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Turbulent systems exhibit a remarkable multi-scale complexity, in which spatial structures induce scale-dependent statistics with strong departures from Gaussianity. In Fourier space, this is reflected by pronounced phase synchronization. A quantitative relation between real-space structure, statistics, and phase synchronization is currently missing. Here, we address this problem in the framework of a minimal phase-coupling model, which enables a detailed investigation by means of dynamical systems theory and multi-scale high-resolution simulations. We identify the spectral power-law steepness, which controls the phase coupling, as the control parameter for tuning the non-Gaussian properties of the system. Whereas both very steep and very shallow spectra exhibit close-to-Gaussian statistics, the strongest departures are observed for intermediate slopes comparable to the ones in hydrodynamic and Burgers turbulence. We show that the non-Gaussian regime of the model coincides with a collapse of the dynamical system to a lower-dimensional attractor and the emergence of phase synchronization, thereby establishing a dynamical-systems perspective on turbulent intermittency., 6 pages, 3 figures. Main changes: Improved introduction and model discussion to emphasize the deterministic system that supports a turbulent attractor. Improved figures and text to introduce a time-dependent order parameter

Evolution algebras are non-associative algebras that describe non-Mendelian hereditary processes and have connections with many other areas. In this paper, we obtain necessary and sufficient conditions for a given algebra A to be an evolution algebra. We prove that the problem is equivalent to the so-called SDC problem, that is, the simultaneous diagonalisation via congruence of a given set of matrices. More precisely we show that an n-dimensional algebra A is an evolution algebra if and only if a certain set of n symmetric n&times, n matrices {M1,&hellip, Mn} describing the product of A are SDC. We apply this characterisation to show that while certain classical genetic algebras (representing Mendelian and auto-tetraploid inheritance) are not themselves evolution algebras, arbitrarily small perturbations of these are evolution algebras. This is intringuing, as evolution algebras model asexual reproduction, unlike the classical ones.

A new algorithm based in Creme Filter, is proposed for breast cancer detection. The images obtained show micro calcifications with better contrast, allowing a better prognosis. The algorithm has only one parameter free, that permitting to observe texture when parameter is changed.

We study numerically the region of convergence of the normal form transformation for the case of the Charney–Hasagawa–Mima (CHM) equation to investigate whether certain finite amplitude effects can be described in normal form coordinates. Specifically, we consider a Galerkin truncation of the CHM equation to four Fourier modes, and ask: Can the finite amplitude phenomenon known as precession resonance (Bustamante et al., Phys. Rev. Lett., vol. 113, 2014, 084502) be described by the normal form variables of wave turbulence theory? The answer is no, due to the failure of convergence of the normal form transformation at the wave amplitudes required for precession resonance. We show this by first characterising precession resonance in the original system, then, searching for this resonance in the normal form equations of motion. We find a large discrepancy between the dynamics of the original system and the transformed normal form system. This prompts an investigation into the convergence of the transformation in the state space of wave amplitudes. We find that the numerically calculated region of convergence is a region of finite wave amplitudes, well beyond the usual limit of weak nonlinearity. However, the amplitudes at the boundary of this region of convergence (i.e. the amplitudes at which the normal form transformation begins to diverge) match closely with the amplitudes at which precession resonance occurs in the original system. We conclude that the precession resonance mechanism cannot be explained by the usual methods of normal forms in wave turbulence theory, so a more general theory for intermediate nonlinearity is required.

In this paper is presented the evaluation of an algorithm for detection of microcalcifications in breast images, allowing us early breast cancer prevention. The Creme Filter is an algorithm with only one parameter “n” that permits to observe the texture dependence when the parameter is changed. The evaluation is done by a cancer expert medical doctor. This algorithm is a new tool for better detecting microcalcifications characteristic of breast cancer.

