Operational Calculus Research Articles

Local additivity revisited

We make a number of simplifications in Gour and Friedland’s proof of local additivity of the minimum output entropy of a quantum channel. We follow them in reframing the question as one about the entanglement entropy of bipartite states associated with a dB × dE matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their underlying strategy. We also extend this result to the maximum relative entropy with respect to a fixed reference state, which has important implications for studying the superadditivity of the capacity of a quantum channel to transmit classical information.

An Explicit Expression of the Unit Step Function

In this paper, an analytical form of the Unit Step Function (or Heaviside Step Function) is presented. In particular, this piecewise – defined function, which constitutes a fundamental concept of Operational Calculus and is also involved in many other areas of applied and engineering mathematics, is explicitly performed in a very simple manner by the aid of purely algebraic representations. The novelty of this work, when compared with other analytical approximations to this discontinuous function, is that the proposed exact formula is not performed in terms of non – elementary special functions, e.g. Gamma function or Error function. In addition, this function does not contain any infinitesimal quantities and also is not the limit of a sequence of functions with a pointwise or uniform convergence. Hence, it may be much more appropriate and useful to the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

Mikusiński’s operational calculus for general conjugated fractional derivatives

Mikusiński’s operational calculus for general conjugated fractional derivatives

Addendum to “fourth-order tensor calculus operations and application to continuum mechanics”

The authors provided a novel explicit expression for the dot derivative of an isotropic tensor function in the work by Kellermann et al. An alternate proof is provided in this addendum without the power series assumption.

Fourth-order tensor calculus operations and application to continuum mechanics

This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.

An operational calculus generalization of Ramanujan's master theorem

We give a formal extension of Ramanujan's master theorem using operational methods. The resulting identity transforms the computation of an integral of a product on the half-line to the computation of a Laplace transform. Since the identity is purely formal, we show consistency of this operational approach with various standard calculus results, followed by several examples, including a Feynman diagram evaluation, to illustrate the power of the extension. We then briefly discuss the connection between Ramanujan's master theorem and identities of Hardy and Carr before extending the latter identities in the same way we extended Ramanujan's. Finally, we generalize our results, producing additional interesting identities as a corollary.

Design the arbitrary order calculus operator by a simulated hyperbolic function for analytical applications

The existing problem of data processing by derivative, especially the higher order derivative, is that current operators lack robustness and accuracy for data with noise. The difference between the frequency distribution of the signal and the noise, as well as the low-frequency suppression from derivatives, is an obvious contrast that can be visually observed in the frequency domain. This contrast must be harmonious to become the better operator and requires an accurate description of the distribution of the processed data. Therefore, this paper proposed the profiles of the three basic types of spectrum peaks – Lorenz, Voigt, or Gaussian are unified in the frequency domain, for actual spectra which are single peaks or composed of multiple peaks, the envelope can be approximated with an exponential expression. Since the distribution of noise is flat, a spectrum with noise can be enveloped by a function that is the sum of an exponential function and a constant. A more precise response function for separating noise can be constructed on the envelope, which can be well simulated by the hyperbolic secant (Sech) function. The simulated hyperbolic function (SH) switched by the inverse Fourier transform is an available convolution kernel in the time domain. Around this kernel, the numerical calculus (Grünwald-Letnikov) and the corresponding adjustment can realize an arbitrary order calculus for the raw spectrum. Based on that, we presented an executable MATLAB script, and verified it interactively in the mathematical analytics, with or without noise. The results showed that the computation's accuracy and precision fully satisfied the operators' requirements. Moreover, the method has been applied in two cases of analytical chemistry to show its potential for a wide range of applications. A higher-order derivative that is free from noise interference aids the peak fit to obtain more specific results. The fractional order derivative spectroscopy yields less uncertainty for direct quantification in a multi-component system.

SEE Transform in Difference Equations and Differential-Difference Equations Compared With Neutrosophic Difference Equations

The Sadiq-Emad-Emann (SEE) transform, also known as operational calculus, has gained significant importance as a fundamental component of the mathematical knowledge necessary for physicists, engineers, mathematicians, and other scientific professionals. This is because the SEE transform offers accessible and efficient resources for resolving several applications and challenges encountered in diverse engineering and science domains. This study aims to introduce the fundamental principles of SEE transformation and establish the validity of two statements and associated attributes. The objective of this study is to use the aforementioned qualities in order to determine the solution of difference and differential-difference equations, with neutrosophic versions of difference and differential difference equations. In addition, we are able to get very effective and expeditious precise answers.

