Density Argument Research Articles

CLUSTERIZATION OF DATA ARRAYS BASED ON COMBINED OPTIMIZATION OF DISTRIBUTION DENSITY FUNCTIONS AND THE EVOLUTIONARY METHOD OF CAT SWARM

Context. The task of clustering arrays of observations of an arbitrary nature is an integral part of Data Mining, and in the more general case of Data Science, a huge number of approaches have been proposed for its solution, which differ from each other both in a priori assumptions regarding the physical nature of the data and the problem, and in the mathematical apparatus. From a computational point of view, the clustering problem turns into a problem of finding local extrema of a multiextremal function of the vector density argument using gradient procedures that are repeatedly launched from different points of the initial data array. It is possible to speed up the process of searching for these extremes by using the ideas of evolutionary optimization, which includes algorithms inspired by nature, swarm algorithms, population algorithms, etc.
 Objective. The purpose of the work is to introduce a data clustering procedure based on the peaks of the data distribution density and the evolutionary method of cat swarms, that combines the main advantages of methods for working with data in conditions of overlapping classes, is characterized by high-quality clustering, high speed and accuracy of the obtained results.
 Method. The method for clustering data arrays based on the combined optimization of distribution density functions and the evolutionary method of cat swarms was proposed. The advantage of the proposed approach is to reduce the time for solving optimization problems in conditions where clusters are overlap.
 Results. The results of the experiments confirm the effectiveness of the proposed approach in clustering problems under the condition of classes that overlap and allow us to recommend the proposed method for use in practice to solve problems of automatic clustering big data.
 Conclusions. The method for clustering data arrays based on the combined optimization of distribution density functions and the evolutionary method of cat swarm was proposed. The advantage of the proposed approach is to reduce the time for solving optimization problems in conditions where clusters are overlap. The method is quite simple from the numerical implementation and is not critical for choosing an optimization procedure. The experimental results confirm the effectiveness of the proposed approach in clustering problems under conditions of overlapping clusters.

The T1 theorem for the generalized product Calderón-Zygmund operator on product endpoint function spaces over RD spaces

The Littlewood-Paley characterization of the product Lipschitz endpoint function spaces and the density argument in the weak sense is established on product RD spaces, where the measures satisfy the doubling condition and the reverse doubling condition, and the quasi-metric satisfies the Hölder regularities. Furthermore, we get the necessary and sufficient condition $T_1(1)=T_2(1)=0$ for the generalized product Calderón-Zygmund operator on product Lipschitz spaces.

Calderón–Zygmund operators on multiparameter Lipschitz spaces of homogeneous type

Abstract The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of general product singular integral operators introduced by Han, Li and Lin [Y. Han, J. Li and C.-C. Lin, Criterion of the L 2 L^{2} boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 2016, 3, 845–907] on the multiparameter Lipschitz spaces of homogeneous type M ~ = M 1 × ⋯ × M n {\tilde{M}=M_{1}\times\cdots\times M_{n}} . Each factor space M i {M_{i}} , 1 ≤ i ≤ n {1\leq i\leq n} , is a space of homogeneous type in the sense of Coifman and Weiss. These operators generalize those studied by Journé on the Euclidean space and include operators studied by Nagel and Stein on Carnot–Carathéodory spaces. The main tool used here is the discrete Littlewood–Paley–Stein theory and almost orthogonality together with a density argument for the product Lipschitz spaces in the weak sense.

The weak Frenet frame of non-smooth curves with finite total curvature and absolute torsion

We deal with a notion of weak binormal and weak principal normal for non-smooth curves of the Euclidean space with finite total curvature and total absolute torsion. By means of piecewise linear methods, we first introduce the analogous notation for polygonal curves, where the polarity property is exploited, and then make use of a density argument. Both our weak binormal and normal are rectifiable curves which naturally live in the projective plane. In particular, the length of the weak binormal agrees with the total absolute torsion of the given curve. Moreover, the weak normal is the vector product of suitable parameterizations of the tangent indicatrix and of the weak binormal. In the case of smooth curves with positive curvature, the weak binormal and normal yield (up to a lifting) the classical notions of binormal and normal.

Calderón-Zygmund operators on Lipschitz spaces over RD spaces

. Suppose that (X, d, µ) is a metric measure space of homogeneous type in the sense of Coifman and Weiss with a reverse doubling condition. By establishing the Littlewood-Paley characterization of Lipschitz spaces, a density argument in the weak sense and almost orthogonality estimate we prove that a Caldeŕon-Zygmund operator T is bounded on Lipschitz space if and only if T 1 = 0.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We find that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal diffusion are asymptotically compatible. We verify these findings through extensive numerical experiments.

Calderón–Zygmund Operators on Homogeneous Product Lipschitz Spaces

The purpose of this paper is to establish a necessary and sufficient condition for the boundedness of product Calderon–Zygmund singular integral operators introduced by Journe on the product Lipschitz spaces. The key idea used in this paper is to develop the Littlewood–Paley theory for the product spaces which includes the characterization of a special product Besov space and a density argument for the product Lipschitz spaces in the weak sense.

