Second Derivative Of Function Research Articles

Você esqueceu mesmo uma função Delta?

This is the third article in a series dedicated to examining the use of the delta function in the physics literature. Here we discuss the appearance of a delta function in the second derivative of the wave function if its derivative is discontinuous. When this happens it is often claimed in the literature that the proposed wave function must be eigenfunction of an operator that contains a delta function and, in some cases, it is rejected. We study in this paper what we believe are alternatives ways to deal with this situation.

Convexity of energy functions of harmonic maps homotopic to covering maps of surfaces

We study the strict convexity of the energy function of harmonic maps at their critical points from a Riemann surface to a Riemann surface, or to the product of negatively curved surfaces. When the target is a Riemann surface and when the map is of nonzero degree, we obtain a precise formula for the second derivative of the energy function along a Weil–Petersson geodesic, which implies that the energy function is strictly convex at its critical points. When the target is the product of two surfaces where each projection of the harmonic map is homotopic to a covering map, we also prove the strict convexity of the associated energy function. As an application we prove that the energy function has a unique critical point in these cases.

Christov expansion method for nonlocal nonlinear evolution equations

Christov functions are a complete orthonormal set of functions on L 2(-∞,∞) that allow us to expand derivatives, nonlinear products, and nonlocal (integro-differential) terms back into the same basis. These properties are beneficial when solving nonlinear evolution equations using Galerkin spectral methods. In this work, we demonstrate such a “Christov expansion method” for the Benjamin–Ono (BO) equation. In the BO equation, the dispersion term is nonlocal, given by the Hilbert transform of the second spatial derivative of the unknown function. The Hilbert transform of the Christov functions can be computed using complex integration and Cauchy’s residue theorem to obtain simple relations. Then, a Galerkin spectral expansion can be used to the solve the BO equation. Time integration is performed using a Crank–Nicolson-type scheme. Importantly, the Christov expansion method yields a banded matrix for the spatial discretization, even though the spatial terms are nonlocal. To demonstrate the approach and its implementation, we perform numerical experiments showing the steady propagation of single and the overtaking interaction of multiple BO solitary waves.

Adaptive control of a class of uncertain nonlinear systems using brain emotional learning and Legendre polynomials

In this paper, an adaptive controller for a class of uncertain nonlinear systems is presented using a combination of Legendre polynomials and brain emotional learning-based intelligent controller (BELBIC). Recently, some versions of BELBIC have been presented with the aim of satisfying the universal approximation property using Gaussian basis function. However, the size of regressor vector is too large that imposes a heavy computational load to the processor. The novelty of this paper is presenting a new version of BELBIC with less computational burden using Legendre polynomials. Moreover, there are very few tuning parameters in Legendre polynomials. Another contribution of this paper is editing the stability analysis presented in recent related works. Due to the intrinsic non-differentiability of the adaptation rules of BELBIC, the second time derivative of Lyapunov function is undefined and thus, the Barbalat’s lemma cannot be applied to verify the asymptotic convergence of the error function. Therefore, bounded-input-bounded-output (BIBO) stability can only be claimed for this controller. Simulation results on different case studies show that Legendre polynomials can improve the universal approximation property of BELBIC with less tuning parameters. Moreover, in the absence of the robust control term in the control law, the performance Legendre polynomials will not deteriorate, while the performance degrade in Gaussian basis function is quite considerable.

Lambert series of logarithm, the derivative of Deninger’s function and a mean value theorem for

Abstract An explicit transformation for the series $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ , or equivalently, $\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re $(y)>0$ , which takes y to $1/y$ , is obtained for the first time. This series transforms into a series containing the derivative of $R(z)$ , a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi _1(z)$ (the derivative of $R(z)$ ) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$ , all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as $y\to 0$ which was also not known before. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )e^{-\delta t}\, dt$ as $\delta \to 0$ .

