Similarity Invariants Research Articles

Shock wave similarity invariants. I. Convergence laws of spherical and cylindrical shock waves in a constant-density gas

Using the similarity and dimensional methods, dimensionless similarity invariants of shock wave similarity are determined, using which convergence laws of strong spherical and cylindrical shock waves in a constant-density gas are determined.

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach— Part 1: The Over-Sampled Case

In Abramovich [“Bounds on Maximum Likelihood Ratio-Part I: Application to Antenna Array Detection-Estimation With Perfect Wavefront Coherence,” IEEE Trans. Signal Process., vol. 52, pp. 1524-1536, June 2004], it was demonstrated, for multivariate complex Gaussian distribution, that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual covariance matrix R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> does not depend on this matrix and is fully specified by the matrix dimension M and the number of independent training samples T. This invariance property hence enables one to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically “as likely” as R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . This “expected likelihood” quality assessment allowed significant improvement of MUSIC DOA estimation performance in the so-called “threshold area,” and for diagonal loading and TVAR model order selection in adaptive detectors. Recently, the so-called complex elliptically symmetric (CES) distributions have been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario (T ≥ M) while Part II deals with the undersampled scenario (T ≤ M).

Multipartite entanglement evolution under separable operations

We study how multipartite entanglement evolves under the paradigm of separable operations, which include the local operations and classical communication as a special case. We prove that the average ``decay'' of entanglement induced by a separable operation is measure independent (among SL-invariant ones) and state independent: The ratio between the average output entanglement and the initial entanglement is solely a function of the separable operation, regardless of the input state and of the SL-invariant entanglement measure being used. We discuss the ``disentangling power'' of a quantum channel and show that it exhibits a similar state invariance as the average entanglement decay ratio. Our Rapid Communication significantly extends the bipartite results of Horodecki and Horodecki, Quantum Inf. Comput. 10, 901 (2010), Konrad et al., Nat. Phys. 4, 99 (2008), and Tiersch et al., Phys. Rev. Lett. 101, 170502 (2008) as well as the multipartite one of Gour, Phys. Rev. Lett. 105, 190504 (2010), all of the previous work being restricted to one-sided or particular noise models.

Similarity invariants of operators on a class of Gowers-Maurey spaces

This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, Σ dc spaces, where an infinite dimensional Banach space X is called a Σ dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T 1 and T 2 on a Σ dc space are similar if and only if the K 0-group of the commutant algebra of the direct sum T 1⊕T 2 is isomorphic to the group of integers ℤ. On a Σ dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in (ΣSI)(X), it further gives a necessary and sufficient condition that two operators in (ΣSI)(X) are similar by using the ordered K 0-groups. It also proves that every operator in (ΣSI)(X) has a unique (SI) decomposition up to similarity on a Σ dc space X, where (ΣSI)(X) denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.

Camera calibration method based on parallelogram similarity invariants

In order to make full use of the characteristics of parallelogram and calibrate the camera online, on the basis of studying the projective properties of a parallelogram, a camera calibration method based on parallelogram similarity invariants is presented. At least three images of the space parallelogram taken from different angles were used to establish the constraint equations on the image of the absolute conic and parallelogram similarity invariants, and the camera's intrinsic and extrinsic parameters were computed by solving the constraint equations. The first two parameters of radial distortion were also estimated according to the projective property of straight line. Experimental results on simulated data and real images demonstrate that the proposed algorithm is effective and flexible.

Thermodynamic time invariances for dual kinetic experiments: Nonlinear single reactions and more

For dual kinetic experiments, i.e., experiments performed in batch or steady-state plug flow reactors from reciprocal initial conditions, thermodynamic (space-)time invariances for all reversible single reactions of the first and second order have been found explicitly.In all analyzed cases, quotient-like functions of concentrations can be defined which equal the equilibrium constant of the reaction during the whole course of the experiment, and not only at the end, i.e., under equilibrium conditions. The obtained invariances can be used as simple fingerprints for distinguishing the types of reactions.For multistep reactions: (a) a similar invariance was obtained for the two-step catalytic reaction (single route complex reaction) under pseudo-steady-state assumption; (b) for the two-step non-steady-state reaction, an approximation for the thermodynamic invariance was found to be valid in two domains, (1) at the very beginning of the reaction (2) at the end of the reaction, near equilibrium conditions. We hypothesize that such two-domain validity of the thermodynamic invariance is a general feature of dual kinetic experiments performed in complex chemical systems.

