.538). The AUC of the cluster score did not reveal statistically significant differences between Matrix and SAP for either early or moderate to advanced stages of glaucoma (P = 0.831, 0.237). Both Matrix and SAP showed good diagnostic performance with glaucoma defined as structural loss. Matrix and SAP data showed similar discrimination capability for different stages of glaucoma determined by cluster analysis.

Matrix maps for substitution sequences in the biquaternion representation

A biquaternion representation for any $2\ifmmode\times\else\texttimes\fi{}2$ matrix substitution sequence is constructed. In this representation, a $2\ifmmode\times\else\texttimes\fi{}2$ matrix is equivalent to a biquaternion which can be resolved into a scalar part and a vector part, representing the trace map (scalar map) and the generalized antitrace map (vector map) of the matrix, respectively. Then a geometrical description for the recursion process is developed. This approach is attested to be a powerful tool for studying one-dimensional aperiodic structures. The minimum dimension of the generalized antitrace map in a general case is shown to be exactly nine. The applications of the biquaternion representation in one-dimensional tight-binding electronic systems are explored. The commutativity between cluster matrices is introduced for seeking extended states in the periodic-doubling sequence. The cyclicity of matrix maps for the Fibonacci sequence is restudied, and more critical wave functions and long-range scaling properties in this system can be found. The geometric structure of the Thue-Morse sequence is examined by presenting an invariant symmetry axis and two invariant planes in its vector map.

Factors affecting light energy transfer in some samarium complexes

The photophysical properties of samarium ion in presence of 7‐Acetoxy Coumarin 3‐Carboxylic acid (ACC) or 1‐(4‐methoxyphenyl) 3‐(4‐t‐butyl phenyl) 1,3‐propandione (MBPK), which act as antenna of near‐UV radiation, are studied in different homogeneous, and heterogeneous media as well as in rigid polymethylmethacrylate matrix (PMMA). High quantum yield value of the sensitized emission of (Sm3+ ‐(ACC)) and (Sm3+ ‐(MBPK)) in N,N‐dimethylformamide and PMMA matrix are obtained pointing to an efficient ligandto‐ metal energy transfer. Moreover, luminescence mappings for pattern recognition analysis that could be profitably exploited as a useful identification method in clinical chemistry and biochemistry have been obtained from which the nature of the solvent and/or the ligand is clearly identified by inspection of the corresponding excitation/emission matrix maps.

Matrix maps of quasi-convex null sequences

In [7] Stieglitz and Tietz identify the space qα of all quasi-convex convergent sequences as a BK-space. They characterize all infinite matrices which map qα into an arbitrary FK-space. In [6] they do so for matrices which map a particular class of sequence spaces into qα. In [10] Zygmund introduces q2 in connexion with convergence factors of Fourier series. Dawson considers in [3] and [4] matrix maps of the space q0α of all quasi-convex null sequences. In Section 2 we characterize all matrices which map q0α into an arbitrary FK-space. Prior to that, a particular matrix map on q0α gives us the BK-topology on q0α. As an application we characterize in Section 3 the matrices which map q0α into the FK-spaces considered by Stieglitz and Tietz in [8]. Based on [6], we determine the matrices which map these spaces into q0α. Using methods similar to those in [7] our results in Section 2 depend on Theorems 2.1 and 4.1 in [5] due to Jakimovski and Livne. Theorem 2.1 gives for suitable pairs of sequence spaces necessary and sufficient conditions for an infinite matrix to map one space into the second one. In Theorem 4.1 a special sequence which is useful in applications of quasi-convexity is constructed. We close our paper with two remarks concerning three results in [8].

Hidden dimers and the matrix maps: Fibonacci chains revisited

The existence of cycles of the matrix maps in Fibonacci class of lattices is well established. We show that such cycles are intimately connected with the presence of interesting positional correlations among the constituent ``atoms'' in a one-dimensional quasiperiodic lattice. We particularly address the transfer model of the classic golden mean Fibonacci chain where a six cycle of the full matrix map exists at the center of the spectrum [Kohmoto et al., Phys. Rev. B 35, 1020 (1987)]. In addition, we show that our prescription leads to a determination of other energy values for a mixed model of the Fibonacci chain, for which the full matrix map may have similar cyclic behavior. Apart from the standard transfer model of a golden mean Fibonacci chain, we address a variant of it and the silver mean lattice, where four cycles of the matrix map is already known to exist. The underlying positional correlations for all such cases are discussed in detail.

