Predicting the Critical Number of Layers for Hierarchical Support Vector Regression

Hierarchical support vector regression (HSVR) models a function from data as a linear combination of SVR models at a range of scales, starting at a coarse scale and moving to finer scales as the hierarchy continues. In the original formulation of HSVR, there were no rules for choosing the depth of the model. In this paper, we observe in a number of models a phase transition in the training error—the error remains relatively constant as layers are added, until a critical scale is passed, at which point the training error drops close to zero and remains nearly constant for added layers. We introduce a method to predict this critical scale a priori with the prediction based on the support of either a Fourier transform of the data or the Dynamic Mode Decomposition (DMD) spectrum. This allows us to determine the required number of layers prior to training any models.

The Decoupling Cosmology Theory

By putting forward the basic hypothesis "Energy Possesses No Gravitation", the energy equation and the motion equation of each stage of the development and evolution of the cosmos are obtained by solving the equivalence principle integrally in the flat space or inertial coordinate system. By comprehensive studying the energy equation and motion equation, the expressions of the cosmic critical scale and the initial cluster of nebulae critical scale (namely the galaxies critical scale), were found and given; During the process of matter and energy decoupling, about 3.3 billion initial cluster of nebulae of the critical scale were generated, and the cosmic scale was at least "Expand Expansion (2E)" at 1,300 times long, and the isotropic homogenization was basically realized above the critical scale; by assuming that the cosmos was born in the "Resonance" of the quantum fluctuation of static photons, deriving the early cosmos grew up at the speed of light and produced elementary matter particles at the speed of light, giving the early cosmos "Light-speed Growth Theory (LGT)" without the Big Bang, inflation and singularity theory; It was found that the rotation during the expansion of the gravitational field of the gas matter was the origin of rotational motion and the de-homogenization of anisotropy in galaxies; It was found that the plasma gas photon decoupling energy was the dark energy of accelerated expansion of the cosmos, the missing mass is the illusion of photon decoupling energy beyond the critical scale, the expressions of the photon decoupling energy and the ultimate expansion velocity of the cosmos are derived, and the exact values which are in good agreement with the Hubble constant were given; proposed and proved through multiple evidences that isotropic homogeneous matter field possesses no gravitation; the Hubble constant and cosmic age expression were derived, and the main parameters of the cosmos, such as the critical scale, were given according to the estimation of the relevant motion equation; By macro energy conservation of the cosmos, it was concluded that the gravitational potential energy was the contraction potential formed after the cosmic thermal expansion energy was converted into inertial rotation energy, the thermal expansion energy, the inertial rotation energy and the gravitational potential energy were transformed in turn and equivalent, and predicts that neutrino thermal expansion energy is the main energy of the formation of gravitational potential energy. Finally, by comparing Friedman-Lemaitre solution equal to the energy equations of the various periods of the cosmos, the expressions of curvature constant K and cosmic factor Λ were given, and the "Phenomenon of Gravitational Lens" is explained by the experiment of "Pseudo-gravitational Lens Effect".The core of this theory is through the basic assumption of "Energy Possesses No Gravitation", to separate matter and energy from their motion in coupled study of gravitational field, so it can be named "The Decoupling Cosmology Theory (DCT)". This theory can be supported by Hubble constant and Cosmic Microwave Background (CMB), and can also be verified by the nuances of the isotropic in different directions of the cosmos, the dark matter and critical scales in the galaxies, and the experiments of "Pseudo-gravitational Lens Effect" etc.

An extended theory of gravity in a modified Riemann’s geometry

In this work, we study the evolution of an isotropic universe in an extended theory of gravity obtained geometrically by transforming the normal-gauge Lyra displacement vector field [Formula: see text] as a complex vectorial function depending on a dynamical scalar field [Formula: see text]. By using the latest observational data, we observe that for [Formula: see text] the universe starts accelerating at the critical scale factor [Formula: see text] which corresponds to a redshift of [Formula: see text]. We also find that the dark energy fluid considered in this model is a generalized fluid with equation of state [Formula: see text].

Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes

One of the main goals of this paper is to extend some of the mathematical techniques of some previous papers by the authors showing that some very useful phenomenological properties which can be observed at the nano-scale can be simulated and justified mathematically by means of some homogenization processes when a certain critical scale is used in the corresponding framework. Here the motivating problem in consideration is formulated in the context of the reverse osmosis. We consider, on a part of the boundary of a domain {\Omega\subset\mathbb{R}^{n}} , a set of very small periodically distributed semipermeable membranes having an ideal infinite permeability coefficient (which leads to Signorini-type boundary conditions) on a part {\Gamma_{1}} of the boundary. We also assume that a possible chemical reaction may take place on the membranes. We obtain the rigorous convergence of the problems to a homogenized problem in which there is a change in the constitutive nonlinearities. Changes of this type are the reason for the big success of the nanocomposite materials. Our proof is carried out for membranes not necessarily of radially symmetric shape. The definition of the associated critical scale depends on the dimension of the space (and it is quite peculiar for the special case of {n=2} ). Roughly speaking, our result proves that the consideration of the critical case of the scale leads to a homogenized formulation which is equivalent to having a global semipermeable membrane, at the whole part of the boundary {\Gamma_{1}} , with a “finite permeability coefficient of this virtual membrane”, which is the best we can get, even if the original problem involves a set of membranes of any arbitrary finite permeability coefficients.

Coexistence of magneto-rotational and Jeans instabilities in an axisymmetric nebula

Aims. We analyze the magneto-rotational instability (MRI) effects on gravitational collapse and its influence on the instability critical scale. Methods. In particular, we study an axisymmetric nonstratified differentially rotating cloud, embedded in a small magnetic field, and we perform a local linear stability analysis, including the self gravity of the system. Results. We demonstrate that the linear evolution of the perturbations is characterized by the emergence of an anisotropy degree of the perturbed mass densities. Starting with spherical growing overdensities, we see that they naturally acquire an anisotropy of order unity in their shape. Despite the linear character of our analysis, we infer that such a seed of anisotropy can rapidly grow in a nonlinear regime, leading to the formation of filament-like structures. However, we show how such an anisotropy is essentially an intrinsic feature of the Jean instability, and how MRI only plays a significant role in fixing the critical scale of the mode spectrum. We then provide a characterization of the present analysis in terms of the cosmological setting, in order to provide an outlook of how the present results could concern the formation of large-scale structures across the Universe.