Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 04/2007, Volume 76, Issue 258, pp. 867 - 877

In this paper we derive two formulas for divided differences of a function of a function. Both formulas lead to other divided difference formulas, such as...

Integers | Interpolation | Composite functions | Product rule | Quotient rule | Chain rule | Polynomials | Mathematical functions | Coefficients | Divided differences | Faà di Bruno's formula | INTERPOLATION | MATHEMATICS, APPLIED | chain rule | Faa di Bruno's formula | divided differences

Integers | Interpolation | Composite functions | Product rule | Quotient rule | Chain rule | Polynomials | Mathematical functions | Coefficients | Divided differences | Faà di Bruno's formula | INTERPOLATION | MATHEMATICS, APPLIED | chain rule | Faa di Bruno's formula | divided differences

Journal Article

Topology and its Applications, ISSN 0166-8641, 02/2018, Volume 235, pp. 375 - 427

In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully...

Abelian categories | Faà di Bruno formula | Chain rule | Functor calculus | Cartesian differential categories | MATHEMATICS | MATHEMATICS, APPLIED | Faadi Bruno formula

Abelian categories | Faà di Bruno formula | Chain rule | Functor calculus | Cartesian differential categories | MATHEMATICS | MATHEMATICS, APPLIED | Faadi Bruno formula

Journal Article

Turkish Journal of Mathematics, ISSN 1300-0098, 2014, Volume 38, Issue 3, pp. 558 - 575

We give moment equalities for sums of independent and identically distributed random variables including, in particular, centered and specifically symmetric...

Bootstrap | Faà di bruno's chain rule | Integer partitions | Marcinkiewicz-Zygmund inequalities | Moments | Self-normalized sums | MATHEMATICS | Faa. di Bruno's chain rule | bootstrap | integer partitions | self-normalized sums

Bootstrap | Faà di bruno's chain rule | Integer partitions | Marcinkiewicz-Zygmund inequalities | Moments | Self-normalized sums | MATHEMATICS | Faa. di Bruno's chain rule | bootstrap | integer partitions | self-normalized sums

Journal Article

BIT Numerical Mathematics, ISSN 0006-3835, 2010, Volume 50, Issue 3, pp. 577 - 586

In this paper we derive a formula for divided differences of composite functions of several variables with respect to rectangular grids of points. Letting the...

Chain rule | Divided difference | Calculus of several variables | Faa di Bruno formula | INTERPOLATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | DIBRUNO,FAA | DI-BRUNO FORMULA

Chain rule | Divided difference | Calculus of several variables | Faa di Bruno formula | INTERPOLATION | COMPUTER SCIENCE, SOFTWARE ENGINEERING | MATHEMATICS, APPLIED | DIBRUNO,FAA | DI-BRUNO FORMULA

Journal Article

Discrete Mathematics, ISSN 0012-365X, 03/2011, Volume 311, Issue 6, pp. 387 - 392

Fa di Bruno's formula is the higher chain rule for differentiation. By means of Gessel's q-composition we derive a q-analogue of Fa di Bruno's determinant...

q-analogue | Complete Bell polynomial | Fa di Bruno's formula | Determinant | MATHEMATICS | DIVIDED DIFFERENCE FORM | Faa di Bruno's formula | Bells | Composite functions | Mathematical analysis | Chains | Determinants | Derivatives | Differentiation

q-analogue | Complete Bell polynomial | Fa di Bruno's formula | Determinant | MATHEMATICS | DIVIDED DIFFERENCE FORM | Faa di Bruno's formula | Bells | Composite functions | Mathematical analysis | Chains | Determinants | Derivatives | Differentiation

Journal Article

The ANZIAM Journal, ISSN 1446-1811, 1/2007, Volume 48, Issue 3, pp. 327 - 341

The Faa di Bruno formule for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial...

Faà di Bruno formula | Multivariate Taylor series | Integral remainder term | Differentiation theory | Multivariate composite functions | POLYNOMIALS | MATHEMATICS, APPLIED | multivariate composite functions | CHAIN RULE | Faa di Bruno formula | multivariate Taylor series | integral remainder term | PARTITIONS | DIFFERENTIATION | differentiation theory

Faà di Bruno formula | Multivariate Taylor series | Integral remainder term | Differentiation theory | Multivariate composite functions | POLYNOMIALS | MATHEMATICS, APPLIED | multivariate composite functions | CHAIN RULE | Faa di Bruno formula | multivariate Taylor series | integral remainder term | PARTITIONS | DIFFERENTIATION | differentiation theory

Journal Article

Journal of Pure and Applied Algebra, ISSN 0022-4049, 10/2019, Volume 223, Issue 10, pp. 4191 - 4225

Differential categories were introduced by Blute, Cockett, and Seely as categorical models of differential linear logic and have since led to abstract...

Faà di Bruno formula | Hurwitz series rings | Differential categories | Differential algebras | FORMS | MATHEMATICS | MATHEMATICS, APPLIED | MODELS | Faa di Bruno formula | Analysis | Algebra | Mathematics - Category Theory

Faà di Bruno formula | Hurwitz series rings | Differential categories | Differential algebras | FORMS | MATHEMATICS | MATHEMATICS, APPLIED | MODELS | Faa di Bruno formula | Analysis | Algebra | Mathematics - Category Theory

Journal Article

International Journal of Mathematical Education in Science and Technology, ISSN 0020-739X, 09/2009, Volume 40, Issue 6, pp. 842 - 845

Moments and cumulants are expressed in terms of each other using Bell polynomials. Inbuilt routines for the latter make these expressions amenable to use by...

