Vortex reconnections in classical and quantum fluids

We review recent rigorous results on the phenomenon of vortex reconnection in classical and quantum fluids. In the context of the Navier–Stokes equations in $$\mathbb {T}^3$$ T 3 we show the existence of global smooth solutions that exhibit creation and destruction of vortex lines of arbitrarily complicated topologies. Concerning quantum fluids, we prove that for any initial and final configurations of quantum vortices, and any way of transforming one into the other, there is an initial condition whose associated solution to the Gross–Pitaevskii equation realizes this specific vortex reconnection scenario. Key to prove these results is an inverse localization principle for Beltrami fields and a global approximation theorem for the linear Schrödinger equation.

Application of improved and optimized fuzzy neural network in classification evaluation of top coal cavability

Longwall top coal caving technology is one of the main methods of thick coal seam mining in China, and the classification evaluation of top coal cavability in longwall top coal caving working face is of great significance for improving coal recovery. However, the empirical or numerical simulation method currently used to evaluate the top coal cavability has high cost and low-efficiency problems. Therefore, in order to improve the evaluation efficiency and reduce evaluation the cost of top coal cavability, according to the characteristics of classification evaluation of top coal cavability, this paper improved and optimized the fuzzy neural network developed by Nauck and Kruse and establishes the fuzzy neural network prediction model for classification evaluation of top coal cavability. At the same time, in order to ensure that the optimized and improved fuzzy neural network has the ability of global approximation that a neural network should have, its global approximation is verified. Then use the data in the database of published papers from CNKI as sample data to train, verify and test the established fuzzy neural network model. After that, the tested model is applied to the classification evaluation of the top coal cavability in 61,107 longwall top coal caving working face in Liuwan Coal Mine. The final evaluation result is that the top coal cavability grade of the 61,107 longwall top coal caving working face in Liuwan Coal Mine is grade II, consistent with the engineering practice.

A two-stage approximation strategy for piecewise smooth functions in two and three dimensions

Given values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The insight used here is that the behavior near the boundaries, or near a singularity curve, is fully characterized and identified by the values of certain differences of the data across the boundary and across the singularity curve. We refer to these values as the signature of $f$. In this paper, we aim at using these values in order to define the approximation. That is, we look for an approximation whose signature is matched to the signature of $f$. Given function data on a grid, assuming the function is piecewise smooth, first, the singularity structure of the function is identified. For example, in the two-dimensional case, we find an approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously, we find the approximations to the different segments of $f$. A system of equations derived from the principle of matching the signature of the approximation and the function with respect to the given grid defines a first stage approximation. A second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.

Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection

We prove the existence of smooth solutions to the Gross–Pitaevskii equation on $$\mathbb {R}^3$$ R 3 that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the $$t^{1/2}$$ t 1 / 2 and change of parity laws. We are mostly interested in solutions tending to 1 at infinity, which have finite Ginzburg–Landau energy and physically correspond to the presence of a background chemical potential, but we also consider the cases of Schwartz initial data and of the Gross–Pitaevskii equation on the torus. In the proof, the Gross–Pitaevskii equation operates in a nearly linear regime, so the result applies to a wide range of nonlinear Schrödinger equations. Indeed, an essential ingredient in the proofs is the development of novel global approximation theorems for the Schrödinger equation on $$\mathbb {R}^n$$ R n . Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime $$D\times \mathbb {R}$$ D × R . This hinges on frequency-dependent estimates for the Helmholtz–Yukawa equation that are of independent interest.

Blending type Approximations by Kantorovich variant of α-Schurer operators

In the present manuscript, we present a new sequence of operators, i:e:, -Bernstein-Schurer-Kantorovich operators depending on two parameters 2 [0; 1] and > 0 foe one and two variables to approximate measurable functions on [0:1+q]; q > 0. Next, we give basic results and discuss the rapidity of convergence and order of approximation for univariate and bivariate of these sequences in their respective sections . Further, Graphical and numerical analysis are presented. Moreover, local and global approximation properties are discussed in terms of rst and second order modulus of smoothness, Peetre’s K-functional and weight functions for these sequences in dierent spaces of functions.

Chaos Theory Methods for Contact-Center Dinamic Control

The work is devoted to searching for optimal control methods for contact center, in particular, methods for predicting the load for further calculation of required number of operators. If the number of operators is always more than required, then the owners of the contact center will incur financial losses. If there are too few employees, the quality of service will decline. Predicting the load of the contact center is required in order to bring the optimal number of operators to work in advance. It is proposed to apply chaos theory to predict the incoming load of a contact center. Positive value of the Lyapunov index indicates the chaotic behavior of the input flow of the load. To predict the load, the methods of linear and nonlinear forecasting and the method of global approximation are used. The paper presents the results of comparing these methods for the problem of predicting the incoming load of contact center.

Attribute-Associated Neuron Modeling and Missing Value Imputation for Incomplete Data

The imputation of missing values is an important research content in incomplete data analysis. Based on the auto associative neural network (AANN), this paper conducts regression modeling for incomplete data and imputes missing values. Since the AANN can estimate missing values in multiple missingness patterns efficiently, we introduce incomplete records into the modeling process and propose an attribute cross fitting model (ACFM) based on AANN. ACFM reconstructs the path of data transmission between output and input neurons and optimizes the model parameters by training errors of existing data, thereby improving its own ability to fit relations between attributes of incomplete data. Besides, for the problem of incomplete model input, this paper proposes a model training scheme, which sets missing values as variables and makes missing value variables update with model parameters iteratively. The method of local learning and global approximation increases the precision of model fitting and the imputation accuracy of missing values. Finally, experiments based on several datasets verify the effectiveness of the proposed method.

Saliva Influence on the Mechanical Properties of Advanced CAD/CAM Composites for Indirect Dental Restorations

This study aims to evaluate the microstructural and mechanical properties of three commercial resin-based materials available for computer-aid design and manufacturing (CAD/CAM)-processed indirect dental restoration: LavaTM Ultimate Restorative (LU), 3M ESPE; Brilliant Crios (BC), COLTENE and CerasmartTM (CS), GC Dental Product. The three types of resin-based composite CAD/CAM materials were physically and mechanically tested under two conditions: directly as received by the manufacturer (AR) and after storage under immersion in artificial saliva (AS) for 30 days. A global approximation to microstructure and mechanical behaviour was evaluated: density, hardness and nanohardness, nanoelastic modulus, flexural strength, fracture toughness, fracture surfaces, and microstructures and fractography. Moreover, their structural and chemical composition using X-ray fluorescence analysis (XRF) and field emission scanning electron microscopy (FESEM) were investigated. As a result, LU exhibited slightly higher mechanical properties, while the decrease of its mechanical performance after immersion in AS was doubled compared to BC and CS. Tests of pristine material showed 13 GPa elastic modulus, 150 MPa flexural strength, 1.0 MPa·m1/2 fracture toughness, and 1.0 GPa hardness for LU, 11.4 GPa elastic modulus; 140 MPa flexural strength, 1.1 MPa·m1/2 fracture toughness, and 0.8 GPa hardness for BC; and 8.3 GPa elastic modulus, 140 MPa flexural strength, 0.9 MPa·m1/2 fracture toughness, and 0.7 GPa hardness for CS. These values were significantly reduced after one month of immersion in saliva. The interpretation of the mechanical results could suggest, in general, a better behaviour of LU compared with the other two despite it having the coarsest microstructure of the three studied materials. The saliva effect in the three materials was critically relevant for clinical use and must be considered when choosing the best solution for the restoration to be used.