Non-zero Derivative Research Articles

The fourth-order expansion of the exchange hole and neural networks to construct exchange-correlation functionals.

The curvature Qσ of spherically averaged exchange (X) holes ρX,σ(r, u) is one of the crucial variables for the construction of approximations to the exchange-correlation energy of Kohn-Sham theory, the most prominent example being the Becke-Roussel model [A. D. Becke and M. R. Roussel, Phys. Rev. A 39, 3761 (1989)]. Here, we consider the next higher nonzero derivative of the spherically averaged X hole, the fourth-order term Tσ. This variable contains information about the nonlocality of the X hole and we employ it to approximate hybrid functionals, eliminating the sometimes demanding calculation of the exact X energy. The new functional is constructed using machine learning; having identified a physical correlation between Tσ and the nonlocality of the X hole, we employ a neural network to express this relation. While we only modify the X functional of the Perdew-Burke-Ernzerhof functional [Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], a significant improvement over this method is achieved.

On the Set of Points at which an Increasing Continuous Singular Function has a Nonzero Finite Derivative

Sánchez, Viader, Paradís and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{\sigma}$ null subset of $[0,1]$.

Method for long-term coherent-noncoherent signal accumulation with non-zero higher derivatives range to radar target

A method of long-term combined accumulation of the reflected signal is justified, which provides for its division into disjoint subsets, coherent accumulation in subsets using one of the fast algorithms and subsequent incoherent accumulation of the squares of the modules of the results of processing the subsets. A distinctive method’s feature is the use with incoherent accumulation of maxima of the squares of the moduli of the coherent processing results, that are selected from the range / radial velocity regions in accordance with a given hypothesis about the minimum and maximum values of the target radial velocity and the radial acceleration detection channel setting.The efficiency of the method was confirmed by simulation modeling. Using the theories of ordinal statistics and the method of moments, a method for calculating the probability of correct detection is developed. Estimates of processing losses are made in comparison with coherent and incoherent accumulation algorithms for a signal reflected from a point target, for the case when there is no range and frequency migration. Estimates for the required number of receiver channels are given.

The Rayleigh wave scattering on a rectangular lattice of the solid roughness discontinuities

The problem of the surface acoustic Rayleigh wave scattering on a deterministic three-dimensional roughness, occupying a finite size rectangular region of an isotropic solid free surface, is solved in the Rayleigh-Born approximation of the perturbation theory in a roughness amplitude. Formula for the displacement field in the scattered Rayleigh wave at a big distance from the roughness, as compared to rough region sizes L1,2 along the x1,2- axes respectively, and asymptotic formulas for this displacement field in the Bragg, i.e. short-wavelength λ≪ L1,2 limit, where λ is the wavelength, are derived. The new laws of scattering are obtained. They are caused by a strong modulation of scattering by the roughness form. They exceed the fundamental physical conception, that a wave scattering in the short-wavelength limit takes place on a medium discontinuities, by the statement, that a wave strongly senses the structure of a medium in the near vicinity of discontinuities as well as the form-factor of the discontinuities lattice. This form-factor is a dependence of the discontinuity amplitude, i.e. of a difference of the left and right limit values of a roughness non-zero derivative, including one of zero order, in coordinate at a point of discontinuity, on a number of this discontinuity in a lattice. This exceeded physical conception violates the classical Laue-Bragg-Wulff laws of scattering.

Long-time coherent accumulation algorithms for reflected signal with non-zero higher derivatives of the range to radar target in the spectral domain

The purpose of the article is to present computationally economical algorithms for long-term coherent accumulation of signals reflected from a point target with compensation for range and frequency migration and accumulation of signals in the spectral region. The algorithms include intra-period processing with simultaneous correction of range and frequency migration and inter-period processing with coherent accumulation of signals at the output of intra-period processing. In the first variant of the algorithm, intra-period processing is implemented by calculating the spectra of the received signals in each repetition period, multiplying the samples of the spectra by the samples of the amplitude-phase-frequency characteristic of the matched filter of a single signal and correcting phase coefficients determined by the number of the repetition period and the values of the range derivatives, and the inverse Fourier transform of the transformed spectra. The difference between the second version of the algorithm at the stage of intraperiod processing is the correction of only the quadratic and subsequent components of the range and frequency migration and the use of the keystone transformation, which eliminates the linear range migration. Coherent accumulation for both variants is realized due to the fast Fourier transform of the signal samples over the repetition periods for all samples over the range. The concept of “rough speed resolutionˮ is introduced, which determines the arrangement of channels when compensating for range migration. The uncertainty function in the coordinates “velocity–acceleration” is obtained. The equivalence of the two variants of the algorithm is shown and estimates for the required number of receiver channels are given. The simulation results confirming the operability of the proposed algorithms are presented.

