Extremal Decomposition Research Articles

On products of inner radii of non-overlapping domains containing certain segment points

This paper is devoted to an open problem: find the maximum of the product ∏ k = 1 n r ( B k , a k ) ( r ( B , a ) is an inner radius of the domain B with respect to a point a) over all non-overlapping domains B k ⊂ C and points a k ∈ B k ∩ [ − 1 ; 1 ] , k = 1 , … , n , where n>4. Also we consider similar problems on extremal decomposition of the complex plane for the case when almost all points are on the real axis and one of the points is infinitely distant, the case when one of the points lies outside the straight line and the case when an inner radius of the domain containing one of the points of a certain segment is taken in some positive real power γ. The purpose of this paper is to obtain sharp estimates for these cases.

Quantifying the intrinsic randomness in sequential measurements

In the standard Bell scenario, when making a local projective measurement on each system component, the amount of randomness generated is restricted. However, this limitation can be surpassed through the implementation of sequential measurements. Nonetheless, a rigorous definition of random numbers in the context of sequential measurements is yet to be established, except for the lower quantification in device-independent scenarios. In this paper, we define quantum intrinsic randomness in sequential measurements and quantify the randomness in the Collins–Gisin–Linden–Massar–Popescu inequality sequential scenario. Initially, we investigate the quantum intrinsic randomness of the mixed states under sequential projective measurements and the intrinsic randomness of the sequential positive-operator-valued measure (POVM) under pure states. Naturally, we rigorously define quantum intrinsic randomness under sequential POVM for arbitrary quantum states. Furthermore, we apply our method to one-Alice and two-Bobs sequential measurement scenarios, and quantify the quantum intrinsic randomness of the maximally entangled state and maximally violated state by giving an extremal decomposition. Finally, using the sequential Navascues–Pironio–Acin hierarchy in the device-independent scenario, we derive lower bounds on the quantum intrinsic randomness of the maximally entangled state and maximally violated state.

Photocatalytic Degradation of Chlorinated Hydrocarbons: The By-Product of the Petrochemical Industry Using Ag-Cu/Graphite Bimetallic Carbon Nitride

In this study, the author improved and modified g-C3N4 by doping it with the metals Ag and Cu, which changed the photochemical properties of g-C3N4, narrowed the band gap, and improved the photocatalytic performance regarding quantum efficiency. Organic hydrocarbons such as 1,2-dichloroethane (DCE) are very stable prepared materials produced as intermediates to obtain polyvinyl chloride, and the prepared photo-catalyst is an innovative method for extreme decomposition of chlorinated hydrocarbons. However, some significant results were obtained using different analysis techniques. X-ray diffraction (XRD) and Fourier-transform infrared spectroscopy (FTIR) showed that the addition of Ag and Cu-NPS partially altered the structure of pure graphitic carbon nitride (g-C3N4-Pure). Scanning electron microscopy (TEM) analysis revealed that the morphological features of Ag-Cu/g-C3N4 contain quantum dots of Ag and Cu nanoparticles in addition to 2d-g-C3N4. The better separation of the photo-generated charge carriers is attributed to better photoactivity in the case of 0.3 g Ag-Cu/g-C3N4 with a reaction time of less than 30 min. Furthermore, the Ag-Cu/g-C3N4 recycling experiment showed that the catalyst remained stable after three stages of the pyrolysis experimental cycle. Another clear indicator of DCE degradation is the measurement using the titration of the Cl ions released by the decomposition.

Extremal decompositions for Nordhaus-Gaddum theorems

A Nordhaus-Gaddum theorem states bounds on p(G)+p(G‾) and p(G)⋅p(G‾) for some graph parameter p(G). We consider the sum upper bound for degeneracy, chromatic number, fractional and circular chromatic number, list chromatic number, span (for L(2,1) labeling), and point partition number. Viewing {G,G‾} as a decomposition of Kn, we describe a strategy to determine the extremal decompositions for these parameters. This produces short proofs of several existing results as well as several new theorems.