We provide a solution to the problem of simultaneous $diagonalization$ $via$ $congruence$ of a given set of $m$ complex symmetric $n\times n$ matrices $\{A_{1},\ldots,A_{m}\}$, by showing that it can be reduced to a possibly lower-dimensional problem where the question is rephrased in terms of the classical problem of simultaneous $diagonalization$ $via$ $similarity$ of a new related set of matrices. We provide a procedure to determine in a finite number of steps whether or not a set of matrices is simultaneously diagonalizable by congruence. This solves a long standing problem in the complex case.

In this paper, we study the multi-asset Black–Scholes model from a structural point of view. For this, we interpret the multi-asset Black–Scholes equation as a multidimensional Schrodinger one particle equation. The analysis of the classical Hamiltonian and Lagrangian mechanics associated with this quantum model implies that, in this system, the canonical momentums cannot always be written in terms of the velocities. This feature is a typical characteristic of the constrained system that appears in the high-energy physics. To study this model in the proper form, one must apply Dirac’s method for constrained systems. The results of the Dirac’s analysis indicate that in the correlation parameters space of the multi-assets model, there exists a surface (called the Kummer surface Σ K , where the determinant of the correlation matrix is null) on which the constraint number can vary. We study in detail the cases with N = 2 and N = 3 assets. For these cases, we calculate the propagator of the multi-asset Black–Scholes equation and show that inside the Kummer Σ K surface the propagator is well defined, but outside Σ K the propagator diverges and the option price is not well defined. On Σ K the propagator is obtained as a constrained path integral and their form depends on which region of the Kummer surface the correlation parameters lie. Thus, the multi-asset Black–Scholes model is an example of a variable constrained dynamical system, and it is a new and beautiful property that had not been previously observed.

The Schwabe cycle of solar activity exhibits modulations and frequency fluctuations on slow time scales of centuries and millennia. Plausible physical explanations for the cause of these long-term variations of the solar cycle are still elusive, with possible theories including stochasticity of alpha effect and fluctuations of the differential rotation. It has been suggested recently in the literature that there exists a possible relation between the spatio-temporal structure of Solar cycle and the nonlinear dynamics of magnetohydrodynamic Rossby waves at the solar tachocline, including both wave-wave and wave-mean flow interactions. Here we extend the nonlinear theory of MHD Rossby waves presented in a previous article to take into account long term modulation effects due to a recently discovered mechanism that allows significant energy transfers throughout different wave triads: the precession resonance mechanism. We have found a large number of Rossby-Haurwitz wave triads whose frequency mismatches are compatible with the solar cycle frequency. Consequently, by analyzing the reduced dynamics of two triads coupled by a single mode (five-wave system), we have demonstrated that in the amplitude regime in which precession resonance occurs, the energy transfer throughout the system yields significant long-term modulations on the main $\sim 11$yr period associated with intra-triad energy exchanges. We further show that such modulations display an inverse relationship between the characteristic wave amplitude and the period of intra-triad energy exchanges, which is consistent with the Waldmeier's law for the solar cycle. In the presence of a constant forcing and dissipation, the five-wave system in the precession resonance regime exhibits irregular amplitude fluctuations with some periods resembling the Grand Minimum states.

The dynamics of a spherical body with a non-uniform mass distribution rolling on a plane were discussed by Sergey Chaplygin, whose 150th anniversary we celebrate this year. The Chaplygin top is a non-integrable system, with a colourful range of interesting motions. A special case of this system was studied by Edward Routh, who showed that it is integrable. The Routh sphere has centre of mass offset from the geometric centre, but it has an axis of symmetry through both these points, and equal moments of inertia about all axes orthogonal to the symmetry axis. There are three constants of motion: the total energy and two quantities involving the angular momenta. It is straightforward to demonstrate that these quantities, known as the Jellett and Routh constants, are integrals of the motion. However, their physical significance has not been fully understood. In this paper, we show how the integrals of the Routh sphere arise from Emmy Noether's invariance identity. We derive expressions for the infinitesimal symmetry transformations associated with these constants. We find the finite version of these symmetries and provide their geometrical interpretation. As a further demonstration of the power and utility of this method, we find the Noether symmetries and corresponding Noether integrals for a system introduced recently: the Chaplygin ball on a rotating turntable, confirming that the known integrals are directly obtained from Noether's theorem.