A note on dielectrophoresis

A story about dipole molecules in liquid (Physics Today, April 2020, page 17) has prompted me to share some of my own experiences.When an electrically polarized object, like a dipole, is in a nonuniform electric field, it experiences a force. That phenomenon is known as dielectrophoresis. It is possible to create a field with constant, nonzero second derivative. A dipole particle in such a field would experience a constant force that can thus be made to move in a fixed direction under a constant force.I created such a field in a shallow device I had made at glass manufacturer Owens-Illinois, my employer, and used yeast and blood cells to study it. The cells did not move unless the field had a frequency of at least 1 MHz. I did not try much higher frequencies. In 1975–76 I took my experiment to the hematology laboratory of Massachusetts General Hospital. Unfortunately, the hoped-for dispersion of velocities among cell types was not observed, and the study ended.However, water molecules themselves have a dipole moment. When polarized light was passed vertically through the horizontal device, applying the 1 MHz field would switch the transparency on and off.Apparently, the yeast or blood cells were passively carried along a current of water, driven by the nonuniform field. It is a mystery why there was no motion of the cells (yeast or blood) until the frequency was increased to 1 MHz. Perhaps clusters of water molecules, which are imagined to explain the high boiling point of water, are not dipoles, and the field at or above 1 MHz breaks up some of the aggregates, allowing the dipole moment of H2O to be sensed. Section:ChooseTop of page <<© 2021 American Institute of Physics.

The Method of Rotated Solutions: A Highly Efficient Procedure for Code Verification

Abstract Code verification provides mathematical evidence that the source code of a scientific computing software platform is free of bugs and that the numerical algorithms are consistent. The most stringent form of code verification requires the user to demonstrate agreement between the formal and observed orders of accuracy. The observed order is based on a determination of the discretization error, and therefore requires the existence of an analytical solution. One drawback of analytical solutions based on traditional engineering problems is that most derivatives appearing in the related governing equations (or mathematical model) are identically zero, which limits their scope during code verification. The method of rotated solutions (MRS) is introduced herein as a methodology that utilizes coordinate transformations to generate additional nonzero derivatives in the governing equations and numerical solutions. These transformations extend the utility of even the simplest one-dimensional solutions to be able to perform more thorough evaluations. This paper outlines the rotated solutions methodology and provides an example that demonstrates and confirms the usefulness of this new technique.

Cherenkov-channeling radiation by relativistic muons in crystals

In this work we analyse Cherenkov radiation by relativistic muons, positive and negative, channeled in optically transparent diamond and silicon crystals in comparison with ordinary Cherenkov radiation. We have shown that the maxima in the spectral angular distributions for both types of radiation are revealed at the derivative extrema for the media refractive index, while, due to the difference in scattering of positively and negatively charged particles at crystal channeling, the number of Cherenkov photons emitted by channeled positive muons might be over the one for negative muons. We have demonstrated that Cherenkov radiation by quasi free projectiles is described as one limiting approximation of a general expression for Cherenkov radiation by channeled projectiles, which takes into account non-zero derivative of the refractive index. The last may result in essential increase of radiation intensity.

On the Angular Derivative of Comb Domains

The angular derivative problem is to provide geometric conditions on the boundary of a simply connected domain $$\Omega $$ near $$\zeta \in \partial \Omega $$ which are equivalent to the existence of a non-zero angular derivative at $$\zeta $$ for the conformal map of $$\Omega $$ onto the half plane. Rodin and Warschawski (Math Z 153:1–17, 1977), proposed a conjecture regarding the existence of an angular derivative at $$+\,\infty $$ for a certain class of comb domains. The purpose of this paper is to show how a theorem of Burdzy (Math Z 192:89–107, 1986) can be used to give an affirmative answer to the necessity part of the Rodin–Warschawski conjecture on comb domains.