Generalized Fibonacci lattices: dynamical maps, energy spectra and wavefunctions

Electronic properties of the generalized Fibonacci lattices are studied using the dynamical system technique, for which the off-diagonal tight-binding model is employed. The matrix and trace maps are obtained and investigated in a unified way. It is found that the energy spectra are Cantor-like and the wavefunctions are critical at many energies. For some systems, it is also shown that there are extended and localized wavefunctions. In addition, according to the degree of spatial extension or localization, two other types of wavefunctions are further distinguished, of which one has the tendency to be extended and the other has the tendency to be localized.

Matrix maps and the isomorphic structure of BK spaces

This note gives a characterization of BK spaces that contain isomorphic copies of c 0 {c_0} in terms of matrix maps and a sufficient condition for a matrix map from l ∞ {l_\infty } into a BK space to be a compact operator. The primary tool used in this note is the Bessaga-Pelczynski characterization of Banach spaces which contain isomorphic copies of c 0 {c_0} . It is shown that weakly compact matrix maps on l ∞ {l_\infty } are compact and that, if E E is a BK space such that there is a matrix A A such that c 0 ⊆ E A {c_0} \subseteq {E_A} and E A {E_A} is not strongly conull, then E E must contain an isomorphic copy of c 0 {c_0} .

Matrix Maps and the Isomorphic Structure of BK Spaces

This note gives a characterization of BK spaces that contain isomorphic copies of ${c_0}$ in terms of matrix maps and a sufficient condition for a matrix map from ${l_\infty }$ into a BK space to be a compact operator. The primary tool used in this note is the Bessaga-Pelczynski characterization of Banach spaces which contain isomorphic copies of ${c_0}$. It is shown that weakly compact matrix maps on ${l_\infty }$ are compact and that, if $E$ is a BK space such that there is a matrix $A$ such that ${c_0} \subseteq {E_A}$ and ${E_A}$ is not strongly conull, then $E$ must contain an isomorphic copy of ${c_0}$.

Boundaries between basins of attraction in matrix maps.

The properties of the boundary between basins of attraction of the logistic map in 2\ifmmode\times\else\texttimes\fi{}2 real matrices are explored. The relationship between the Julia set and the boundary is analyzed. Numerical values for the capacity dimension are given, and a mechanism for the origin of a transient scaling behavior is proposed.

Recreation boating spatial patterns: Theory and management

Abstract The activity patterns of recreation boating on an intensively used small urban lake are measured. Matrix maps were used to collect data and to analyze the results. Distinct patterns were observed, with competition for space and conflict among different types of boats. A theory of reciprocal choreography is proposed to help explain the findings. Management options are proposed.

Matrix Maps of Bounded Sequences in a Banach Space

Matrix Maps of Bounded Sequences in a Banach Space

Matrix maps of bounded sequences in a Banach space

Matrices of bounded linear operators are applied to bounded sequences in a Banach space. The main result is related to Knopp’s core theorem for scalar sequences and matrices. From the main result, necessary and sufficient conditions are given for a matrix of operators to map bounded sequences into null sequences in a Banach space.

