scholarly journals Inverse Derivative Operator and Umbral Methods for the Harmonic Numbers and Telescopic Series Study

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 781
Author(s):  
Giuseppe Dattoli ◽  
Silvia Licciardi ◽  
Rosa Maria Pidatella
Keyword(s):  
Differential Operators ◽  
Integral Operators ◽  
Integral Point ◽  
Point Of View ◽  
Harmonic Numbers ◽  
Integral Calculus ◽  
Negative Order ◽  
Derivatives Of ◽  
Series Study ◽  
Infinitely Differentiable Function

The formalism of differ-integral calculus, initially developed to treat differential operators of fractional order, realizes a complete symmetry between differential and integral operators. This possibility has opened new and interesting scenarios, once extended to positive and negative order derivatives. The associated rules offer an elegant, yet powerful, tool to deal with integral operators, viewed as derivatives of order-1. Although it is well known that the integration is the inverse of the derivative operation, the aforementioned rules offer a new mean to obtain either an explicit iteration of the integration by parts or a general formula to obtain the primitive of any infinitely differentiable function. We show that the method provides an unexpected link with generalized telescoping series, yields new useful tools for the relevant treatment, and allows a practically unexhausted tool to derive identities involving harmonic numbers and the associated generalized forms. It is eventually shown that embedding the differ-integral point of view with techniques of umbral algebraic nature offers a new insight into, and the possibility of, establishing a new and more powerful formalism.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Some 2-substituted derivatives of 5-(2-amino-6-hydroxy-4-oxo-3,4-dihydro-5-pyrimidinyl)pentanoic acid

1981 ◽  
Vol 46 (6) ◽  
pp. 1397-1404 ◽  
Author(s):  
Rudolf Kotva ◽  
Jiří Křepelka ◽  
Antonín Černý ◽  
Vojtěch Pujman ◽  
Miroslav Semonský
Keyword(s):  
Point Of View ◽  
Pentanoic Acid ◽  
Antineoplastic Effect ◽  
Derivatives Of

On condensation of 6-substituted trialkyl esters of 2-carboxy-1,7-heptanedioic acids XIII-XXIII with guanidineand subsequent saponification 2-substituted 5-(2-amino-6-hydroxy-4-oxo-3,4-dihydro-5-pyrimidinyl)pentanoic acids II-XII were prepared. From the pharmacological point of view some of the substances prepared had a potentiating effect on the antileukemia effect of 5-fluorouracil in mice and the antineoplastic effect manifested by a diminution of the tumours in animals with experimental tumours.

Some misinterpretations and lack of understanding in differential operators with no singular kernels

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 594-612 ◽  
Author(s):  
Abdon Atangana ◽  
Emile Franc Doungmo Goufo
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Initial Value Problems ◽  
Differential Operators ◽  
Integral Operators ◽  
Initial Value ◽  
Complex Processes ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Theoretical Results

AbstractHumans are part of nature, and as nature existed before mankind, mathematics was created by humans with the main aim to analyze, understand and predict behaviors observed in nature. However, besides this aspect, mathematicians have introduced some laws helping them to obtain some theoretical results that may not have physical meaning or even a representation in nature. This is also the case in the field of fractional calculus in which the main aim was to capture more complex processes observed in nature. Some laws were imposed and some operators were misused, such as, for example, the Riemann–Liouville and Caputo derivatives that are power-law-based derivatives and have been used to model problems with no power law process. To solve this problem, new differential operators depicting different processes were introduced. This article aims to clarify some misunderstandings about the use of fractional differential and integral operators with non-singular kernels. Additionally, we suggest some numerical discretizations for the new differential operators to be used when dealing with initial value problems. Applications of some nature processes are provided.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

SUSY coherent states and enhanced canonical quantizations for Pauli model

2021 ◽  
pp. 2150105
Author(s):  
Jean Vignon Hounguevou ◽  
Daniel Sabi Takou ◽  
Gabriel Y. H. Avossevou
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Coherent States ◽  
Differential Operators ◽  
Canonical Quantization ◽  
Point Of View ◽  
Supersymmetric Quantum Mechanics ◽  
Geometric Properties

In this paper, we study coherent states for a quantum Pauli model through supersymmetric quantum mechanics (SUSYQM) method. From the point of view of canonical quantization, the construction of these coherent states is based on the very important differential operators in SUSYQM call factorization operators. The connection between classical and quantum theory is given by using the geometric properties of these states.

scholarly journals Comparative analysis of efficiency of democracy in Ukraine according to the indices of democracy The Economist Democracy Index, Freedom In the World and Polity IV