A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity with body force term in the Navier-Stokes equations. The class consists of stream function  characterized by equation , in polar coordinates ,  where ,  and  are continuously differentiable functions, derivative of  is non-zero but double derivative of  is zero. We find exact solutions, for a suitable component of body force, considering two cases based on velocity profile. The first case fixes both the functions ,  and provides viscosity as function of temperature. Where as the second case fixes the function , leaves  arbitrary and provides viscosity and temperature for the arbitrary function . In both the cases, we can create infinite set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution in the presence of body force. Â

Aharonov-Bohm effect without contact with the solenoid

We add a scalar potential to the 2D Aharonov-Bohm (AB) model which properly diverges both at the solenoid border and at infinity so that the resulting operator is essentially self-adjoint and has a discrete spectrum; the former property is interpreted as no contact of the particle with the solenoid border since there is no need of boundary conditions. We study gauge transformations to get the usual periodic behavior of the AB properties as a function of the magnetic flux. The presence of the AB effect is proven through the ground state energy, which is shown to be smooth in case it is simple and with a nonzero derivative if the ground state is real valued; such properties are verified in the case of circular solenoids, for which it is shown to be a nonconstant periodic function with a minimum at integer and a maximum at half-integer circulations (at half-integer circulations, it is doubly degenerated).

Solvable Diffusion Models with Linear and Mean-Reverting Nonlinear Drifts

We present new extensions to a method for constructing families of solvable one-dimensional time-homogeneous diffusions whose transition densities and other related quantities are obtainable in analytically closed form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines a smooth monotonic differentiable map with nonzero derivative (diffeomorphism) and a measure change via a Doob-$h$ transform. This gives rise to new multiparameter solvable diffusions that are divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The first class of models, having linear drift and nonlinear (state-dependent) volatility functions, is useful for pricing equity and foreign exchange (FX) options in finance, while the second class of diffusions contains new models that are mean-reverting and are applicable to pricing interest-rate and other path-dependent derivatives such as volatility index (VIX) options. As specific examples of the first class of affine drift models, we present explicit results for two new families of models that arise from the squared Bessel process (the Bessel family) and the Ornstein--Uhlenbeck diffusion (the OU family). For the second class of nonlinear drift models, we give examples of solvable subfamilies called the Bessel family of mean-reverting diffusions and derive closed-form integral formulas for conditional expectations of certain functionals. In particular, we derive a new closed-form analytical formula for the Laplace transform with respect to the strike price of a standard call VIX option. We then succeed in Laplace inverting the expression to obtain numerically exact VIX call option prices with realistic implied volatilities with respect to strike and maturity. Moreover, we accurately calibrate the OU family of models to FX option market data, exhibiting pronounced implied volatility smiles across several strikes and maturities.

A Function With Continuous Nonzero Derivative Whose Inverse Is Nowhere Continuous

SummaryA function with continuous nonzero derivative whose inverse is nowhere continuous is given. The function satisfies all the premises of the inverse function theorem except one: it is not defined on an open set.

Local influence diagnostics for the test of mean–variance efficiency and systematic risks in the capital asset pricing model

In this paper we consider the capital asset pricing model under the multivariate normal distribution for modeling asset returns. We develop and implement local influence diagnostic techniques not based on likelihood displacement. The main interest is to consider the test of mean–variance efficiency of a given portfolio as objective function. Sensitivity of maximum likelihood estimators of slopes is also considered. For this purpose, first and second-order local influence measures are calculated under a case weights perturbation scheme. In particular, for our objective functions, which have nonzero first derivative at the critical point, the proposed second-order measures are scale invariant, unlike the normal curvature. The performance of the proposed measures is illustrated by using two real data sets as well as a simulation study and a simulated data set. Empirical results seem to indicate that the Wald test statistic is less sensitive to atypical returns than maximum likelihood estimators of the slopes. By the other hand, there is some evidence about the convenience of jointly using first and second-order influence measures, to detect months/days with outlying returns, principally in Wald test statistic.

Tracking control for nonminimum phase MIMO micro aerial vehicle – a dynamic sliding manifold approach

Nonminimum phase output tracking has been addressed for a fixed-wing micro aerial vehicle. Nonminimum phase characteristics in an aircraft result from the fact that the process of generating upward vertical motion produces an initial downward force, causing the aircraft to lose altitude momentarily, and vice versa. This phenomenon leads to unstable internal dynamics in the plant model. A dynamic sliding-surface-based sliding mode control is chosen to stabilize the internal dynamics and to provide asymptotic output tracking error convergence to zero with desired eigenvalue placement. The control is designed in such a way that the output of the plant model should track a nonlinear time-varying reference trajectory generated at every time instant with a finite number of nonzero time derivatives. Design of second-order sliding-mode-based super twisting controller which ensures finite-time stability of the internal dynamics is proposed in this work. The proposed control methodology is applied in the design of an autolanding controller for a micro aerial vehicle with nonminimum phase behavior. MATLAB-based simulation results and discussions are presented to evaluate the performance of the controller and robustness of the sliding mode with respect to matched and unmatched disturbances. The proposed control algorithm has been successfully implemented in hardware-in-the-loop simulation and results are given.