We apply Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider $\alpha$ and $\beta$ Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We demonstrate that resonances are responsible for the irreversible transfer of energy among the Fourier modes. We predict that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity. Our methodology is not limited to only these systems and can be applied to the case of a finite number of modes, such as in the original FPUT experiment, or to the thermodynamic limit, i.e. when the number of modes approach infinity. In the latter limit, we perform state of the art numerical simulations and show that the results are consistent with theoretical predictions. We suggest that the route to thermalization, based only on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.

This article presents the result of applying the Cremme filter to the radiographic images of the knee. In addition, texture indices and the invariant moment of each image are calculated, observing that an index n of the analytical form of the filter has an effect on the resulting images. In particular, the contrast and the ISM, as the univariate moment 4 at n =4 presents a minimum. This can improve the diagnosis for the detection of osteoarthritis of the knee, giving an index that can be studied to optimize the application of the filter.

The recognition of objects and the proper handling of them is a common problem in service robots. This article presents a set of computational vision algorithms to improve an object detection system using an RGBD sensor. Within this set of algorithms is the computational vision algorithm to identify a plane, a second algorithm to determine the position of an object and a third algorithm that performs the Principal Components Analysis to identify the orientation of the objects and to perform a correct manipulation. Several tests are presented that validate our approach. This group of algorithms is used by the service robot Justina at the upcoming RoboCup 2018 international tournament [1].

We report results on the explicit parameterisation of discrete Rossby-wave resonant triads of the Charney-Hasegawa-Mima equation in the small-scale limit (i.e. large Rossby deformation radius), following up from our previous solution in terms of elliptic curves [6]. We find an explicit parameterisation of the discrete resonant wavevectors in terms of two rational variables. We show that these new variables are restricted to a bounded region and find this region explicitly. We argue that this can be used to reduce the complexity of a direct numerical search for discrete triad resonances. Also, we introduce a new direct numerical method to search for discrete resonances. This numerical method has complexity O ( N 3 ) (up to logarithmic factors), where N is the largest wavenumber in the search. We apply this new method to find all discrete irreducible resonant triads in the wavevector box of size 5000, in a calculation that took about 10.5 days on a 16-core machine. Finally, based on our method of mapping to elliptic curves, we discuss some dynamical implications regarding the spread of quadratic invariants across scales via resonant triad interactions, through the analysis of sharp bounds on the relative sizes of the resonantly interacting wavevectors.

In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the exchange of energy between normal modes. In the weakly nonlinear regime, those interactions may be responsible for the equipartition of energy in Fourier space. Here we consider analytically resonant wave-wave interactions for the celebrated Fermi-Pasta-Ulam-Tsingou (FPUT) system. Using a number-theoretical approach based on cyclotomic polynomials, we show that the problem of finding exact resonances for a system of N particles is equivalent to a Diophantine equation whose solutions depend sensitively on the number of particles N, in particular on the set of divisors of N. We provide an algorithm to construct all possible resonances for N particles, based on two basic methods: pairing-off and cyclotomic, which we introduce and use to build up explicit solutions to the 4-wave, 5-wave and 6-wave resonant conditions. Our results shed some light in the understanding of the long-standing FPUT paradox, regarding the sensitivity of the resonant manifolds with respect to the number of particles N and the corresponding time scale of the interactions leading to an eventual thermalisation. In this light we demonstrate that 6-wave resonances always exist for any N, while 5-wave resonances exist if N is divisible by 3 and N ⩾ 9. It is known that in the discrete case 4-wave resonances do not produce energy mixing across the spectrum, so we investigate whether 5-wave resonances can produce energy mixing across a significant region of the Fourier spectrum by looking at the structure of the interconnected network of Fourier modes that can interact nonlinearly via resonances. We obtain that the answer depends on the set of odd divisors of N which are not divisible by 3: the size of this set determines the number of dynamically independent components, corresponding to independent constants of motion (energies). We show that 6-wave resonances connect all these independent components, providing in principle a restoring mechanism for full-scale thermalisation.