Parameterization of kernels of the Volterra series for systems given by nonlinear differential equations

The presented article is devoted on an issue regarding to the transformation of nonlinear models of a certain class to the Volterra functional series. The new identification method based on analytical input and output of a system was developed. The key task of representing kernels was achieved by Frechet functional derivative. Explicit calculation of the functional derivative for the output was solved using mean-value theorem, while the sifting property of the Dirac function was used in order to solve derivative of input. Attention is given to the calculation of the second-order functional derivative. The procedure for adding linear kernels to the composition of a quadratic kernel is described. All techniques of the method to other components of the model are described in detail. The method's outcome is differential equation, which allows representation of kernels. An illustrative example of the transformation of the nonlinear differential Riccati equation is considered. The form of a subsystem consisting of linear and quadratic kernels for adding to a complex system is shown. Linear and quadratic kernels were parameterized using operational calculus within the example. The agreement of the obtained analytical results, with the frequency characteristics, which was obtained by the test signals method, is shown.

The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching

Abstract It is more difficult to give Laplace transform directly in a defined form or derive it by Fourier transform in mathematics teaching. The article gives a solution for solving high exponential series sum by using Laplace transform. With the help of Laplace transform, calculus operations can be transformed into complex plane algebra operations. The application of the algorithm to the option hedging strategy verifies the applicability of the algorithm proposed in this article.

Pestov identities and X-ray tomography on manifolds of low regularity

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $ (M,g) $ with $ g \in C^{1,1} $. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $ C^{1,1} $-regularity is optimal on the Hölder scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.

SYMMETRICALLY CONFORMABLE FRACTIONAL DIFFERENTIAL OPERATORS BY COMPUTATIONAL NUMERICAL MODELING WITH SPECIAL FUNCTION

The [Formula: see text]-convoluted operators related to the [Formula: see text]-Whittaker function, confluent hypergeometric function of the first kind, have been developed using the [Formula: see text]-symbol calculus in which this sort of calculus presents a generalization of the gamma function. [Formula: see text]-symbol fractional calculus is employed to generalize and extend many differential and integral operators of fractional calculus. Based on this premise, a new geometric formula for normalized functions in the symmetric domain known as the open unit disk using the conformable fractional differential operator has been presented in this study. Thus, our technique entails investigating the most well-known geometric properties of this new operator, such as the subordination features and coefficient bounds so that the theory of differential subordination can be adjusted accordingly. By means of this technique, numerical results have been investigated for the proposed method. To this end, a few prominent corollaries of our primary findings as standout instances have been pointed out based on the positivity of the solutions, computational and numerical analyses.

Analysis of the Risk Factors for Massive Hemorrhage after PCNL in the Oblique Supine Position.

Percutaneous nephrolithotomy (PCNL) is a proven and efficient treatment method; Nevertheless, it is essential to note that there is still a risk of significant bleeding. The purpose of this paper is to explore the risk factors for massive hemorrhage after PCNL in the oblique supine position and provide a basis for the development of measures to prevent massive hemorrhage. The clinical data of 97 patients who underwent PCNL in the oblique supine position at Changshu No. 2 People's Hospital from January 2019 to December 2020 were retrospectively analyzed. Patients were placed in the massive hemorrhage group if their hemoglobin levels decreased by ≥20 g/L 24 h after the operation, and the other patients were placed in the nonmassive hemorrhage group. Differences in sex, age, body mass index (BMI), hypertension, diabetes, surgical side, perirenal fat stranding (PFS), calculus long diameter, surgical access, and operation time were compared between the two groups to determine the risk factors for massive bleeding. Multivariable logistic regression analysis was used to determine the risk factors for massive hemorrhage after PCNL. There were no significant differences in sex, BMI, hypertension, diabetes, surgical side, or calculus long diameter between the two groups (p > 0.05), and there were statistically significant differences in age, PFS, surgical access, and operation time (p < 0.05). Multivariate logistic regression analysis indicated that PFS and extensive surgical access were independent risk factors (p < 0.05). PFS and extensive surgical access were independent risk factors. Carefully reading computed tomography (CT) films before surgery and reducing the size of the surgical access area are important measures for reducing the risk of massive hemorrhages.

Torsional oscillations of armature shaft of generator of main diesel locomotive in diesel start-up mode

The article considers a mechanical model of torsional vibrations in the starting system of a diesel generator (DG) with two concentrated mass moments of inertia and a previously unaccounted factor of torsional vibrations between the traction generator and the crankshaft of a diesel locomotive of the UzTE16M type. For this model, the Lagrange method is used to derive a system of equations for mass oscillations along the generalized coordinates of "kinematic" and elastic oscillations between the masses of the armature of the traction generator and diesel engine with a variable mass moment of inertia. For the resulting system of equations, by the method of operational calculus, solutions were made for the accepted functions of mass moments of inertia, moments of driving forces, and resistances. Based on the developed model, it is recommended to determine the angular velocity of the diesel crankshaft which is currently used.