A rotation-free Hellinger-Reissner meshfree thin plate formulation naturally accommodating essential boundary conditions

A rotation-free Hellinger-Reissner meshfree thin plate formulation is proposed to naturally accommodate the essential boundary conditions in a variationally consistent way. In this approach, the Galerkin weak form is established based upon the Hellinger-Reissner variational principle, where the bending moment is expressed as the second order smoothed gradients which inherently embed the integration constraint and fulfill the variational consistency condition. Owing to the Hellinger-Reissner variational principle, the essential boundary conditions naturally arise in the weak form. Accordingly, the enforcement of essential boundary conditions has a similar form as that of the Nitsche's method, i.e., both have standard consistent and stabilized terms. Compared with the Nitsche's method, the costly second and higher order derivatives of traditional meshfree shape functions are replaced by the fast-evaluated second order smoothed gradients and their derivatives. Meanwhile, the stabilized term in proposed method does not involve any artificial parameter and thus eliminates the stabilization parameter-dependent issue in the Nitsche's formulation. Several examples are presented to illustrate the convergence, accuracy and efficiency of the proposed rotation-free Hellinger-Reissner meshfree thin plate formulation.

Optimal substitution of black soldier fly larvae for fish in broiler chicken diets

In pursuit of finding the optimum substitution level of black soldier fly larvae meal (BSFLM) as an alternative protein source for fish meal (FM) in broiler diets, the effect of partial or complete substitution on feed utilization efficiency, growth performance, carcass characteristics, meat quality and marginal economic benefits was assessed. 150 Cobb500 day old chicks were randomly allocated to five treatment diets in which graded levels of BSFLM substituted. FM; SD1 (control diet with only FM), SD2 (250), SD3 (500), SD4 (750), and SD5 (1000 g/kg DM), for 28 days. Subsequently, a conventional FM-based finisher diet was fed until day 42. Weight gain, feed intake and feed conversion ratio (FCR), were determined and the second derivative of the FCR regression function used to determine the optimal inclusion of BSFLM. Further, marginal analysis was used to evaluate the marginal economic benefit of including BSFLM. The response to increasing amounts of BSFLM in the diet followed a quadratic trend (P<0.05) with a reduction in feed intake, FCR and weight gain (P<0.05) occurring after 500 g/kg substitution and reaching the nadir at complete substitution. However, birds exhibited a compensatory growth when switched to fish-based finisher diet (day 28–42). No effect of substitution was observed on carcass parts except for a decrease in dressing percentage and an increase in visceral organ weights (P<0.05) at complete substitution. The optimal substitution based on FCR was calculated to be 540 g/kg. Proximate and fatty acid analysis of muscle indicated that Substitution of FM with BSFLM, reduced (p<0.05) protein and Omega 3 fatty acids. Additionally, increasing BSFLM in the diet up to 500 g/kg DM resulted in a 23% decrease in feed cost per bird and higher net marginal benefits. Therefore, replacement of FM with up to 540 g/kg BSFLM in starter diets, does not compromise growth and meat quality in broiler chickens.

The nth Power Fourier Spectrum Analysis for the Generalized Seismic Wavelets

The generalized seismic wavelets (GSW) are defined by fractional derivatives of the Gaussian function, whose asymmetry allows them to represent seismic signals more accurately than the commonly used symmetrical Ricker wavelet. The latter is a particular case with a second derivative of the Gaussian function. To better obtain the GSW, which could be well-matched with seismic signals, this paper proposes the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th power Fourier spectrum analysis method for GSW. Firstly, based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th power Fourier spectrum of GSW, the proposed method builds the mathematical relationship between frequency characteristics (e.g., central frequency and bandwidth) and the statistical properties (e.g., mean frequency and deviation). Secondly, according to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -th power Fourier spectrum, we propose a weighting calculation method for the derivative order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i> of the Gaussian function. This method could be used for estimating GSW matched the seismic first arrival record, which is conducive to improving the accuracy of seismic imaging, inversion, and Q analysis. In theory, our proposed weighting method has better robustness and noise resistance than the traditional spectrum analysis method based on the power or amplitude spectrum. The experiment of synthetic noise-including first arrival record and vertical seismic profiling (VSP) field data, shows the effectiveness of the proposed approach.