Isometries and relative entropy preserving maps on density operators

In this article we consider certain kinds of metrics and relative entropies on sets of density operators acting on a finite-dimensional complex Hilbert space. The metrics are the ones induced by the von Neumann–Schatten p-norms. We describe the general form of ‘a priori’ nonsurjective isometries of the space of all density operators with respect to those distances. In the second part of this article we study mappings preserving entropic quantities. We determine the structure of ‘a priori’ nonsurjective maps on the set of all density operators which leave a certain measure of relative entropy invariant and also characterize the surjective maps on the set of all invertible density operators with similar invariance properties.

Löwner evolution driven by a stochastic boundary point

We consider evolution in the unit disk in which the sample paths are represented by the trajectories of points evolving randomly under the generalized Loewner equation. The driving mechanism differs from the SLE evolution, but nevertheless solutions possess similar invariance properties.

Connection between dynamics and thermodynamics of liquids on the melting line

The dynamics of a large number of liquids and polymers exhibit scaling properties characteristic of a simple repulsive inverse power-law potential, most notably the superpositioning of relaxation data as a function of the variable TV^{γ}, where T is temperature, V the specific volume, and γ a material constant. A related scaling law T{m}V{m}{Γ}, with the same exponent Γ = γ, links the melting temperature T{m} and volume V{m} of the model IPL liquid; liquid dynamics is then invariant at the melting point. Motivated by a similar invariance of dynamics experimentally observed at transitions of liquid crystals, we determine dynamic and melting-point scaling exponents γ and Γ for a large number of nonassociating liquids. Rigid, spherical molecules containing no polar bonds have Γ = γ; consequently, the reduced relaxation time, viscosity, and diffusion coefficient are each constant along the melting line. For other liquids γ > Γ always; that is, the dynamics is more sensitive to volume than is the melting point, and for these liquids the dynamics at the melting point slows down with increasing T{m} (that is, increasing pressure).

Conservation laws and invariants of motion for nonlinear internal waves: part II

In this paper, we derive three conservation laws and three invariants of motion for the generalized Gardner equation. These conserved quantities for internal waves are the momentum, energy, and Hamiltonian. The approach used for the derivation of these conservation laws and their associated invariants of motion is direct and does not involve the use of variational principles. It can be easily applied for finding similar invariants of motion for other general types of KdV, Gardner, and Boussinesq equations. The stability and conservation properties of discrete schemes for the simulations of internal waves propagation can be assessed and monitored using the analytical expressions of the constants of motion that are derived in this work.

ASIFT: An Algorithm for Fully Affine Invariant Comparison

If a physical object has a smooth or piecewise smooth boundary, its images obtained by cameras in varying positions undergo smooth apparent deformations. These deformations are locally well approximated by affine transforms of the image plane. In consequencethe solid object recognition problem has often been led back to the computation of affine invariant image local features. The similarity invariance (invariance to translation, rotation, and zoom) is dealt with rigorously by the SIFT method The method illustrated and demonstrated in this work, AffineSIFT (ASIFT), simulates a set of sample views of the initial images, obtainable by varying the two camera axis orientation parameters, namely the latitude and the longitude angles, which are not treated by the SIFT method. Then it applies the SIFT method itself to all images thus generated. Thus, ASIFT covers effectively all six parameters of the affine transform. Source Code The source code (ANSI C), its documentation, and the online demo are accessible at the IPOL web page of this article 1 .

Measure preserving extensions and minimal wandering rates

For conservative ergodic (infinite) measure preserving transformations we show that existence and asymptotics of minimal wandering rates are preserved if we pass to extensions or factors. Consequently, they are similarity invariants.

On Transforming a Spatial Graph into a Plane Graph

This talk is an improved revision of the talk (see [1]) given at the workskp Knots and soft-matter physics, Kyoto, August, 2008 on a complexity of a spatial graph with an emphasis on a transformation of spatial graph into a plane graph. In a research of proteins, molecules, or polymers, it is important to understand geometrically and topologically spatial graphs possibly with degree one vertices including knotted arcs. For every spatial graph without degree one vertices, we introduce the complexity and related topological invariants, called the warping degree, the γ-warping degree, and the (γ,Γ)-warping degree by revising the similar invariants in [1]. The concept of complexity explains a path from any given spatial graph to a plane graph. Similarly, the unknotting number and related concepts γ -unknotting number, Γ -unknotting number, (γ ,Γ )-unknotting number (generalizing the usual unknotting number of a knot) are explained. These invariants are used to define semi-topological invariants for a spatial graph with degree one vertices, meaningful even for a knotted arc.

P122 The use of impulse oscillometry (IOS) to study fractal scaling and sample entropy in airway resistance time series in severe asthma

IntroductionSevere asthma affects airway calibre and can be monitored using IOS. Sample entropy (SampEn) is a measure of complexity and is defined as the probability that sequences of patterns (template...