Static and dynamic effects of chirality in dielectric media

Chiral dielectrics are considered from the perspective of continuum representations of spatial heterogeneity. Static effects in isotropic chiral dielectrics are predicted, provided the electric field has nonzero third spatial derivatives. The effects are compared with static chiral phenomena in Cosserat elastic materials which obey generalized continuum constitutive equations. Dynamic monopole-like magnetic induction is predicted in chiral dielectric media.

Relativity and the Solar Wind: The Maxwell-Equation Origins of the Solar-Wind Motional Electric Field

The motional electric field of the solar wind as seen by the Earth is examined theoretically and with spacecraft measurements. As it flows outward from the sun, the solar-wind plasma carries a spatially structured magnetic field with it. To calculate the motional electric field of the solar wind the spatially structured magnetic field is Lorentz transformed; for a full physical understanding, it is also necessary to Lorentz transform the current densities and charge densities in the solar wind. Referring to Maxwell’s equations, two related questions are asked: 1) Is the source of the solar-wind motional electric field charge density in the solar wind, time derivatives of current densities in the solar wind, or both? 2) Is the solar-wind motional electric field at Earth an electrostatic field, an induction field, or a superposition of the two? A Helmholtz decomposition of the motional electric field of the solar wind is made into a divergence-origin (electrostatic) and a curl-origin (induction) electric field. The global electric field associated with the outward advection of the global Parker-spiral magnetic field is found to be electrostatic with its origin being a distributed charge density in the solar-wind plasma. The electrostatic versus induction nature of the time-varying electric field associated with the advection of mesoscale magnetic structure varies with time as differently shaped magnetic structures in the solar-wind plasma pass the Earth; the mesoscale structure of the solar-wind plasma contains sheets of space charge and sheets wherein the current density has nonzero time derivatives.

On the relationship of natural frequency, damping, and feedforward acceleration with transient vibrations of track-seek control

The relationship between natural frequency, damping, the waveform of a feedforward acceleration input, and the transient vibrations of track-seek control is derived theoretically and a target for vibration design of the head actuator of a hard disk drive (HDD) is discussed from the viewpoint of the transient vibrations. When the acceleration feedforward input of the track-seek control is expressed as a polynomial function of time, the transient response of a single degree of freedom system is solved by applying integral by parts to the Duhamel's integral repeatedly. The results show the transient vibrations are generated at the initial and terminal times of the input and their amplitude depends on the lowest orders of non-zero derivatives of the polynomial. They also show that increasing the damping ratio effectively reduces the transient vibrations for fourth and sixth order polynomials but not for fifth and seventh order polynomials. The results were confirmed by performing a continuous time simulation and analyzing the shock response spectrum. Also discussed is the relationship between the transient vibrations and the seek time and track pitch, which are key parameters associated with the data access time and the HDD capacity. This relationship indicated the requirements needed for vibration design of the head actuator to increase HDD performance and capacity.

A singular function with a non-zero finite derivative on a dense set with Hausdorff dimension one

This article closes a trilogy on the existence of singular functions with non-zero finite derivatives. In two previous papers, the authors had exhibited a continuous strictly increasing singular function from [0,1] into [0,1] with a derivative that takes non-zero finite values at two different zero-measure sets: first, at the points of an uncountable set; then at the points of a dense set in [0,1]. In the present paper, the possibilities are further stretched as the construction is improved to extend it to an uncountable dense set whose intersection with any interval (a,b) has Hausdorff dimension one. Another feature of this third article is the construction of the required function using the most paradigmatic of the singular functions: the Cantor–Lebesgue one.

On Analysis and Judgment of Balance for Boolean Functions by E-Derivative

Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools and deeply into the internal structure of Boolean, we study the issues of the analysis and judgment of balance for Boolean functions. We get that the linear functions and the nonzero derivative of the product of two linear functions are balanced functions, and the product of two linear functions are not balanced functions. We also obtain the quadratic homogeneous Booleans are not all balanced function. Besides, we deduce the theorem which determine the sum of linear function and balanced function whether it is a balanced function. What is more, the features of balance of Boolean functions can be reflected easily and effectively by e-derivative.