Functional Taylor Expansion Research Articles

A novel semi-analytic algorithm for evaluation of nearly singular integrals in boundary element analysis

A novel semi-analytic method is proposed to evaluate various nearly singular integrals accurately. Different from the traditional semi-analytical method, the Taylor expansion for shape functions, Jacobian, and so on in our method is based on the nearest point from a source point to an integral element rather than the projection point from the source point to the integral element. Therefore, it can more accurately approximate the distance from the source point to Gaussian integration points. Then, approximate expressions for the integrand functions of two-dimensional potential problems are derived. The effectiveness of the proposed method is verified by evaluating the calculation accuracy of physical quantities at points in different geometric models, and comparing it with the traditional semi-analytical method and distance transform method.

FRACTIONAL TAYLOR EXPANSION OF CLASSICAL AND LAGUERRE-TYPE FUNCTIONS

Beginning with the fractional exponential, which is frequently used in applications, fractional Taylor expansions of several elementary functions in both the ordinary and Laguerrian cases are introduced, showing that some basic formulas continue to apply to new expansions. Some graphs obtained using the computer algebra system Mathematica© are given for illustration purposes.

Asymptotic expansion of solutions for the Robin-Dirichlet problem of Kirchhoff-Carrier type with Balakrishnan-Taylor damping

In this paper, we consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type with Balakrishnan-Taylor damping. First, under suitable conditions on the initial data, the local existence and uniqueness of a weak solution are proved. Next, an asymptotic expansion of solutions in a small parameter with high order is established. The used main tools are the linearization method for nonlinear terms together with the Faedo-Galerkin method, and the key lemmas of the expansion of high-order polynomials and the Taylor expansion for multi-variable functions.

Kinetic derivation of Cahn-Hilliard fluid models

A compressible Cahn-Hilliard fluid model is derived from the kinetic theory of dense gas mixtures. The fluid model involves a van der Waals and Cahn-Hilliard gradient energy, a generalized Korteweg's tensor, a generalized Dunn and Serrin heat flux, and Cahn-Hilliard-type diffusive fluxes. Starting from the BBGKY hierarchy for gas mixtures, a Chapman-Enskog method is used-with a proper scaling of the generalized Boltzmann equations-as well as higher-order Taylor expansions of pair distribution functions. A Euler and van der Waals model is obtained at zeroth order, while the Cahn-Hilliard fluid model is obtained at first order, involving viscous, heat, and diffusive fluxes. The Cahn-Hilliard extra terms are associated with intermolecular forces and pair interaction potentials.

A Fast and Simple Face Detection Algorithm Using Neural Network and Its Implementation on FPGA

Face recognition is one of the interesting types of biometric which determines the presence or absence of human faces in the picture. In this paper, a face recognition system is presented that benefits from an optimized architecture based on the MLP neural network. The proposed method considerably improves the speed and the accuracy of detection compared to traditional architectures of neural network. To reduce the overall computation, neural network is organized so that to be able to rule out the majority of the non-image areas located in the image’s background before applying the main algorithm. An important advantage of this new architecture is its homogeneous structure that makes it suitable for optimized implementation on a hardware platform. In this work, FPGA is used as the platform for implementation of the proposed algorithms. The implementation was done considering Taylor expansion of the activation functions. The performance of the proposed method and the implemented system was evaluated on the BioID dataset. Accomplishment of the proposed method is high precision while reducing training time and total calculations, together with appropriate robustness. Finally, a comparison with other face recognition methods has been done to show the performance of the presented system. The comparison result shows that the proposed system outperforms the other mentioned methods.