Quaternion Fourier–Mellin moments for color images

The quaternion Fourier–Mellin moments for describing color images are introduced, which can be seen as the generalization of traditional Fourier–Mellin moments for gray-level images. Then, the quaternion Fourier–Mellin moment invariants are derived, which could be a useful tool in color object recognition tasks that require the similarity invariance. In addition, the problem of color image registration using quaternion Fourier–Mellin moments is discussed. The registration method can match color images differing in rotation and scaling. The advantage of our method is that it can process color image directly, without losing color information. Experimental results validate the effectiveness of the method we proposed.

Barycentric Coordinates on Surfaces

AbstractThis paper introduces a method for defining and efficiently computing barycentric coordinates with respect to polygons on general surfaces. Our construction is geared towards injective polygons (polygons that can be enclosed in a metric ball of an appropriate size) and is based on replacing the linear precision property of planar coordinates by a requirement in terms of center of mass, and generalizing this requirement to the surface setting. We show that the resulting surface barycentric coordinates can be computed using planar barycentric coordinates with respect to a polygon in the tangent plane. We prove theoretically that the surface coordinates properly generalize the planar coordinates and carry some of their useful properties such as unique reconstruction of a point given its coordinates, uniqueness for triangles, edge linearity, similarity invariance, and smoothness; in addition, these coordinates are insensitive to isometric deformations and can be used to reconstruct isometries. We show empirically that surface coordinates are shape‐aware with consistent gross behavior across different surfaces, are well‐behaved for different polygon types/locations on variety of surface forms, and that they are fast to compute. Finally, we demonstrate effectiveness of surface coordinates for interpolation, decal mapping, and correspondence refinement.

Dissipativity preserving balancing for nonlinear systems — A Hankel operator approach

In this paper we present a version of balancing for nonlinear systems which is dissipative with respect to a general quadratic supply rate that depends on the input and the output of the system. We discuss an approach that allows us to apply the theory of balancing based upon Hankel singular value analysis. In order to do that we prove that the available storage and the required supply of the original system are the controllability and the observability functions of a modified, asymptotically stable, system. Then Hankel singular value theory can be applied and the axis singular value functions of the modified system equal the nonlinear extensions of “similarity invariants” obtained from the required supply and available storage of the original system. Furthermore, we also consider an extension of normalized comprime factorizations and relate the available storage and required supply with the controllability and observability functions of the factorizations. The obtained relations are used to perform model order reduction based on balanced truncation, yielding dissipative reduced order models for the original systems. A second order electrical circuit example is included to illustrate the results.

Alternatives to Target Entropy and Alpha Angle in SAR Polarimetry

The purpose of this paper is to discuss two polarimetric parameters which are widely used in synthetic aperture radar (SAR) polarimetry, namely, target entropy and alpha angle. We propose alternative parameters based on our analysis on how they are connected to covariance matrix similarity invariants and how they can be physically interpreted in optical polarimetry. The proposed alternatives can be computed by a fairly simple algorithm and even by the use of software without complex mathematics abilities. As an example, a NASA/Jet Propulsion Laboratory Airborne SAR L-band image of the San Francisco Bay is used to compare the proposed parameter schemes with the original entropy and alpha. A coherent rationale for these alternative parameters is formulated in order to provide insight to polarimetric parameter interpretation.

Dimensionless similarity invariants and criteria of cylindrical Z-pinch compression regimes in a gas

A dimensionless equation containing two dimensionless complexes Z and d0 was derived from the consideration of the energy balance at the early stage of cylindrical Z-pinch development in a gas. The requirement of the equation invariance under the transformation of stretching (compression) of coordinates and time imply that the Ξ and d0 complexes are dimensionless similarity invariants of the indicated discharge. These invariants were shown to be quantitative criteria of realization of single, double, triple, etc. cylindrical Z-pinch plasma column compression regimes.

Similarity class of a matrix with a prescribed submatrix

In this article we study the possible similarity classes of a square matrix when an arbitrary submatrix is prescribed. As the main result, we solve the even more general problem of describing the possible strict equivalence classes of a regular pencil when a subpencil is prescribed. This result improves the result by [Cabral and Silva, Similarity invariants of completions of submatrices, Lin. Alg. Appl. 169 (1992), 151–161.], since an explicit solution is obtained without any existential quantifiers involved, over an algebraically closed field. In fact, the sufficiency of the conditions obtained by [Gohberg, Kaashoek and Van Schangen, Eigenvalues of completions of submatrices, Lin. Multilin. Alg. 25 (1989), 55–70.] is proved. In the proof, we use various results and techniques including matrix pencils completions, Kronecker canonical form, Littlewood–Richardson coefficients, Young diagrams, the solution of the Carlson problem and localization techniques.