Is multidimensional scaling suitable for mapping the input respiratory impedance in subjects and patients?

This paper presents the application of multidimensional scaling (MDS) analysis to data emerging from noninvasive lung function tests, namely the input respiratory impedance. The aim is to obtain a geometrical mapping of the diseases in a 3D space representation, allowing analysis of (dis)similarities between subjects within the same pathology groups, as well as between the various groups. The adult patient groups investigated were healthy, diagnosed chronic obstructive pulmonary disease (COPD) and diagnosed kyphoscoliosis, respectively. The children patient groups were healthy, asthma and cystic fibrosis. The results suggest that MDS can be successfully employed for mapping purposes of restrictive (kyphoscoliosis) and obstructive (COPD) pathologies. Hence, MDS tools can be further examined to define clear limits between pools of patients for clinical classification, and used as a training aid for medical traineeship.

Scaling of semivariograms and the kriging estimation of field-measured properties

Two methods were evaluated for scaling a set of semivariograms into a unified function for kriging estimation of field-measured properties. Scaling is performed using sample variances and sills of individual semivariograms as scale factors. Theoretical developments show that kriging weights are independent of the scaling factor which appears simply as a constant multiplying both sides of the kriging equations. The scaling techniques were applied to four sets of semivariograms representing spatial scales of 30 x 30 m to 600 x 900 km. Experimental semivariograms in each set successfully coalesced into a single curve by variances and sills of individual semivariograms. To evaluate the scaling techniques, kriged estimates derived from scaled semivariogram models were compared with those derived from unscaled models. Differences in kriged estimates of the order of 5% were found for the cases in which the scaling technique was not successful in coalescing the individual semivariograms, which also means that the spatial variability of these properties is different. The proposed scaling techniques enhance interpretation of semivariograms when a variety of measurements are made at the same location. They also reduce computational times for kriging estimations because kriging weights only need to be calculated for one variable. Weights remain unchanged for all other variables in the data set whose semivariograms are scaled.

Precision and scaling in morphogen gradient read-out.

Morphogen gradients infer cell fate as a function of cellular position. Experiments in Drosophila embryos have shown that the Bicoid (Bcd) gradient is precise and exhibits some degree of scaling. We present experimental results on the precision of Bcd target genes for embryos with a single, double or quadruple dose of bicoid demonstrating that precision is highest at mid-embryo and position dependent, rather than gene dependent. This confirms that the major contribution to precision is achieved already at the Bcd gradient formation. Modeling this dynamic process, we investigate precision for inter-embryo fluctuations in different parameters affecting gradient formation. Within our modeling framework, the observed precision can only be achieved by a transient Bcd profile. Studying different extensions of our modeling framework reveals that scaling is generally position dependent and decreases toward the posterior pole. Our measurements confirm this trend, indicating almost perfect scaling except for anterior most expression domains, which overcompensate fluctuations in embryo length.

Solid phase crystallization under continuous heating: kinetic and microstructure scaling laws

The kinetics and microstructure of solid-phase crystallization under continuous heating conditions and random distribution of nuclei are analyzed. An Arrhenius temperature dependence is assumed for both nucleation and growth rates. Under these circumstances, the system has a scaling law such that the behavior of the scaled system is independent of the heating rate. Hence, the kinetics and microstructure obtained at different heating rates differ only in time and length scaling factors. Concerning the kinetics, it is shown that the extended volume evolves with time according to αex = [exp(κCt′)]m+1, where t′ is the dimensionless time. This scaled solution not only represents a significant simplification of the system description, it also provides new tools for its analysis. For instance, it has been possible to find an analytical dependence of the final average grain size on kinetic parameters. Concerning the microstructure, the existence of a length scaling factor has allowed the grain-size distribution to be numerically calculated as a function of the kinetic parameters

Nonlinear principal and canonical directions from continuous extensions of multidimensional scaling

A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.

On convergence of transforms based on parabolic scaling

This thesis studies properties of transforms based on parabolic scaling, like Curvelet-, Contourlet-, Shearlet- and Hart-Smith-transform. Essentially, two di erent questions are considered: How these transforms can characterize H older regularity and how non-linear approximation of a piecewise smooth function converges. In study of Hölder regularities, several theorems that relate regularity of a function f : R2 → R to decay properties of its transform are presented. Of particular interest is the case where a function has lower regularity along some line segment than elsewhere. Theorems that give estimates for direction and location of this line, and regularity of the function are presented. Numerical demonstrations suggest also that similar theorems would hold for more general shape of segment of low regularity. Theorems related to uniform and pointwise Hölder regularity are presented as well. Although none of the theorems presented give full characterization of regularity, the su cient and necessary conditions are very similar. Another theme of the thesis is the study of convergence of non-linear M ─term approximation of functions that have discontinuous on some curves and otherwise are smooth. With particular smoothness assumptions, it is well known that squared L2 approximation error is O(M-2(logM)3) for curvelet, shearlet or contourlet bases. Here it is shown that assuming higher smoothness properties, the log-factor can be removed, even if the function still is discontinuous.