Neural network perturbation theory and its application to the Born series

Deep Learning using the eponymous deep neural networks (DNNs) has become an attractive approach towards various data-based problems of theoretical physics in the past decade. There has been a clear trend to deeper architectures containing increasingly more powerful and involved layers. Contrarily, Taylor coefficients of DNNs still appear mainly in the light of interpretability studies, where they are computed at most to first order. However, especially in theoretical physics numerous problems benefit from accessing higher orders, as well. This gap motivates a general formulation of neural network (NN) Taylor expansions. Restricting our analysis to multilayer perceptrons (MLPs) and introducing quantities we refer to as propagators and vertices, both depending on the MLP's weights and biases, we establish a graph-theoretical approach. Similarly to Feynman rules in quantum field theories, we can systematically assign diagrams containing propagators and vertices to the corresponding partial derivative. Examining this approach for S-wave scattering lengths of shallow potentials, we observe NNs to adapt their derivatives mainly to the leading order of the target function's Taylor expansion. To circumvent this problem, we propose an iterative NN perturbation theory. During each iteration we eliminate the leading order, such that the next-to-leading order can be faithfully learned during the subsequent iteration. After performing two iterations, we find that the first- and second-order Born terms are correctly adapted during the respective iterations. Finally, we combine both results to find a proxy that acts as a machine-learned second-order Born approximation.

On three-stage temperature dependence of elastic wave velocities for rocks

Abstract For most rocks, the typical temperature behavior of elastic wave velocities generally features a three-stage nonlinear characteristic that could be expressed by a reverse S-shape curve with two inflexion points. The mechanism regulating the slow-to-fast transition of elastic constants remains elusive. The physics of critical points seems related to the multimineral composition of rocks with differentiated thermodynamic properties. Based on laboratory experiments for several rocks with different levels of heterogeneity in compositions, we conduct theoretical and empirical simulations by nonlinear thermoelasticity methods and a S-shape model, respectively. The classical theory of linear thermoelasticity based on the Taylor expansion of strain energy functions has been widely used for crystals, but suffers from a deficiency in describing thermal-associated velocity variations for rocks as a polycrystal mixture. Current nonlinear thermoelasticity theory describes the third-order temperature dependence of velocity variations by incorporating the fourth-order elastic constants. It improves the description of temperature-induced three-stage velocity variations in rocks, but involves with some divergences around two inflexion points, especially at high temperatures. The S-shape model for empirical simulations demonstrates a more accurate depiction of thermal-associated three-stage variations of P-wave velocities. We investigate the physics of the parameters ${a_1}$ and ${b_1}$ in the S-shape model. These fitting parameters are closely related to thermophysical properties by being proportional to the specific heat and thermal conductivity of rocks. We discuss the mechanism that regulates the slow-to-fast transition in the three-stage nonlinear behavior for various rocks.

Thermodynamic consistency by a modified Perkus–Yevick theory using the Mittag-Leffler function

Closure relations that satisfies thermodynamical consistencies are very important, for it implies physical consistent Ornstein–Zernike equation solutions. The method proposed by Percus (1962), which uses the functional Taylor expansion to obtain closure relations, is presented along with the particular generating functional considered to retrieve the Percus–Yevick (PY) approximation. The main assumption for this particular case, which takes into account the low density limit, is discussed. Based on this argument, it was proposed a modified generating functional with no prior considerations about the system density. As an example, the two-parameters Mittag-Leffler function was used. Since it generalizes the exponential function, the parameters determined at low densities retrieves the PY approximation. For higher densities it was possible to obtain parameters which lead to pressure consistent results, in good agreement with those found in literature.

Radial Distribution Function for a Hard Sphere Liquid: A Modified Percus-Yevick and Hypernetted-Chain Closure Relations

Establishment of the radial distribution function by solving the Ornstein-Zernike equation is still an important problem, even more than a hundred years after the original paper publication. New strategies and approximations are common in the literature. A crucial step in this process consists in defining a closure relation which retrieves correlation functions in agreement with experiments or molecular simulations. In this paper, the functional Taylor expansion, as proposed by J. K. Percus, is applied to introduce two new closure relations: one that modifies the Percus‑Yevick closure relation and another one modifying the Hypernetted-Chain approximation. These new approximations will be applied to a hard sphere system. An improvement for the radial distribution function is observed in both cases. For some densities a greater accuracy, by a factor of five times compared to the original approximations, was obtained.