Several studies and projects carried out at the University of Valparaiso are focused on the condition of the elderly, particularly those who present conditions of vulnerability and deterioration. In this context, the actions of the academy are consistent with the policies defined by Chilean Government. This is generated by the current conditions of demographic and epidemiological transition, which on the one hand extend people's lives and on the other hand increase morbidity and deterioration in a more chronic than acute way. The potential of biomedical engineering can be proved by through the problem solving of the older adult in subjects of prostration, disability, monitoring of chronic diseases, and in the monitoring of treatments adherence (drugs); together with the development of technical aids for daily living; besides, due the relevance of professional work in several lines such as telemedicine, Home Care, new type of instrumentation specifically for adults, and the management of technologies, greatly impacts the diagnosis, care, treatment and support to a healthy and active aging, of a population sector at a time of greater presence and importance.

In this work, the results of the behavior of Bosso Filter are presented. The filter depends on the different values of parameters a and Λ , characteristics of the Bosso Filter. When the discrete equation is obtained, a third parameter N appears. This parameter corresponds to the number of pixels in the kernel. A possible application is mammography for breast's cancer detection. To analyze the behavior of the filter, IDL Software is used on two dimensional images, to observe useful regions of operation.

We present a theoretical and numerical study of Fourier space triad phase dynamics in one-dimensional stochastically forced Burgers equation at Reynolds number $\mathrm{Re} \approx 2.7 \times 10^4$. We demonstrate that Fourier triad phases over the inertial range display a collective behaviour characterised by intermittent periods of synchronisation and alignment, reminiscent of Kuramoto model (Kuramoto 1984) and directly related to collisions of shocks in physical space. These periods of synchronisation favour efficient energy fluxes across the inertial range towards small scales, resulting in strong bursts of dissipation and enhanced coherence of Fourier energy spectrum. The fast time scale of the onset of synchronisation relegates energy dynamics to a passive role: this is further examined using a reduced system with the Fourier amplitudes fixed in time -- a phase-only model. We show that intermittent triad phase dynamics persists without amplitude evolution and we broadly recover many of the characteristics of the full Burgers system. In addition, for both full Burgers and phase-only systems the physical space velocity statistics reveals that triad phase alignment is directly related to the non-Gaussian statistics typically associated with structure-function intermittency in turbulent systems., Comment: Accepted for publication in Journal of Fluid Mechanics

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock'n'roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.

A common objective in mobile robotics is to find an optimal trajectory to reach a final destination from a starting point location. Obstacles are likely to be encountered in the robot's trajectory that need to be avoided. In this article, a path planning approach for mobile robots that uses the algorithms of potential field in conjunction with Local Minimal Avoidance (LMA) is presented in order to avoid local minima and to achieve an optimal robot's navigation.