A new scale-invariant ratio and finite-size scaling for the stochastic susceptible-infected-recovered model

The critical behavior of the stochastic susceptible-infected-recovered model on a square lattice is obtained by numerical simulations and finite-size scaling. The order parameter as well as the distribution in the number of recovered individuals is determined as a function of the infection rate for several values of the system size. The analysis around criticality is obtained by exploring the close relationship between the present model and standard percolation theory. The quantity UP, equal to the ratio U between the second moment and the squared first moment of the size distribution multiplied by the order parameter P, is shown to have, for a square system, a universal value 1.0167(1) that is the same for site and bond percolation, confirming further that the SIR model is also in the percolation class.

Universal scaling in Bardeen-Cooper-Schrieffer superconductivity in two dimensions in non-s waves

The solutions of a renormalized BCS model are studied in two space dimensions for s, p and d waves for finite-range separable potentials. The gap parameter, the critical temperature T-c, the coherence length xi and the jump in specific heat at T-c as a function of the zero-temperature condensation energy exhibit universal scalings. In the weak-coupling limit, the present model yields a small xi and large T-c, appropriate for high-T-c cuprates. The specific heat, penetration depth and thermal conductivity as functions of temperature show universal scaling for p and d waves.

Semiclassical scaling functions of sine-Gordon model

We present an analytic study of the finite size effects in sine-Gordon model, based on the semi-classical quantization of an appropriate kink background defined on a cylindrical geometry. The quasi-periodic kink is realized as an elliptic function with its real period related to the size of the system. The stability equation for the small quantum fluctuations around this classical background is of Lame type and the corresponding energy eigenvalues are selected inside the allowed bands by imposing periodic boundary conditions. We derive analytical expressions for the ground state and excited states scaling functions, which provide an explicit description of the flow between the IR and UV regimes of the model. Finally, the semiclassical form factors and two-point functions of the basic field and of the energy operator are obtained, completing the semiclassical quantization of the sine-Gordon model on the cylinder. (C) 2004 Elsevier B.V. All rights reserved.

Universal scaling in a trapped Fermi super-fluid in the BCS-unitarity crossover

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Scaling and universality in two dimensions: three-body bound states with short-ranged interactions

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Scaling dynamics for a particle in a time-dependent potential well

Some dynamical properties for a classical particle confined in an infinitely deep box of potential containing a periodically oscillating square well are studied. The dynamics of the system is described by using a two-dimensional non-linear area-preserving map for the variables energy and time. The phase space is mixed and the chaotic sea is described using scaling arguments. Scaling exponents are obtained as a function of all the control parameters, extending the previous results obtained in the literature. (c) 2012 Elsevier B.V. All rights reserved.

A family of stadium-like billiards with parabolic boundaries under scaling analysis

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Statistical properties of a dissipative kicked system: Critical exponents and scaling invariance

A new universal empirical function that depends on a single critical exponent (acceleration exponent) is proposed to describe the scaling behavior in a dissipative kicked rotator. The scaling formalism is used to describe two regimes of dissipation: (i) strong dissipation and (ii) weak dissipation. For case (i) the model exhibits a route to chaos known as period doubling and the Feigenbaum constant along the bifurcations is obtained. When weak dissipation is considered the average action as well as its standard deviation are described using scaling arguments with critical exponents. The universal empirical function describes remarkably well a phase transition from limited to unlimited growth of the average action. (C) 2012 Elsevier B.V. All rights reserved.

Dynamical properties of a particle in a wave packet: Scaling invariance and boundary crisis

Some dynamical properties present in a problem concerning the acceleration of particles in a wave packet are studied. The dynamics of the model is described in terms of a two-dimensional area preserving map. We show that the phase space is mixed in the sense that there are regular and chaotic regions coexisting. We use a connection with the standard map in order to find the position of the first invariant spanning curve which borders the chaotic sea. We find that the position of the first invariant spanning curve increases as a power of the control parameter with the exponent 2/3. The standard deviation of the kinetic energy of an ensemble of initial conditions obeys a power law as a function of time, and saturates after some crossover. Scaling formalism is used in order to characterise the chaotic region close to the transition from integrability to nonintegrability and a relationship between the power law exponents is derived. The formalism can be applied in many different systems with mixed phase space. Then, dissipation is introduced into the model and therefore the property of area preservation is broken, and consequently attractors are observed. We show that after a small change of the dissipation, the chaotic attractor as well as its basin of attraction are destroyed, thus leading the system to experience a boundary crisis. The transient after the crisis follows a power law with exponent -2. (C) 2011 Elsevier Ltd. All rights reserved.