On Bessel-Maitland Function and m-Parameter Mittag-Leffler Function Associated with Fractional Calculus Operators and its Applications

On Bessel-Maitland Function and m-Parameter Mittag-Leffler Function Associated with Fractional Calculus Operators and its Applications

Some new Grüss inequalities associated with generalized fractional derivative

&lt;abstract&gt;&lt;p&gt;In this paper, we prove several new integral inequalities for the $ k $-Hilfer fractional derivative operator, which is a fractional calculus operator. As a result, we have a whole new set of fractional integral inequalities. For the generalized fractional derivative, we also use Young's inequality to find new forms of inequalities. Such conclusions for this novel and generalized fractional derivative are extremely useful and valuable in the domains of differential equations and fractional differential calculus, both of which have a strong connections to real-world situations. These findings may stimulate additional research in a variety of fields of pure and applied sciences.&lt;/p&gt;&lt;/abstract&gt;

Certain new applications of Faber polynomial expansion for some new subclasses of $ \upsilon $-fold symmetric bi-univalent functions associated with $ q $-calculus

&lt;abstract&gt;&lt;p&gt;In this article, we define the $ q $-difference operator and Salagean $ q $-differential operator for $ \upsilon $-fold symmetric functions in open unit disk $ \mathcal{U} $ by first applying the concepts of $ q $-calculus operator theory. Then, we considered these operators in order to construct new subclasses for $ \upsilon $-fold symmetric bi-univalent functions. We establish the general coefficient bounds $ |a_{\upsilon k+1}| $ for the functions in each of these newly specified subclasses using the Faber polynomial expansion method. Investigations are also performed on Feketo-Sezego problems and initial coefficient bounds for the function $ h $ that belong to the newly discovered subclasses. To illustrate the relationship between the new and existing research, certain well-known corollaries of our main findings are also highlighted.&lt;/p&gt;&lt;/abstract&gt;

Impact on the beam of the final length of the moving mass

There have been considered solving direct problem of deforming of isotropic, elastic and hingedly supported beam of finite length. A roller moving at a constant speed along the axis of the beam acts on the beam. The roller has a cylindrical shape of a certain radius and a length that is greater than or equal to the width of the beam. The differential equations of beam motion are analyzed from the point of view of the influence of their components and especially the right-hand parts of the equations on the dynamic behavior of the beam in the case of using certain common materials of the beam and roller, and an option to reduce the equations to a more simplified form is proposed. The unknown functions included in the equations are sought in the form of Fourier series. This allows us to reduce the original equations to ordinary differential equations, which are solved using the Laplace transform. Expressions for coefficients in Fourier series are found using operational calculus and the residue theory. The results of the first numerical experiment on the study of the influence of the roller speed on beam deflections are presented in the form of curves in the figure. For a specific calculated mechanical system in the form of a steel roller, which moves along a steel beam at a constant speed under zero initial conditions, the research results are presented in the form of graphs of beam deflections for different speeds of the roller. The second numerical experiment was carried out to study the propagation of vibrational waves of the beam in the case of motion of the roller at a sufficiently high speed. For this, the figure shows the combined shapes of the beam and the position of the roller at different moments of action of the moving mass. The behavior of the beam at a high speed of movement of the roller was analyzed and a comparison of the deflections of the beam with the deflections of the static model of the beam was made. Further directions for the development of the problem in applied fields of technology and in inverse problems of identifying unknown parameters by indirect manifestations are outlined.

Investigation for the k-analogue of τ-Gauss hypergeometric matrix functions and associated fractional calculus

In this manuscript, we present a new definition of ( k , τ ) -Gauss hypergeometric matrix function and study its analytical properties, like derivative formulas and integral representations. Furthermore, as an application we establish k -fractional calculus operators for the novel matrix function. We also give some new and known results as special cases of our proposed generalization of the Wright hypergeometric matrix function.

Anisotropic global microlocal analysis for tempered distributions

We study an anisotropic version of the Shubin calculus of pseudodifferential operators on \({\textbf{R}}^{d}\). Anisotropic symbols and Gabor wave front sets are defined in terms of decay or growth along curves in phase space of power type parametrized by one positive parameter that distinguishes space and frequency variables. We show that this gives subcalculi of Shubin’s isotropic calculus, and we show a microlocal as well as a microelliptic inclusion in the framework. Finally we prove an inclusion for the anisotropic Gabor wave front set of chirp type oscillatory functions with a real polynomial phase function.