Efficient Taylor expansion computation of multidimensional vector functions on GPU

The Taylor expansion [19] is used in many applications for a value estimation of scalar functions of one or two variables in the neighbour point. Usually, only the first two elements of the Taylor expansion are used, i.e. a value in the given point and derivatives estimation. The Taylor expansion can be also used for vector functions, too. The usual formulae are well known, but if the second element of the expansion, i.e. with the second derivatives are to be used, mathematical formulations are getting too complex for efficient programming, as it leads to the use of multi-dimensional matrices. This contribution describes a new form of the Taylor expansion for multidimensional vector functions. The proposed approach uses “standard” formalism of linear algebra, i.e. using vectors and matrices, which is simple, easy to implement. It leads to efficient computation on the GPU in the three dimensional case, as the GPU offers fast vector-vector computation and many parts can be done in parallel.

The motivic zeta functions of a matroid

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincar\'{e} duality in the sense of Adiprasito-Huh-Katz. In the process, we obtain a formula for the Hilbert series of the cohomology ring of a matroid, in the sense of Feichtner-Yuzvinsky. We then show that our motivic zeta functions specialize to the topological zeta functions for matroids introduced by van der Veer, and we compute the first two coefficients in the Taylor expansion of these topological zeta functions, providing affirmative answers to two questions posed by van der Veer.

On the Taylor Expansion of Value Functions

We introduce a framework for approximate dynamic programming that we apply to discrete time chains on $\mathbb{Z}_+^d$ with countable action sets. Our approach is grounded in the approximation of the (controlled) chain's generator by that of another Markov process. In simple terms, our approach stipulates applying a second-order Taylor expansion to the value function to replace the Bellman equation with one in continuous space and time where the transition matrix is reduced to its first and second moments. In some cases, the resulting equation (which we label {\bf TCP}) can be interpreted as corresponding to a Brownian control problem. When tractable, the TCP serves as a useful modeling tool. More generally, the TCP is a starting point for approximation algorithms. We develop bounds on the optimality gap---the sub-optimality introduced by using the control produced by the Taylored equation. These bounds can be viewed as a conceptual underpinning, analytical rather than relying on weak convergence arguments, for the good performance of controls derived from Brownian control problems. We prove that, under suitable conditions and for suitably large initial states, (i) the optimality gap is smaller than a $1-\alpha$ fraction of the optimal value, where $\alpha\in (0,1)$ is the discount factor, and (ii) the gap can be further expressed as the infinite horizon discounted value with a lower-order per period reward. Computationally, our framework leads to an aggregation approach with performance guarantees. While the guarantees are grounded in PDE theory, the practical use of this approach requires no knowledge of that theory.

Taylor expansion for matrix functions of vector variable using the Kronecker product

Taylor expansion is one of the many mathematical tools that is applied in Mechanics and Engineering. In this paper, using the partial derivative of a matrix with respect to a vector and the Kronecker product, the formulae of Taylor series of vector variable for scalar functions, vector functions and matrix functions are built and demonstrated. An example regarding the linearization of the differential equations of an elastic manipulator is presented using Taylor expansion.

An analytical method to determine the complex refractive index of an ultra-thin film by ellipsometry

We proposed a completely analytical method to determine the complex refractive index of an ultra-thin film on an arbitrary substrate by ellipsometry. The method is based on directly solving the polynomial equation derived from the 2nd-order Taylor expansion of ellipsometric functions without any prior knowledge about electronic transitions of the material. Numerical simulations demonstrate that the proposed method can be applied to deal with ultra-thin films with a wide thickness range from a single atomic layer to 5 nm over an ultra-broad spectral range from deep ultraviolet to infrared regions. With the proposed method, complex refractive indices of typical two-dimensional (2D) materials, including the monolayer graphene and 1–3 layer WSe2, were experimentally investigated over an ultra-wide spectral range (193–1690 nm). The proposed method shows great potential in the accurate determination of complex refractive indices of ultra-thin films especially the emerging 2D materials.