SSS Small Target Detection via Combining Weighted Sparse Model With Shadow Characteristics

The acquirement of seafloor small target information is one of the most important tasks of a side-scan sonar (SSS) survey. Thus, SSS small target detection becomes fundamental work for SSS applications which holds vital importance for marine engineering, maritime military and so on. However, existing methods cannot take the prior shadow information into consideration well, which would easily miss small targets. In this paper, a novel SSS small target detection method considering shadow characteristics is proposed. First, we give a detailed analysis of the SSS imaging theory as well as the prior information about the characteristics of shadows. Then, considering the prior information of the SSS short-shadow, the second partial derivative of the Gaussian function is specifically introduced for the construction of a weighted item. After that, incorporating the weighted item with the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</sub> -norm, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm, and low-rank constraints on the noise, the target, as well as the background, respectively, a weighted sparse detection model is proposed. To further take the long-shadows into consideration, a long-shadow detection method and its corresponding target detection method are proposed. By combining the two detection results, we get the comprehensive detection result. Experiments based on SSS images in different scenarios proved the validity of the proposed method.

Effect of Asymmetric Finite Difference Formulas on the Orders of Central Difference Approximations for the Second Derivative of a Periodic Function

Effect of Asymmetric Finite Difference Formulas on the Orders of Central Difference Approximations for the Second Derivative of a Periodic Function

A convenient gauge for virial identities in axial symmetry

Virial identities are a useful mathematical tool in General Relativity. Not only have they been used as a numerical accuracy tool, but they have also played a significant role in establishing no-go and no-hair theorems while giving some physical insight into the considered system from an energy balance perspective. While the calculation of these identities tends to be a straightforward application of Derrick's scaling argument [1], the complexity of the resulting identity is system dependent. In particular, the contribution of the Einstein-Hilbert action, due to the presence of second order derivatives of the metric functions, becomes increasingly complex for generic metrics. Additionally, the Gibbons-Hawking-York term needs to be taken into account [2]. Thankfully, since the gravitational action only depends on the metric, it is expected that a “convenient” gauge that trivializes the gravitational action contribution exists. While in spherical symmetry such a gauge is known (the m−σ parametrization), such has not been found for axial symmetry. In this letter, we propose a “convenient” gauge for axial symmetry and use it to compute an identity for Kerr black holes with scalar hair.

Decision-making in a state cybersecurity

Nowadays, the issue of cyber security as a component of information security of the state is extremely relevant for Ukraine and the World community. Even so, it should be taken into account that the use of cyberspace expands people's communication capabilities, promotes the development of information technologies, research and novelty, and stimulates the development of industry and the economy. Nevertheless, the advantages of modern cyberspace can lead to new threats to people, society, national and international security. Along with initiatives of natural (unintentional) origin, the number and power of cyberattacks is increasing. These cyberattacks are inspired by the interests of individuals, groups, states, and associations of states. Increasing the effectiveness of mathematical modeling of state cyber security information systems can be ensured by modeling both the complex system and the subsystems that are the essential part of it. It stimulates the progress of models and algorithms that allow solving complex problems of system management and processing of the information flow. Furthermore, it is necessary to limit the tolerance interval of the solution for each task of the complex system of cyber security information systems. The following steps should be taken into account: – assignment of directive terms for solving problems; – information related tasks of information processing and transmission. The analysis of the research shows the weakness of a conventional approach to the complex solution of the problem of the synthesis of mathematical models and algorithms for determining the time of decision-making in the system of both security and information cyber security; methods of countering threats, mathematical methods of modeling, creating and evaluating models of both cyberattacks and cybersecurity and solving the problem of synthesis of mathematical models and algorithms for determining the time of decision-making in the system of both security and cyber security information systems. The defined issue is an unperformed part of the general problem of ensuring information security in complex systems of technical protection and information cyber security. The tasks belong to the class of problems of nonlinear mathematical programming. Performing them, methods of the theory of nonlinear programming can be used. The main priority of the tasks is its large dimension, which has a number of tasks that are performed in the network and are introduced in the information correlation. One of the current approaches to perform non-linear programming problems of large dimensions is the use of approximation methods of the theory of non-linear programming, the essence is that the solution of the finite-dimensional nonlinear complementarity problem is carried out as a result of solving a sequence of problems of a simpler type, which require significantly optimization operating costs than the original problem. Linear approximation is not always effective as it allows to obtain only a necessarily approximate value. Lately, the new research has introduced an approach to eliminate this fault with the help of auxiliary quadratic equation. The minimization method takes a special place among these methods. Unlike other methods of this class, it converges to any initial approximation and does not require assumptions about the convexity of functions, does not require firm positive certainty of the matrix of second derivatives of Lagrange functions, and has a fairly simple structure of the auxiliary quadratic problem. Hence, the linearization method has a linear rate of ascent. However, there is a modification of the method that means that at a considerable distance from the extremum point, the speed of convergence is linear, and with sufficient proximity to it, it is quadratic. Solving quadratic problems is not exact and only those restrictions with the maxima acceptable limits can be taken into account. This feature minimizes the dimension of auxiliary problems and thereby minimizes the computer complexity of the original nonlinear problem. The described algorithm is implemented in the form of applied programs for evaluating the effectiveness of algorithms for determining the time of decision-making by cyber security information systems. Thus, the given results can be used in the development of effective decision-making time determination algorithms based on mathematical models for decision-making support systems of the cyber security information system, as well as for modeling complex technical systems and assessing the effective use of several computing information systems. Key words: cyber security, cyber defense, cyber attacks, cyber defense system, information protection, state protection, cyber space, communication channels.

