scholarly journals The Decomposition Algorithm for Skew-Symmetrizable Exchange Matrices

10.37236/2447 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Weiwen Gu
Keyword(s):  
Decomposition Algorithm ◽  
Symmetric Matrices ◽  
Combinatorial Algorithm ◽  
Median Graph ◽  
The One ◽  
Mutation Type

Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admit unfoldings to skew-symmetric matrices. We develop a combinatorial algorithm that determines if a given skew-symmetrizable matrix is of such type. This algorithm generalizes the one in Weiwen Gu's Decomposition Algorithm for Median Graph of Triangulation of a Bordered 2D Surface. As a corollary, we use this algorithm to determine if a given skew-symmetrizable matrix has finite mutation type.

scholarly journals Tolerance and Nature of Residual Refraction in Symmetric Power Space as Principal Lens Powers and Meridians Change

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Herven Abelman ◽  
Shirley Abelman
Keyword(s):  
Symmetric Power ◽  
Symmetric Matrices ◽  
Eigenvalues And Eigenvectors ◽  
Real Scalar ◽  
Power Change ◽  
Refractive Correction ◽  
Dioptric Power ◽  
The One ◽  
Residual Power

Unacceptable principal powers in well-centred lenses may require a toric over-refraction which differs in nature from the one where correct powers have misplaced meridians. This paper calculates residual (over) refractions and their natures. The magnitude of the power of the over-refraction serves as a general, reliable, real scalar criterion for acceptance or tolerance of lenses whose surface relative curvatures change or whose meridians are rotated and cause powers to differ. Principal powers and meridians of lenses are analogous to eigenvalues and eigenvectors of symmetric matrices, which facilitates the calculation of powers and their residuals. Geometric paths in symmetric power space link intended refractive correction and these carefully chosen, undue refractive corrections. Principal meridians alone vary along an arc of a circle centred at the origin and corresponding powers vary autonomously along select diameters of that circle in symmetric power space. Depending on the path of the power change, residual lenses different from their prescription in principal powers and meridians are pure cross-cylindrical or spherocylindrical in nature. The location of residual power in symmetric dioptric power space and its optical cross-representation characterize the lens that must be added to the compensation to attain the power in the prescription.

scholarly journals A Decomposition Algorithm for the Oriented Adjacency Graph of the Triangulations of a Bordered Surface with Marked Points

10.37236/578 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Weiwen Gu
Keyword(s):  
Oriented Graph ◽  
Decomposition Algorithm ◽  
Adjacency Graph ◽  
Connected Graphs ◽  
Marked Points ◽  
Adjacency Graphs ◽  
Mutation Type

In this paper we consider an oriented version of adjacency graphs of triangulations of bordered surfaces with marked points. We develop an algorithm that determines whether a given oriented graph is an oriented adjacency graph of a triangulation. If a given oriented graph corresponds to many triangulations, our algorithm finds all of them. As a corollary we find out that there are only finitely many oriented connected graphs with non-unique associated triangulations. We also discuss a new algorithm which determines whether a given quiver is of finite mutation type. This algorithm is linear in the number of nodes and is more effective than the previously known one.

Sparsity induced by covariance transformation: some deterministic and probabilistic results

2021 ◽  
Vol 477 (2247) ◽  
pp. 20200756
Author(s):  
Jakub Rybak ◽  
Heather S. Battey
Keyword(s):  
Positive Definite Matrix ◽  
Theoretical Work ◽  
Covariance Matrices ◽  
Symmetric Matrices ◽  
Natural Basis ◽  
Matrix Logarithm ◽  
Covariance Models ◽  
The One ◽  
Eigen Decomposition ◽  
Scientific Fields

Motivated by statistical challenges arising in modern scientific fields, notably genomics, this paper seeks embeddings in which relevant covariance models are sparse. The work exploits a bijective mapping between a strictly positive definite matrix and its orthonormal eigen-decomposition, and between an orthonormal eigenvector matrix and its principle matrix logarithm. This leads to a representation of covariance matrices in terms of skew-symmetric matrices, for which there is a natural basis representation, and through which sparsity is conveniently explored. This theoretical work establishes the possibility of exploiting sparsity in the new parametrization and converting the conclusion back to the one of interest, a prospect of high relevance in statistics. The statistical aspects associated with this operation, while not a focus of the present work, are briefly discussed.