The Boundedness of Calderón–Zygmund Operators on Lipschitz Spaces Over Spaces of Homogeneous Type

Let $$(\mathcal {X},\rho ,\mu )$$ be Ahlfors-1 regular metric spaces. By developing the Littlewood–Paley characterization of Lipschitz spaces over $$(\mathcal {X},\rho ,\mu )$$ and establishing a density argument in the weak sense, the authors give a necessary and sufficient condition for the boundedness of Calderon–Zygmund operators on the Lipschitz spaces.

Phase diagram of antimony up to 31 GPa and 835 K

X-ray powder diffraction experiments using resistively heated diamond anvil cells have been conducted in order to establish the phase behavior of antimony up to 31 GPa and 835 K. The dip in the melting curve at 5.7 GPa and 840 K is identified as the triple point between the Sb-I, incommensurate Sb-II, and liquid phases. No evidence of the previously reported simple cubic phase was observed. Determination of the phase boundary between Sb-II and Sb-III suggests the existence of a second triple point in the region of 13 GPa and 1200 K. The incommensurate composite structure of Sb-II was found to remain ordered to the highest temperatures studies---no evidence of disordering of the guest-atom chains was observed. Indeed, the modulation reflections that arise from interactions between the host and guest subsystems were found to be present to the highest temperatures, suggesting such interactions remain relatively strong in Sb even in the presence of increased thermal motion. Finally, we show that the incommensurately modulated structure recently reported as giving an improved fit to diffraction data from incommensurate Ba-IV can be rejected as the structure of Sb-II using a simple density argument.

The Quiet Eye and Motor Expertise: Explaining the "Efficiency Paradox".

It has been consistently reported that experts show longer quiet eye (QE) durations when compared to near-experts and novices. However, this finding is rather paradoxical as motor expertise is characterized by an economization of motor-control processes rather than by a prolongation in response programming, a suggested explanatory mechanism of the QE phenomenon. Therefore, an inhibition hypothesis was proposed that suggests an inhibition of non-optimal task solutions over movement parametrization, which is particularly necessary in experts due to the great extent and high density of their experienced task-solution space. In the current study, the effect of the task-solution space’ extension was tested by comparing the QE-duration gains in groups that trained a far-aiming task with a small number (low-extent) vs. a large number (high-extent) of task variants. After an extensive training period of more than 750 trials, both groups showed superior performance in post-test and retention test when compared to pretest and longer QE durations in post-test when compared to pretest. However, the QE durations dropped to baseline values at retention. Finally, the expected additional gain in QE duration for the high-extent group was not found and thus, the assumption of long QE durations due to an extended task-solution space was not confirmed. The findings were (by tendency) more in line with the density explanation of the inhibition hypothesis. This density argument suits research revealing a high specificity of motor skills in experts thus providing worthwhile options for future research on the paradoxical relation between the QE and motor expertise.

The Jang Equation Reduction of the Spacetime Positive Energy Theorem in Dimensions Less Than Eight

We extend the Jang equation proof of the positive energy theorem due to R. Schoen and S.-T. Yau from dimension $n=3$ to dimensions $3 \leq n <8$. This requires us to address several technical difficulties that are not present when $n=3$. The regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those of R. Schoen and S.-T. Yau. In recent joint work with L.-H. Huang, D. Lee, and R. Schoen we have given a different proof of the full positive mass theorem in dimensions $3 \leq n < 8$. We pointed out that this theorem can alternatively be derived from our density argument and the positive energy theorem of the present paper.

A Dimensionally Continued Poisson Summation Formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.

Interaction between the intergalactic medium and central radio source in the NGC 4261 group of galaxies

Using observations from the Chandra and XMM–Newton X-ray observatories, we examine the interaction between the intragroup medium and central radio source in the nearby NGC 4261 galaxy group. We confirm the presence of cavities associated with the radio lobes and estimate their enthalpy to be ∼2.4 × 1058 erg. The mechanical power output of the jets is ≥1043 erg s−1, at least a factor of 60 greater than the cooling luminosity in the region the lobes inhabit. We identify rims of compressed gas enclosing the lobes, but find no statistically significant temperature difference between them and their surroundings, suggesting that the lobe expansion velocity is approximately sonic (⁠⁠). The apparent pressure of the radio lobes, based on the synchrotron minimum energy density argument, is a factor of 5 lower than that of the intragroup medium. Pressure balance could be achieved if the entrainment of thermal gas provided additional non-radiating particles in the lobe plasma, but the energy required to heat these particles would be ∼20 per cent of the mechanical energy output of the radio source. NGC 4261 has a relatively compact cool core, which should probably be categorized as a galactic corona. The corona is capable of fuelling the active nucleus for considerably longer than the inferred source lifetime, but can be only inefficiently heated by the active galactic nucleus (AGN) or conduction. The expansion of the radio lobes has affected the structure of the gas in the galaxy, compressing and moving the material of the corona without causing significant shock heating, and expelling gas from the immediate neighbourhood of the jets. We discuss the possible implications of this environment for the duration of the AGN outburst and consider mechanisms which might lead to the cessation of nuclear activity.