The Problem of Extremal Decomposition of a Complex Plane with Free Poles on a Circle

In this paper, the problem concerning the maximum product of the inner radii of n mutually non-overlapping domains symmetric with respect to a unit circle and the inner radius of some domain with respect to the zero raised to a positive degree γ is studied. The problem has been solved for \( n=\overline{4,7} \) and some γ > 1.

Research on gearbox impact feature extraction method based on improved ESMD

A gearbox vibration signal contains non-linear impact characteristics and the significant feature information tends to be overwhelmed by other interference components, which make it difficult to extract the typical fault features fully and effectively. Aiming at the key issue of how to effectively extract the impact characteristics, a fault diagnosis method based on improved extreme symmetric mode decomposition (ESMD) and a support vector machine (SVM) is proposed in this paper. The vibration signal is adaptively decomposed into multiple intrinsic mode function (IMF) components by the improved ESMD and then a certain number of components are selected with the maximum kurtosis-envelope spectrum index. The singular spectral entropy, energy entropy and permutation entropy of each component are applied to construct the feature vector set, in which the dimensionality of the set is reduced with the distance separability criterion. Finally, the dimension-reduced feature vector set is input into the SVM for pattern recognition. Dynamic simulation and experimental gearbox research show that the improved ESMD method can extract and identify gearbox fault information effectively.

The problem of extremal decomposition of a complex plane with free poles on a circle

In this paper, the problem concerning the maximum product of the inner radii of $n$ mutually non-overlapping domains symmetric with respect to a unit circle and the inner radius of some domain with respect to the zero raised to a positive degree $\gamma$ is studied. The problem has been solved for $n=\overline{4,7}$ and some $\gamma>1$.

About one problem on extremal decomposition

About one problem on extremal decomposition

Problem of Extreme Decomposition for a Complex Plane with Free Poles on a Circle

Problem of Extreme Decomposition for a Complex Plane with Free Poles on a Circle

Задачи об экстремальном разбиении для p-гармонических радиусов Робена

The theorems on the extremal decomposition of plane domains concerning to the products of Robin's radii are extended to the case of domains in Euclidean space. In some cases, the classical non-overlapping condition is weakened. The proofs are based on the moduli technique for families of curves and dissymmetrization.

Problem on Extremal Decomposition of the Complex Plane with Free Poles

We consider the well-known problem of the geometric theory of functions of a complex variable on non-overlapping domains with free poles on radial systems. The main results of the present work strengthen and generalize several known results for this problem.

Problem on extremal decomposition of the complex plane with free poles

We consider the well-known problem of the geometric theory of functions of a complex variable on non-overlapping domains with free poles on radial systems. The main results of the present work strengthen and generalize several known results for this problem.

Extremal Decomposition of the Complex Plane with Free Poles II

Problems on extremal decomposition of the complex plane with free poles located on an (n;m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed.

A Problem of Extreme Decomposition of the Complex Plane with Free Poles

We study the problem of nonoverlapping domains with free poles on radial systems. Our main results strengthen and generalize several known results obtained in the investigation of this problem earlier.

Extremal decomposition of the complex plane with free poles

Problems on extremal decomposition of the complex plane with free poles located on an (n;m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed.

Solar power forecasting using robust kernel extreme learning machine and decomposition methods

This paper proposes empirical mode decomposition (EMD)-based robust kernel extreme learning machine (RKELM) to achieve a precise predicted value of solar power generation in a smart grid environment. The non-stationary historical solar power data is initially decomposed into various intrinsic mode functions (IMFs) using EMD, which are subsequently passed through the proposed robust Morlet wavelet kernel extreme learning machine (RWKELM) for solar power prediction at different time horizons. Further a reduced kernel matrix version of RWKELM is used to decrease the training time significantly without appreciable loss of forecasting accuracy. By implementing the real time data for validation of the proposed method for short term solar power prediction it can be observed that the proposed EMD-based RWKELM outperforms various other methods, in terms of different performance matrices and execution time. The solar power prediction results on experimental data show the lowest error which proves the highest prediction accuracy.