scholarly journals A minimal-variable symplectic method for isospectral flows

2019 ◽  
Vol 60 (3) ◽  
pp. 741-758 ◽  
Author(s):  
Milo Viviani
Keyword(s):  
Numerical Methods ◽  
Lie Algebras ◽  
Point Vortex ◽  
Symmetric Matrices ◽  
Symplectic Method ◽  
Heisenberg Spin ◽  
Heisenberg Spin Chain ◽  
Numerical Integrator ◽  
Midpoint Method ◽  
The One

AbstractIsospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie–Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the spherical midpoint method on $${{\mathfrak {s}}}{{\mathfrak {o}}}(3)$$ s o ( 3 ) . In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie–Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature.

scholarly journals New Graphs of Finite Mutation Type

10.37236/863 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Harm Derksen ◽  
Theodore Owen
Keyword(s):  
Directed Graph ◽  
Symmetric Matrix ◽  
Cluster Algebras ◽  
Symmetric Matrices ◽  
Finite Graphs ◽  
Isomorphism Classes ◽  
Geometric Type ◽  
Symmetrizable Matrices ◽  
Marked Points ◽  
Mutation Type

To a directed graph without loops or $2$-cycles, we can associate a skew-symmetric matrix with integer entries. Mutations of such skew-symmetric matrices, and more generally skew-symmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph $\Gamma$ is the set of all isomorphism classes of graphs that can be obtained from $\Gamma$ by a sequence of mutations. A graph is called mutation-finite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutation-finite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs "of geometric type". Besides graphs with $2$ vertices, and graphs of geometric type, there are only 9 other "exceptional" mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

Nucleation of Disorder in Fe3Al Alloys

1967 ◽  
Vol 25 ◽  
pp. 312-313
Author(s):  
P. R. Swann ◽  
W. R. Duff ◽  
R. M. Fisher
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Annealing Temperature ◽  
Antiphase Domain ◽  
Effect Of Temperature ◽  
Domain Structures ◽  
Transmission Electron ◽  
Domain Boundaries ◽  
Higher Temperature ◽  
The One

Recently we have investigated the phase equilibria and antiphase domain structures of Fe-Al alloys containing from 18 to 50 at.% Al by transmission electron microscopy and Mössbauer techniques. This study has revealed that none of the published phase diagrams are correct, although the one proposed by Rimlinger agrees most closely with our results to be published separately. In this paper observations by transmission electron microscopy relating to the nucleation of disorder in Fe-24% Al will be described. Figure 1 shows the structure after heating this alloy to 776.6°C and quenching. The white areas are B2 micro-domains corresponding to regions of disorder which form at the annealing temperature and re-order during the quench. By examining specimens heated in a temperature gradient of 2°C/cm it is possible to determine the effect of temperature on the disordering reaction very precisely. It was found that disorder begins at existing antiphase domain boundaries but that at a slightly higher temperature (1°C) it also occurs by homogeneous nucleation within the domains. A small (∼ .01°C) further increase in temperature caused these micro-domains to completely fill the specimen.

Fundamental Research into Thin Films in a Developing Country

1972 ◽  
Vol 30 ◽  
pp. 500-501
Author(s):  
J.A. Eades ◽  
E. Grünbaum
Keyword(s):  
Thin Films ◽  
Growth Process ◽  
Empirical Process ◽  
Nucleation And Growth ◽  
Research Groups ◽  
Film Research ◽  
Fundamental Research ◽  
Controlled Deposition ◽  
The One ◽  
The University

In the last decade and a half, thin film research, particularly research into problems associated with epitaxy, has developed from a simple empirical process of determining the conditions for epitaxy into a complex analytical and experimental study of the nucleation and growth process on the one hand and a technology of very great importance on the other. During this period the thin films group of the University of Chile has studied the epitaxy of metals on metal and insulating substrates. The development of the group, one of the first research groups in physics to be established in the country, has parallelled the increasing complexity of the field.The elaborate techniques and equipment now needed for research into thin films may be illustrated by considering the plant and facilities of this group as characteristic of a good system for the controlled deposition and study of thin films.

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