We revisit the problem of a triad of resonantly interacting nonlinear waves driven by an external force applied to the unstable mode of the triad. The equations are Hamiltonian, and can be reduced to a dynamical system for 5 real variables with 2 conservation laws. If the Hamiltonian, $H$, is zero we reduce this dynamical system to the motion of a particle in a one-dimensional time-independent potential and prove that the system is integrable. Explicit solutions are obtained for some particular initial conditions. When explicit solution is not possible we present a novel numerical/analytical method for approximating the dynamics. Furthermore we show analytically that when H=0 the motion is generically bounded. That is to say the waves in the forced triad are bounded in amplitude for all times for any initial condition with the single exception of one special choice of initial condition for which the forcing is in phase with the nonlinear oscillation of the triad. This means that the energy in the forced triad generically remains finite for all time despite the fact that there is no dissipation in the system. We provide a detailed characterisation of the dependence of the period and maximum energy of the system on the conserved quantities and forcing intensity. When $H \neq 0$ we reduce the problem to the motion of a particle in a one-dimensional time-periodic potential. Poincar\'{e} sections of this system provide strong evidence that the motion remains bounded when $H \neq 0$ and is typically quasi-periodic although periodic orbits can certainly be found. Throughout our analyses, the phases of the modes in the triad play a crucial role in understanding the dynamics., Comment: 14 pages, 7 figures

One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.

We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point type introduced by Gibbon et al. (Physica D, vol. 132, 1999, pp. 497–510). By employing the method of mapping to regular systems, presented by Bustamante (Physica D, vol. 240 (13), 2011, pp. 1092–1099) and extended to the symmetry-plane case by Mulungye et al. (J. Fluid Mech., vol. 771, 2015, pp. 468–502), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^{2}.$ This late-time regime has an atypical spectrum: it is Gaussian rather than exponential in the wavenumbers. The analyticity-strip width decays to zero in a finite time, albeit so slowly that it remains well above the collocation-point scale for all simulation times $t, where $T^{\ast }$ is the singularity time. Reaching such a proximity to singularity time is not possible in the original temporal variable, because floating-point double precision (${\approx}10^{-16}$) creates a ‘machine-epsilon’ barrier. Due to this limitation on the original independent variable, the mapped variables now provide an improved assessment of the relevant blowup quantities, crucially with acceptable accuracy at an unprecedented closeness to the singularity time: $T^{\ast }-t\approx 10^{-140}$.

ERGA/LERMA Universite Paris VI UMR 8112, 3 rue Galil´ ´ee 94200 Ivry, FranceAbstractWe consider statistical ensembles of Schwarzschild black holes and prove that theseensembles describe black holes of non vanishing temperatures. The mean space-times as-sociated to these ensembles are explored through exact computations of their energy distri-butions, total masses and calorific capacities. We discuss our results, with special emphasison their connections with current and near future observations of astrophysical black holes,string theory and cosmology.Key words: Relativistic statistical physics, black holes.

We present a systematic derivation of the Biot-Savart equation from the Nonlinear Schr\"odinger equation, in the limit when the curvature radius of vortex lines and the inter-vortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines, $$ H= \frac{\kappa^2}{8 \pi}\int_{|{\bf s}-{\bf s}'|>\xi_*} \frac{d{\bf s} \cdot d {\bf s}'}{|{\bf s}-{\bf s}'|} \,,$$ with cut-off length $\xi_* \approx 0.3416293 /\sqrt{\rho_0},$ where $\rho_0$ is the background condensate density far from the vortex lines and $\kappa$ is the quantum of circulation.

We present a phenomenological study of the phase dynamics of the one-dimensional stochastically forced Burgers equation, and of the same equation under a Fourier mode reduction on a fractal set. We study the connection between coherent structures in real space and the evolution of triads in Fourier space. Concerning the one-dimensional case, we find that triad phases show alignments and synchronisations that favour energy fluxes towards small scales --a direct cascade. In addition, strongly dissipative real-space structures are associated with entangled correlations amongst the phase precession frequencies and the amplitude evolution of Fourier triads. As a result, triad precession frequencies show a non-Gaussian distribution with multiple peaks and fat tails, and there is a significant correlation between triad precession frequencies and amplitude growth. Links with dynamical systems approach are briefly discussed, such as the role of unstable critical points in state space. On the other hand, by reducing the fractal dimension $D$ of the underlying Fourier set, we observe: i) a tendency toward a more Gaussian statistics, ii) a loss of alignment of triad phases leading to a depletion of the energy flux, and iii) the simultaneous reduction of the correlation between the growth of Fourier mode amplitudes and the precession frequencies of triad phases.