Reheating era in Gauss-Bonnet theories of gravity compatible with the GW170817 event

In the present article we showcase how the reheating era can be described properly in the context of Einstein-Gauss-Bonnet gravity assuming that the primordial gravitational waves propagate with the velocity of light. The equations of the duration of reheating along with the reheating temperature are derived and as demonstrated, their expressions are quite similar to the case of a canonical scalar field where now the second derivative of the Gauss-Bonnet scalar coupling function appears and effectively alters the numerical value of the scalar potential. The appearance of such term is reminiscing of a λR model of gravity where λ is now dynamical. We consider two viable inflationary models of interest, the former involves an error function as scalar Gauss-Bonnet coupling function and the latter a Woods-Saxon scalar potential. It is shown that for both models the aforementioned quantities can be in agreement with theoretical expectations. The only constraint that is needed is the assumption that the second time derivative of the Gauss-Bonnet scalar coupling function is actually lesser than the Planck mass squared, that is ξ¨<MPl28 in order to obtain a viable description. We find that a free parameter of the theory and specifically the potential amplitude for the Woods-Saxon model during the inflationary era, dictates the effective equations of state and therefore the reheating epoch can be described either by a type of stiff matter with EoS parameter equal to unity or by an EoS parameter close to that of the radiation.

A cell-based smoothed radial point interpolation method applied to kinematic limit analysis of thin plates

The cell-based smoothed radial point interpolation method in conjunction with a second order cone programming is proposed for kinematic limit analysis of rigid-perfectly plastic thin plates in this paper. The transverse displacement field is interpolated by the radial point interpolation method (RPIM) and no rotational degrees of freedom are involved. The gradient smoothing technique is utilized to construct smoothed curvature field in every smoothing domain and there is no need to compute the second derivatives of shape functions. The rotational boundary conditions are satisfied in the process of curvature smoothing and the translational boundary conditions can be directly enforced without any special treatment. The limit analysis problem of thin plates is formulated by minimizing the dissipation power subject to a set of equality constraints and this minimization problem can be expressed easily as a standard second order cone programming. It is testified from the computational results that the proposed procedure can provide reasonable and satisfactory upper bound limit load multipliers for thin plates.

Spectroscopic ellipsometry characterization of PbWO4 single crystals

Optical characterization of PbWO4 single crystals grown by Czochralski method was achieved in virtue of spectroscopic ellipsometry experiments carried out in the energy region of 1.0–5.6 eV at room temperature. Tetragonal scheelite structure with lattice parameters of a = b = 5.4619 Å and c = 12.0490 Å was determined for the bulk crystal utilizing from XRD analysis. Analyses of the ellipsometry data presented the photon energy dependencies of complex dielectric function of the crystal. The real part of the dielectric function exhibited increasing behavior with energy in the below 4.1 eV and then decreased immediately. Zero frequency refractive index and dielectric constant were determined to be 2.02 and 4.08, respectively, using Wemple and DiDomenico oscillator model. High frequency dielectric constant was calculated as 4.30 by Spitzer-Fan model. Optical band gap of PbWO4 was found to be 3.24 eV from the dielectric relaxation time spectrum. Moreover, existence of two critical points with energies of 3.70 and 4.58 eV was revealed from the analyses of extinction coefficient and second derivative of the dielectric function. These levels were considered to be due to creation of cation exciton (Pb2+ 6s2 − Pb2+ 6s6p) and transitions in the [WO4]2− group.