The problem of the product of inner radii of four nonoverlapping domains, some of there are symmetric about unit circle

Considered in the paper is one quite general problem of geometric function theory on extremal decomposition of the complex plane, namely to determine the maximum of product of the inner radii of $n$ non-overlapping domains $\{B_{k}\}_{k=1}^{n},$, symmetric with respect to the unit circle, and the power $\gamma$ of the inner radius of a domain $\{B_ {0}\}$, which contains the origin. Starting point of the theory of extremal problems on non-overlapping domains is the result of Lavrent’ev \cite{Lavr} who in 1934 solved the problem of a product of conformal radii of two mutually nonoverlapping simply connected domains. It was the first result of this direction. Goluzin \cite{Goluzin} generalized this problem in the case of an arbitrary finite number of mutually disjoint domains and obtained an accurate evaluation for the case of three domains. Further, Kuzmina \cite{Kuzm} showed that the problem of the evaluation for the case of four domains is reduced to the smallest capacity problems in a certain continuum family and received the exact inequality for $n=4$. For $n\geq5$ full solution of the problem is not obtained at this time. The problem, considered in this paper, stated in \cite{Dubinin-1994} by V.N. Dubinin and earlier in different form by G.P. Bakhtina \cite{Bakhtina-1984}. Let $a_{0}=0$, $|a_{1}|=\ldots=|a_{n}|=1$, $a_{k}\in B_{k}\in \overline{\mathbb{C}}$, where $B_{0},\ldots, B_{n}$ are disjoint domains, and $B_{1},\ldots, B_{n}$ are symmetric about the unit circle. Find the exact upper bound for $r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k,a_k)$, where $r(B_k,a_k)$ is the inner radius of $B_k$ with respect to $a_k$. For $\gamma=1$ and $n\geq2$ this problem was solved by L.V. Kovalev \cite{kovalev-2000,kovalev2-2000} and for $\gamma_{n}=0,38n^{2}$ and $n\geq2$ under the additional assumption that the maximum $\alpha_{0}$ of the angles between neighbouring line segments $[0, a_{k}]$ do not exceed $2\pi/\sqrt{2\gamma}$ it was solved in \cite{BahDenV}. In the present paper this problem is solved for three non-overlapping symmetric domains and for $0<\gamma\leq1.233$ without additional restrictions, moreover, for the first time such $1<\gamma$ are considered for this case. Was proved the lemma, by which it was obtained the estimate of the inner radius of a domain $\{B_ {0}\}$, which contains the origin. Using this lemma and the result of paper \cite{BahDenV}, it was proved that for $\alpha_{0}>2\pi/\sqrt{2\gamma}$ consided product does not exceed some expression.

Extremal decomposition of the complex plane with free poles. II

Problems on extremal decomposition of the complex plane with free poles located on an (n,m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed.

Green energy and extremal decompositions

Green energy and extremal decompositions

Extremal decomposition of the complex plane with free poles

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where \(B_{0}\), ..., \(B_{n}\), \(n\ge 2\), are pairwise disjoint domains in \(\overline{\mathbb{C}}\), \(a_0 = 0\), \(|a_{k}| = 1\), \(k=\overline{1,n}\), and \(\gamma\in (0, n]\) (\(r(B,a)\) is the inner radius of the domain \(B\subset\overline{\mathbb{C}}\) relative to a point \(a\in B\)). For all values of the parameter \(\gamma\in (0, n]\), it is necessary to show that its maximum is attained for a configuration of domains \(B_{k}\) and points \(a_{k}\), \(k=\overline{0,n}\), possessing the \(n\)-fold symmetry. The problem was solved by V.N. Dubinin [1, 2] for \(\gamma=1\) and by G.V. Kuz’mina [4] for \(0 \lt \gamma \lt 1\). L.V. Kovalev [4] obtained its solution for \(n \ge 5\) under the additional assumption that the angles between neighbouring line segments \([0, a_{k}]\) do not exceed \(2\pi /\sqrt{\gamma}\). In particular, this problem will be solved in the present paper for \(n=2\) and \(\gamma\in(1,\,2]\).