In this manuscript we present a method for the quantitative comparison of two surfaces, applicable to temporal and/or spatial extent in one or two dimensions. Often surface comparisons are simply overlaid graphs of results from different methodologies that are qualitative at best; it is the purpose of this work to facilitate quantitative evaluation. The surfaces can be analytical, numerical, and/or experimental, and the result returned by the method, termed surface similarity parameter or normalized error, has been normalized so that its value lies between zero and one. When the parameter has a value of zero, the surfaces are in perfect agreement, whereas a value of one is indicative of perfect disagreement. To provide insight regarding the magnitude of the parameter, several canonical cases are presented, followed by results from breaking water wave experimental measurements with numerical simulations, and by a comparison of a prescribed, periodic, square-wave surface profile and the subsequent manufactured surface.

Motivated by the work on stagnation-point-type exact solutions (with infinite energy) of 3D Euler fluid equations by Gibbon et al. (Physica D, vol. 132 (4), 1999, pp. 497–510) and the subsequent demonstration of finite-time blowup by Constantin (Int. Math. Res. Not. IMRN, vol. 9, 2000, pp. 455–465) we introduce a one-parameter family of models of the 3D Euler fluid equations on a 2D symmetry plane. Our models are seen as a deformation of the 3D Euler equations which respects the variational structure of the original equations so that explicit solutions can be found for the supremum norms of the basic fields: vorticity and stretching rate of vorticity. In particular, the value of the model’s parameter determines whether or not there is finite-time blowup, and the singularity time can be computed explicitly in terms of the initial conditions and the model’s parameter. We use a representative of this family of models, whose solution blows up at a finite time, as a benchmark for the systematic study of errors in numerical simulations. Using a high-order pseudospectral method, we compare the numerical integration of our ‘original’ model equations against a ‘mapped’ version of these equations. The mapped version is a globally regular (in time) system of equations, obtained via a bijective nonlinear mapping of time and fields from the original model equations. The mapping can be constructed explicitly whenever a Beale–Kato–Majda type of theorem is available therefore it is applicable to the 3D Euler equations (Bustamante, Physica D, vol. 240 (13), 2011, pp. 1092–1099). We show that the mapped system’s numerical solution leads to more accurate (by three orders of magnitude) estimates of supremum norms and singularity time compared with the original system. The numerical integration of the mapped equations is demonstrated to entail only a small extra computational cost. We study the Fourier spectrum of the model’s numerical solution and find that the analyticity strip width (a measure of the solution’s analyticity) tends to zero as a power law in a finite time. This is in agreement with the finite-time blowup of the fields’ supremum norms, in the light of rigorous bounds stemming from the bridge (Bustamante & Brachet, Phys. Rev. E, vol. 86 (6), 2012, 066302) between the analyticity-strip method and the Beale–Kato–Majda type of theorems. We conclude by discussing the implications of this research on the analysis of numerical solutions to the 3D Euler fluid equations.

A robust energy transfer mechanism is found in nonlinear wave systems, which favours transfers towards modes interacting via triads with nonzero frequency mismatch, applicable in meteorology, nonlinear optics and plasma wave turbulence. We introduce the concepts of truly dynamical degrees of freedom and triad precession. Transfer efficiency is maximal when the triads' precession frequencies resonate with the system's nonlinear frequencies, leading to a collective state of synchronised triads with strong turbulent cascades at intermediate nonlinearity. Numerical simulations confirm analytical predictions., Comment: This paper supersedes arXiv:1305.5517 [physics.flu-dyn]

It is shown that a transversal potential appears when a current flows along a film with a transversal impurity gradient. The potential is evaluated in the relaxation time approximation for the Boltzmann-Bloch equation, being quadratic in the applied field. Semi-classical Monte Carlo simulations have verified those results.