Positive Definiteness of Symmetric Rank 1 (H-Version) Update for Unconstrained Optimization

Several attempts have been made to modify the quasi-Newton condition in order to obtain rapid convergence with complete properties (symmetric and positive definite) of the inverse of Hessian matrix (second derivative of the objective function). There are many unconstrained optimization methods that do not generate positive definiteness of the inverse of Hessian matrix. One of those methods is the symmetric rank 1( H-version) update (SR1 update), where this update satisfies the quasi-Newton condition and the symmetric property of inverse of Hessian matrix, but does not preserve the positive definite property of the inverse of Hessian matrix where the initial inverse of Hessian matrix is positive definiteness. The positive definite property for the inverse of Hessian matrix is very important to guarantee the existence of the minimum point of the objective function and determine the minimum value of the objective function.

A new fractional infectious disease model under the non-singular Mittag–Leffler derivative

In this manuscript, we consider a fractional mathematical model, which describes the dynamics of infectious disease, under the non-singular Mittag–Leffler derivative. The model under consideration is the extension of the SIRV model, where the infectious class has been divided into two compartments, namely the acute and chronically infectious individuals. First, we obtain the possible equilibrium states of the given model. With the help of the next generation matrix approach, the reproduction number has been calculated for the system to find conditions on the spread or control of the disease. Additionally, a new concept of strength number and analysis of the second derivative of the Lyapunov function has been used for the detection of waves. We investigate the said problem for qualitative analysis and determine at least one solution by applying the approach of fixed point theory. For approximate solution, the technique of iterative fractional-order Adams–Bashforth scheme has been used. Numerical simulation for the proposed scheme has been performed at various fractional-order lying between 0, 1 and for integer-order 1. All the compartments show convergency and stability with growing time. A good comparative result has been given by different fractional orders and achieves stability faster at the low fractional orders.

Spectral CG Algorithm for Solving Fuzzy Nonlinear Equations

The nonlinear conjugate gradient method is an effective technique for solving large-scale minimizations problems, and has a wide range of applications in various fields, such as mathematics, chemistry, physics, engineering and medicine. This study presents a novel spectral conjugate gradient algorithm (non-linear conjugate gradient algorithm), which is derived based on the Hisham–Khalil (KH) and Newton algorithms. Based on pure conjugacy condition The importance of this research lies in finding an appropriate method to solve all types of linear and non-linear fuzzy equations because the Buckley and Qu method is ineffective in solving fuzzy equations. Moreover, the conjugate gradient method does not need a Hessian matrix (second partial derivatives of functions) in the solution. The descent property of the proposed method is shown provided that the step size at meets the strong Wolfe conditions. In numerous circumstances, numerical results demonstrate that the proposed technique is more efficient than the Fletcher–Reeves and KH algorithms in solving fuzzy nonlinear equations.

Improvements of the Newton–Raphson method

The Newton–Raphson (N-R) method is one of the most ubiquitous approaches with several applications in numerous areas to find the optimization solution. It has long been known that in practice to apply the N-R method, it is customarily required to be obtained the second derivative of the objective function. Nevertheless, it is difficult or impossible to find the second derivative of the target function in many situations, leading to the impossibility to obtain the optimization solutions. This paper conducts a numerical method, develops an algorithm, and provides the R code so that one will be able to obtain the optimization solutions when our proposed approach is used. This study investigates numerous simulation studies and a factual data set. Based on the findings in this work, it can be firmly concluded that the N-R method with the numerical method for the second derivative of the objective function is found to be a robust and trustworthy approach.

Stopping spikes, continuation bays and other features of optimal stopping with finite-time horizon

We consider optimal stopping problems with finite-time horizon and state-dependent discounting. The underlying process is a one-dimensional linear diffusion and the gain function is time-homogeneous and difference of two convex functions. Under mild technical assumptions with local nature we prove fine regularity properties of the optimal stopping boundary including its continuity and strict monotonicity. The latter was never proven with probabilistic arguments. We also show that atoms in the signed measure associated with the second order spatial derivative of the gain function induce geometric properties of the continuation/stopping set that cannot be observed with smoother gain functions (we call them continuation bays and stopping spikes). The value function is continuously differentiable in time without any requirement on the smoothness of the gain function.