Providing Motivation for the Chain Rule

1967 ◽  
Vol 60 (8) ◽  
pp. 850-855
Author(s):  
Lyman S. Holden
Keyword(s):  
General Formula ◽  
Differential Calculus ◽  
Geometric Interpretation ◽  
Chain Rule

The purpose of this article is to describe how a certain graphing technique can be used to increase the heuristic appeal of a general formula from the differential calculus. The formula to which reference is made is the formula for the derivative of the composition of two functions (the chain rule). We shall interpret this rule in relation to the graph of the function that is the composition of two functions. Using the geometric interpretation of the derivative as the slope of a curve, it will be possible to provide improved motivation for the rule.

scholarly journals A Framework for Differential Calculus on Persistence Barcodes

2021 ◽  
Author(s):  
Jacob Leygonie ◽  
Steve Oudot ◽  
Ulrike Tillmann
Keyword(s):  
Data Analysis ◽  
Differential Calculus ◽  
Gradient Descent ◽  
Chain Rule ◽  
Topological Data Analysis ◽  
Objective Functions ◽  
A Chain ◽  
Topological Data

AbstractWe define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively, from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

scholarly journals Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
S. Gaboury ◽  
R. Tremblay
Keyword(s):  
General Formula ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Mathematical Physics ◽  
Chain Rule ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Several Variables ◽  
Special Cases

In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell F1 function, and the Lauricella functions of several variables FD(n) are given.

scholarly journals Cumulants of the q-semicircular Law, Tutte Polynomials, and Heaps

2013 ◽  
Vol 65 (4) ◽  
pp. 863-878 ◽  
Author(s):  
Matthieu Josuat Vergès
Keyword(s):  
General Formula ◽  
Hypergeometric Series ◽  
Symmetric Functions ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Combinatorial Interpretation ◽  
Free Cumulants ◽  
Combinatorial Properties ◽  
Semicircular Law ◽  
Tutte Polynomials

AbstractThe q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of itsmoments in terms ofmatchings, where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case q = 0 of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier–Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

scholarly journals The noncommutative geometry of the Landau Hamiltonian: Differential aspects

2021 ◽  
Author(s):  
Giuseppe De Nittis ◽  
Maximiliano Sandoval
Keyword(s):  
Dirac Operator ◽  
Noncommutative Geometry ◽  
Differential Calculus ◽  
Quantum Hall ◽  
Geometric Interpretation ◽  
Quantum Harmonic Oscillator ◽  
C Algebra ◽  
Dense Subalgebra ◽  
Dixmier Trace ◽  
Do So

Abstract In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of R2. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

Mathematical simulation of error functions of decision-making at tolerance control of measuring techniques availability

Metrologiya ◽  
2020 ◽  
pp. 3-15
Author(s):  
Rustam Z. Khayrullin ◽  
Alexey S. Kornev ◽  
Andrew A. Kostoglotov ◽  
Sergey V. Lazarenko
Keyword(s):  
Measurement Error ◽  
Geometric Interpretation ◽  
Measuring Instruments ◽  
Measured Quantity ◽  
Metrological Examination ◽  
Technical Documentation ◽  
Error Functions ◽  
Measuring Techniques ◽  
Tolerance Control ◽  
Laws Of Distribution

Analytical and computer models of false failure and undetected failure (error functions) were developed with tolerance control of the parameters of the components of the measuring technique. A geometric interpretation of the error functions as two-dimensional surfaces is given, which depend on the tolerance on the controlled parameter and the measurement error. The developed models are applicable both to theoretical laws of distribution, and to arbitrary laws of distribution of the measured quantity and measurement error. The results can be used in the development of metrological support of measuring equipment, the verification of measuring instruments, the metrological examination of technical documentation and the certification of measurement methods.

X-ray Structure Refinement and Vibrational Spectroscopy of Ca8Gd2 (PO4)6 O2

2013 ◽  
Vol 12 (10) ◽  
pp. 719-726
Author(s):  
R. Ayadi ◽  
Mohamed Boujelbene ◽  
T. Mhiri
Keyword(s):  
Solid State ◽  
General Formula ◽  
Vibrational Spectroscopy ◽  
Powder Diffraction ◽  
Infrared Spectra ◽  
Solid State Reaction ◽  
Structure Refinement ◽  
Unit Cell ◽  
Cell Group ◽  
X Ray

The present paper is interested in the study of compounds from the apatite family with the general formula Ca10 (PO4)6A2. It particularly brings to light the exploitation of the distinctive stereochemistries of two Ca positions in apatite. In fact, Gd-Bearing oxyapatiteCa8 Gd2 (PO4)6O2 has been synthesized by solid state reaction and characterized by X-ray powder diffraction. The site occupancies of substituents is0.3333 in Gd and 0.3333 for Ca in the Ca(1) position and 0. 5 for Gd in the Ca (2) position.  Besides, the observed frequencies in the Raman and infrared spectra were explained and discussed on the basis of unit-cell group analyses.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

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