ON OZAKI CLOSE-TO-CONVEX FUNCTIONS

Let $f$ be analytic in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and given by $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$. We give sharp bounds for the initial coefficients of the Taylor expansion of such functions in the class of strongly Ozaki close-to-convex functions, and of the initial coefficients of the inverse function, together with some growth estimates.

Side Wall Effects on the Hydrodynamics of a Floating Body by Image Green Function Based on TEBEM

A novel numerical model based on the image Green function and first-order Taylor expansion boundary element method (TEBEM), which can improve the accuracy of the hydrodynamic simulation for the non-smooth body, was developed to calculate the side wall effects on first-order motion responses and second-order drift loads upon offshore structures in the wave tank. This model was confirmed by comparing it to the results from experiments on hydrodynamic coefficients, namely the first-order motion response and second-order drift load upon a hemisphere, prolate spheroid, and box-shaped barge in the wave tank. Then, the hydrodynamics of the KVLCC2 model were also calculated in two wave tanks with different widths. It was concluded that this model can predict the hydrodynamics for offshore structures effectively, and the side wall has a significant impact on the first-order quantities and second-order drift loads, which satisfied the resonant rule.

Synchronization on Lie Groups: Coordination of Blind Agents

This paper presents an algorithm for synchronization of blind agents (agents are unable to observe other agents, i.e., no communication) evolving on a connected Lie group G. We employ the method of extremum seeking control for nonlinear dynamical systems defined on connected Riemannian manifolds to achieve synchronization among the agents. In this approach, each agent updates its position on G by only receiving the synchronization cost function. The results are obtained by employing the notion of geodesic dithers for extremum seeking on Riemannian manifolds and their equivalent version on Lie groups and applying Taylor expansion of smooth functions on Riemannian manifolds. Due to geometrical properties of the synchronization set, we employ the method of quotient manifolds to prove the convergence of the proposed algorithm. The obtained results are applied to synchronization problems on SE(3) to demonstrate the efficacy of the proposed algorithm.

Simple method of fundamental reactive power measurement for ASIC application

This study presents a simple method for measuring the fundamental reactive power concurrently with the other basic power components defined in IEEE Standard 1459-2010. The method has low hardware requirements and is suitable for implementation in low cost devices, e.g. application-specific integrated circuits (ASICs). The algorithm is based on voltage samples that are shifted by a constant number that represents one-quarter of the rated period. The impact of instantaneous frequency variations on the measurement is mitigated by a sine function Taylor expansion. The algorithm is described in detail and the results of an experimental investigation are presented, which consisted of simulations and laboratory research into known testing signals. Next, the method was applied to an actual microgrid with fluctuating load and a high level of distortion, and the results were compared to those of the reference methods. The results confirm the usefulness of the proposed solution for the measurement of power flows in power systems, including highly disturbed ones.

New upper bounds of Ostrowski type integral inequalities utilizing Taylor expansion

In this paper, we have been introduced and tested some significant new bounds of Ostrowski type integral inequalities. In accordance with this purpose we have taken advantageous of the Taylor expansion for functions. Some numerical experiments have been given to show the applicability and accuracy of the proposed method.

Efficient estimation of conditional covariance matrices for dimension reduction

ABSTRACTLet and . In this paper, we propose an estimator of the conditional covariance matrix, , in an inverse regression setting. Based on the estimation of a quadratic functional, this methodology provides an efficient estimator from a semi parametric point of view. We consider a functional Taylor expansion of under some mild conditions and the effect of using an estimate of the unknown joint distribution. The asymptotic properties of this estimator are also provided.