On matrix maps of entire sequences

On matrix maps of entire sequences

Diagonal submatrices of matrix maps

Diagonal submatrices of matrix maps

A Study of Faunal Resemblance Using Numerical Taxonomy and Factor Analysis

This paper is an analysis of patterns of animal distribution by means of ( 1 ) calculation and clustering of a distance matrix, (2) factor analysis and trend surface maps of scores of five centroid factors, and (3) a stereogram of the counties projected on the principal axes. The data, which are the incidences of amphibians, reptiles, and mammals in the 105 counties of Kansas, were used principally for illustrative purposes, and comprise 185 OTU's consisting of 137 species (37 of which were subdivided into 85 subspecies). Five factors were extracted from a matrix of correlations among OTU's and mapped by trend surface analysis. These were shown to be of interest even though they yielded overlapping faunal zones. The predictive powers of the method, by the inclusion of environmental variables, is discussed. The present study is an application of certain multivariate analytical techniques in the description and analysis of faunal distribution patterns. Quantitative estimates of faunal resemblances have been largely limited to coefficients of resemblance, which authors often derive anew for a particular use. Simpson (1960) and Long (1963) have reviewed the various coefficients in the literature. Hagmeier and Stults (1964) and Hagmeier (1966) used a coefficient which measured the relative number of species distributions ending in areas of 50-miles-square. They grouped areas with low values into faunal zones, using areas of high values as borders. Coefficients were calculated between faunal zones and the resulting matrix was clustered by numerical taxonomic methods. R-type factor analyses of species distributions have been used extensively by some ecologists to describe vegetation (see Goodall, 1954). The uses of factor analysis in systematics have been reviewed by Sokal (1965). The major purpose of this analysis is to test the applicability of the methods of numerical taxonomy and trend surface analysis to biogeographic distribution patterns. The analysis is best viewed primarily as an illustrative example and not as a definitive study of the distribution of vertebrates in Kansas. THE DATA The data were taken from distribution maps for Kansas by Hall (1955), for mammals, and Smith (1956), for amphibians and reptiles. Because the data were used only as an example, the maps were assumed to depict accurately the distributions of these animals without further corroboration. The study included 61 species of mammals (17 of which were divided into 2 subspecies each, 6 into 3 subspecies, 1 into 4 subspecies, and 1 into 5 subspecies), 52 species of reptiles (7 of which were divided into 2 subspecies each), and 24 species of amphibians (5 of which were divided into 2 subspecies each), yielding a total of 185 OTU's. The raw data can be obtained by writing to the author. They comprise a data matrix consisting of ones and zeroes, representing presence and absence, respectively, of each species or subspecies within each of the 105 counties. The computations were carried out at the Computation Center of The University of Kansas using the NT-SYS system of multivariate routines developed by F. J. Rohlf, R. L. Bartcher and J. R. L. Kishpaugh for the IBM 7040 and the GE 625 computers. RESULTS USING A PHENOGRAM

𝑀-similarity and isomorphisms in 𝐵₀-spaces

Let X be a Bo-space, i.e. a locally convex Frechet space. A system (xi; fi) with (xi) CX and (fi) CX* is a generalized basis for X if (a) fi(xj) =ij for all i and j, and (b) fi(x) =0 for all i implies x= 0 [2]. The sequence space F(X) = { (f,(x)): x(EX} becomes a Bo-space equivalent to X when equipped with the topology induced by the map F(.) = (fi(.)). Arsove and Edwards proved that a space X, with generalized basis (xi; fi), is isomorphic (i.e., linearly homomorphic) to a space Y if and only if Y has a generalized basis (y?; gi) such that, as sets, F(X) = G(Y). The author discussed [3] a concept dual to that of generalized basis, and obtained isomorphism theorems analogous to the Arsove-Edwards theorem. W. Ruckle [6] discussed matrix maps which preserve bases in Bo-spaces. (A system (xi; fi) is a basis for X if it is a generalized basis, and x = Efi(x)xi for every x in X.) Here we introduce a generalized notion of similarity and obtain results which contain the above mentioned work in the context of B0-spaces. DEFINITION. Let (xi;fi) be a biorthogonal system for a Bo-space X (i.e., fi(xj) = bij for all i, j), and (yt; gi) such a system for a Bo-space Y. We shall say that (xi; f,) and (ye; gi) are M-similar if either (a) there exists a matrix A: F(X)->G(Y) which is one-to-one and onto, or (b) there exists a matrix B: G(Y)->F(X) which is one-to-one and onto. If U is a Bo-space of sequences, let U'= { (ti): Ztiui converges for all (ust) in U }I.