Extending the Applicability and Convergence Domain of a Fifth-Order Iterative Scheme under Hölder Continuous Derivative in Banach Spaces

The most significant contribution made by this study is that the applicability and convergence domain of a fifth-order convergent nonlinear equation solver is extended. We use Hölder condition on the first Fréchet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. This study generalizes the local analysis based on Lipschitz constants. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the involved operator. Finally, numerical experiments ensure that our analysis expands the utility of the considered scheme. In addition, the proposed technique produces a larger convergence domain, in comparison to the earlier study, without using any extra conditions.

Dual Memory LSTM with Dual Attention Neural Network for Spatiotemporal Prediction

Spatiotemporal prediction is challenging due to extracting representations being inefficient and the lack of rich contextual dependences. A novel approach is proposed for spatiotemporal prediction using a dual memory LSTM with dual attention neural network (DMANet). A new dual memory LSTM (DMLSTM) unit is proposed to extract the representations by leveraging differencing operations between the consecutive images and adopting dual memory transition mechanism. To make full use of historical representations, a dual attention mechanism is designed to capture long-term spatiotemporal dependences by computing the correlations between the current hidden representations and the historical hidden representations from temporal and spatial dimensions, respectively. Then, the dual attention is embedded into DMLSTM unit to construct a DMANet, which enables the model with greater modeling power for short-term dynamics and long-term contextual representations. An apparent resistivity map (AR Map) dataset is proposed in this paper. The B-spline interpolation method is utilized to enhance AR Map dataset and makes apparent resistivity trend curve continuous derivative in the time dimension. The experimental results demonstrate that the developed method has excellent prediction performance by comparisons with some state-of-the-art methods.

APPLICATION OF B-SPLINE METHOD IN SURFACE FITTING PROBLEM

Fitting a smooth surface on irregular data is a problem in many applications of data analysis. Spline polynomials in different orders have been used for interpolation and approximation in one or two-dimensional space in many researches. These polynomials can be made by different degrees and they have continuous derivative at the boundaries. The advantage of using B-spline basis functions for obtaining spline polynomials is that they impose the continuity constraints in an implicit form and, more importantly, their calculation is much simpler. In this study, we explain the theory of the least squares B-spline method in surface approximation. Furthermore, we present numerical examples to show the efficiency of the method in linear, quadratic and cubic forms and it’s capability in modeling changes in numerical values. This capability can be used in different applications to represent any natural phenomenon which can’t be experienced by humans directly. Lastly, the method’s accuracy and reliability in different orders will be discussed.

The Loewner Energy of Loops and Regularity of Driving Functions

Loewner driving functions encode simple curves in 2D simply connected domains by real-valued functions. We prove that the Loewner driving function of a $C^{1,\beta }$ curve (differentiable parametrization with $\beta$-Hölder continuous derivative) is in the class $C^{1,\beta -1/2}$ if $1/2<\beta \leq 1$, and in the class $C^{0,\beta + 1/2}$ if $0 \leq \beta \leq 1/2$. This is the converse of a result of Carto Wong [26] and is optimal. We also introduce the Loewner energy of a rooted planar loop and use our regularity result to show the independence of this energy from the basepoint.

The approximation of piecewise smooth functions by trigonometric Fourier sums

We obtain exact order-of-magnitude estimates of piecewise smooth functions approximation by trigonometric Fourier sums. It is shown that in continuity points Fourier series of piecewise Lipschitz function converges with rate $\ln n/n$. If function $f$ has a piecewise absolutely continuous derivative then it is proven that in continuity points decay order of Fourier series remainder $R_n(f,x)$ for such function is equal to $1/n$. We also obtain exact order-of-magnitude estimates for $q$-times differentiable functions with piecewise smooth $q$-th derivative. In particular, if $f^{(q)}(x)$ is piecewise Lipschitz then $|R_n(f,x)| \le c(x)\frac{\ln n}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$ and $\sup_{x \in [0,2\pi]}|R_n(f,x)| \le \frac{c}{n^q}$. In case when $f^{(q)}(x)$ has piecewise absolutely continuous derivative it is shown that $|R_n(f,x)| \le \frac{c(x)}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$. As a consequence of the last result convergence rate estimate of Fourier series to continuous piecewise linear functions is obtained.

Extension of the discrete universality theorem for zeta-functions of certain cusp forms

In the paper, an universality theorem on the approximation of analytic functions by generalized discrete shifts of zeta functions of Hecke-eigen cusp forms is obtained. These shifts are defined by using the function having continuous derivative satisfying certain natural growth conditions and, on positive integers, uniformly distributed modulo 1.