2020 ◽  
Vol 23 (8) ◽  
pp. 78-92
Author(s):  
Kateryna Fedoryshynа
Keyword(s):  
Comparative Analysis ◽  
Civil Liberties ◽  
Integral Point ◽  
Holistic Approach ◽  
Point Of View ◽  
Executive Power ◽  
Time Period ◽  
The World ◽  
Final Score ◽  
Legal Restrictions

This article represents an analysis of efficiency of Ukrainian democracy within the framework of three popural indices of democracy – The Economist Democracy Index, Freedom In the World index and Polity IV. Comparative analysis shows the core factors which bring three different democratic concepts, used in the indices, to the integral unity. Finding correlation between factors of Ukrainian democracy, measured in the indices through a certain time period (2006-2018), helps getting integral look at the problem of non existent universal theoretic base for understanding democracy. The basic idea of the analysis, represented in this article, shows that different factors, used by indices in measuring democracy, do not evenly correlate in practice, though they represent holistic approach to the essence of democracy. Choosing specific theoretical approach of understanding democracy makes it hard for indices to fully measure real democracy. This analysis aims at searching correlation in different basic factors of democratic models, used by indices with different approaches. As the result of the analysis the article ranks a number of basic factors, used in three popular indices of democracy, according to the strength of correlation of these factors with other factors of the index they represent and with the final score of the index. Integral choice of the basic factors, which correlate with the change of Ukraine’s democratic trends according to the three indices, covers several dimensions of democratic model. Ukrainian democratic trends in the specific time period (2006-2018), as the analysis shows, from integral point of view correlates the most with the changes in electoral process and pluralism, civil liberties and legal restrictions of the executive power. Political culture, political participation and individual rights show weak correlation with Ukrainian democratic trends within the period of time, chosen for the analysis.

scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

scholarly journals Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space

1994 ◽  
Vol 135 ◽  
pp. 165-196 ◽  
Author(s):  
Masatake Miyake ◽  
Masafumi Yoshino
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Newton Polygon ◽  
Index Theorem ◽  
Fredholm Property ◽  
Goursat Problem ◽  
Point Of View ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Gevrey Spaces

In the study of ordinary differential equations, Malgrange ([Ma]) and Ramis ([R1], [R2]) established index theorem in (formal) Gevrey spaces, and the notion of irregularity was nicely defined for the study of irregular points. In their studies, a Newton polygon has a great advantage to describe and understand the results in visual form. From this point of view, Miyake ([M2], [M3], [MH]) studied linear partial differential operators on (formal) Gevrey spaces and proved analogous results, and showed the validity of Newton polygon in the study of partial differential equations (see also [Yn]).

scholarly journals DFT Visualization and Experimental Evidence of BHT-Mg-Catalyzed Copolymerization of Lactides, Lactones and Ethylene Phosphates

Polymers ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1641 ◽  
Author(s):  
Ilya Nifant’ev ◽  
Andrey Shlyakhtin ◽  
Maxim Kosarev ◽  
Dmitry Gavrilov ◽  
Stanislav Karchevsky ◽  
...  
Keyword(s):  
Ring Opening ◽  
Density Functional ◽  
Point Of View ◽  
Basis Set ◽  
Random Copolymers ◽  
Key Factors ◽  
Functional Theory ◽  
Cyclic Esters ◽  
Dft Modeling ◽  
Derivatives Of

Catalytic ring-opening polymerization (ROP) of cyclic esters (lactides, lactones) and cyclic ethylene phosphates is an effective way to process materials with regulated hydrophilicity and controlled biodegradability. Random copolymers of cyclic monomers of different chemical nature are highly attractive due to their high variability of characteristics. Aryloxy-alkoxy complexes of non-toxic metals such as derivatives of 2,6-di-tert-butyl-4-methylphenoxy magnesium (BHT-Mg) complexes are effective coordination catalysts for homopolymerization of all types of traditional ROP monomers. In the present paper, we report the results of density functional theory (DFT) modeling of BHT-Mg-catalyzed copolymerization for lactone/lactide, lactone/ethylene phosphate and lactide/ethylene phosphate mixtures. ε-Caprolactone (ε-CL), l-lactide (l-LA) and methyl ethylene phosphate (MeOEP) were used as examples of monomers in DFT simulations by the Gaussian-09 program package with the B3PW91/DGTZVP basis set. Both binuclear and mononuclear reaction mechanistic concepts have been applied for the calculations of the reaction profiles. The results of calculations predict the possibility of the formation of random copolymers based on l-LA/MeOEP, and substantial hindrance of copolymerization for ε-CL/l-LA and ε-CL/MeOEP pairs. From the mechanistic point of view, the formation of highly stable five-membered chelate by the products of l-LA ring-opening and high donor properties of phosphates are the key factors that rule the reactions. The results of DFT modeling have been confirmed by copolymerization experiments.

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