Robust estimation in a nonlinear cointegration model

This paper considers the nonparametric M -estimator in a nonlinear cointegration type model. The local time density argument, which was developed by Phillips and Park (1998) [6] and Wang and Phillips (2009) [9], is applied to establish the asymptotic theory for the nonparametric M -estimator. The weak consistency and the asymptotic distribution of the proposed estimator are established under mild conditions. Meanwhile, the asymptotic distribution of the local least squares estimator and the local least absolute distance estimator can be obtained as applications of our main results. Furthermore, an iterated procedure for obtaining the nonparametric M -estimator and a cross-validation bandwidth selection method are discussed, and some numerical examples are provided to show that the proposed methods perform well in the finite sample case.

Young T-dwarf candidates in IC 348

The determination of the lower-end of the initial mass function (IMF) provides strong constraints on star formation theories. We report here on a search for isolated planetary-mass objects in the 3 Myr-old star-forming region IC 348. Deep, narrowband CH4off and CH4on images were obtained with CFHT/WIRCam over 0.11 sq.deg. in the central part of IC 348 to identify young T-dwarfs from their 1.6 um methane absorption bands. We report three faint T-dwarf candidates with CH4on-CH4off colours > 0.4 mag. Extinction was estimated for each candidate and lies in the range Av ~ 5-12 mag. Comparisons with T-dwarf spectral models, and colour/colour and colour/magnitude diagrams, reject two of the three candidates because of their extreme z' - J blueness. The one remaining object is not thought to be a foreground field dwarf because of a number density argument and also its strong extinction Av - 12 mag, or thought to be a background field T-dwarf which would be expected to be much fainter. Models and diagrams give this object a preliminary T6 spectral type. With a few Jupiter masses, the young T-dwarf candidate reported here is potentially amongst the youngest, lowest mass objects detected in a star-forming region so far. Its frequency is consistent with the extrapolation of current lognormal IMF estimates down to the planetary mass domain.

Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities

We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model.

Almost everywhere convergence of a subsequence of the NɆrlund logarithmic means of Walsh-Kaczmarz-Fourier series

The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Walsh-Kaczmarz-Fourier series is of weak type (1,1) . Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Walsh-Kaczmarz-Fourier series is of weak type (1,1) , provided that the supremum in the maximal operator is taken over special indices. The set of Walsh-Kaczmarz polynomials is dense in L1 , so by the well-known density argument the logarithmic means tκ 2n ( f ) converge a.e. to f for all integrable function f . Mathematics subject classification (2000): 42C10.

Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin-Fourier series

The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Vilenkin-Fourier series is of weak type (1, 1). Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Vilenkin-Fourier series is of weak type (1, 1), provided that the supremum in the maximal operator is taken over special indices. The set of Vilenkin polynomials is dense in L1, so by the well-known density argument the logarithmic means t2n(f) converge a.e. to f for all integrable function f. .

ON THE HYPERBOLIC RELAXATION OF THE CAHN–HILLIARD EQUATION IN 3D: APPROXIMATION AND LONG TIME BEHAVIOUR

In this paper we consider the hyperbolic relaxation of the Cahn–Hilliard equation ruling the evolution of the relative concentration u of one component of a binary alloy system located in a bounded and regular domain Ω of ℝ3. This equation is characterized by the presence of the additional inertial term ∊utt that accounts for the relaxation of the diffusion flux. For this equation we address the problem of the long time stability from the point of view of global attractors. The main difficulty in dealing with this system is the low regularity of its weak solutions, which prevents us from proving a uniqueness result and a proper energy identity for the solutions. We overcome this difficulty by using a density argument based on a Faedo–Galerkin approximation scheme and the recent J. M. Ball's theory of generalized semiflows. Moreover, we address the problem of the approximation of the attractor of the continuous problem with the one of Faedo–Galerkin scheme. Finally, we show that the same type of results hold also for the damped semilinear wave equation when the nonlinearity ϕ is not Lipschitz continuous and has a super critical growth.

Is perturbation DFT approach applicable to purely repulsive fluids?

A recently proposed third order + second order perturbation density functional theory (DFT) approach is tested for the validity and applicability to purely repulsive model fluids subjected to various external fields. Hard core repulsive Yukawa potential, point particle Yukawa potential, and inverse power potential are employed as sample models. Theoretical DFT results are compared with the corresponding simulation data obtained by grand canonical ensemble Monte Carlo simulation. This comparison indicates that the third order + second order perturbation DFT approach is suitable for these purely repulsive fluids only on condition of high accuracy of the imported bulk second order direct correlation function (DCF). However, in this case the origin of the successful performance somewhat differs from that observed for the mean field approximation applied to van der Waals fluids. In the present case it originates from the observation that the bulk second order DCF is strongly dependent on the density argument for the hard-core part, while for the distances exceeding the core dimension this dependence is considerably weaker.