Secondary 60E05 | Primary 62E30 | moments | cumulants; Faa di Bruno's chain rule | Bell polynomials | Cumulants; Faa di Bruno's chain rule | Moments | Probability | Algebra | Mathematical Formulas | Statistics

Secondary 60E05 | Primary 62E30 | moments | cumulants; Faa di Bruno's chain rule | Bell polynomials | Cumulants; Faa di Bruno's chain rule | Moments | Probability | Algebra | Mathematical Formulas | Statistics

Journal Article

Journal of Computational Mathematics, ISSN 0254-9409, 7/2006, Volume 24, Issue 4, pp. 553 - 560

The n-divided difference of the composite function h := f o g of functions f, g at a group of nodes t₀, t₁, ···, tn is shown by the combinations of divided...

Interpolation | Indices of summation | Composite functions | Higher order derivatives | Chain rule | Polynomials | Mathematical functions | Computational mathematics | Newton interpolation | Divided difference | FAÀ di Bruno's formula | Bell polynomial | Composite function | MATHEMATICS | MATHEMATICS, APPLIED | divided difference | Faa di Bruno's formula | composite function

Interpolation | Indices of summation | Composite functions | Higher order derivatives | Chain rule | Polynomials | Mathematical functions | Computational mathematics | Newton interpolation | Divided difference | FAÀ di Bruno's formula | Bell polynomial | Composite function | MATHEMATICS | MATHEMATICS, APPLIED | divided difference | Faa di Bruno's formula | composite function

Journal Article

UTILITAS MATHEMATICA, ISSN 0315-3681, 03/2016, Volume 99, pp. 43 - 62

Faa di Bruno gave a chain rule for the derivatives of a function of a scalar function of a scalar. It is succinctly expressed in terms of Bell polynomials. We...

MATHEMATICS, APPLIED | MODELS | Multivariate Bell polynomials | Faa di Bruno | MULTIVARIATE HERMITE-POLYNOMIALS | Chain rule | STATISTICS & PROBABILITY | Derivatives of a function of a function | DIFFERENTIATION | Partial derivatives

MATHEMATICS, APPLIED | MODELS | Multivariate Bell polynomials | Faa di Bruno | MULTIVARIATE HERMITE-POLYNOMIALS | Chain rule | STATISTICS & PROBABILITY | Derivatives of a function of a function | DIFFERENTIATION | Partial derivatives

Journal Article

Utilitas Mathematica, ISSN 0315-3681, 11/2012, Volume 89, pp. 113 - 127

The chain rule for the derivatives of a composite function g(p) circle ... circle g(1) can be succinctly expressed in terms of the Bell polynomials of the...

Chain rule | Faa di bruno | Derivatives | Composite functions | Bell polynomials | MATHEMATICS, APPLIED | Faa di Bruno | STATISTICS & PROBABILITY

Chain rule | Faa di bruno | Derivatives | Composite functions | Bell polynomials | MATHEMATICS, APPLIED | Faa di Bruno | STATISTICS & PROBABILITY

Journal Article

Utilitas Mathematica, ISSN 0315-3681, 07/2011, Volume 85, pp. 65 - 86

Faa di Bruno gave a chain rule for the derivatives of a function of a function. It is succinctly expressed in terms of Bell polynomials. This has a...

Deriva-tives | Log-normal | Composite functions | Bell polynomials | Faa di Bruno | Chain rule | Hermite polynomials | POLYNOMIALS | MATHEMATICS, APPLIED | STATISTICS & PROBABILITY | Derivatives

Deriva-tives | Log-normal | Composite functions | Bell polynomials | Faa di Bruno | Chain rule | Hermite polynomials | POLYNOMIALS | MATHEMATICS, APPLIED | STATISTICS & PROBABILITY | Derivatives

Journal Article

2014 IEEE Workshop on Statistical Signal Processing (SSP), ISSN 2373-0803, 06/2014, pp. 217 - 219

Volterra series are used for modelling nonlinear systems with memory effects. The n th -order impulse response and the kernels in the series can be determined...

Conferences | FAA | Signal processing | Educational institutions | Calculus | Mathematical model | Kernel

Conferences | FAA | Signal processing | Educational institutions | Calculus | Mathematical model | Kernel

Conference Proceeding

SIAM Review, ISSN 0036-1445, 6/2014, Volume 56, Issue 2, pp. 353 - 370

We show that the ordinary derivative of a real analytic function of one variable can be realized as a Grassmann-Berezin-type integration over the Zeon algebra,...

Differential calculus | Algebra | Mathematical theorems | Analytic functions | Mathematical integrals | Abstract algebra | Hermite polynomials | EDUCATION | Combinatorics | Fibonacci numbers | Mathematical integration | Faà di Bruno formula | Spivey identity | Zeon algebra | Special polynomials | Grassmann-Berezin integration | Combinatorial numbers | Cauchy integral | special polynomials | MATHEMATICS, APPLIED | combinatorial numbers | OPERATOR | NUMBERS | Faa di Bruno formula | ENUMERATION | Usage | Polynomials | Combinatorial number theory | Analysis | Tests, problems and exercises | Cauchy problem | Studies | Integral equations | Mathematical analysis

Differential calculus | Algebra | Mathematical theorems | Analytic functions | Mathematical integrals | Abstract algebra | Hermite polynomials | EDUCATION | Combinatorics | Fibonacci numbers | Mathematical integration | Faà di Bruno formula | Spivey identity | Zeon algebra | Special polynomials | Grassmann-Berezin integration | Combinatorial numbers | Cauchy integral | special polynomials | MATHEMATICS, APPLIED | combinatorial numbers | OPERATOR | NUMBERS | Faa di Bruno formula | ENUMERATION | Usage | Polynomials | Combinatorial number theory | Analysis | Tests, problems and exercises | Cauchy problem | Studies | Integral equations | Mathematical analysis

Journal Article

Nonlinear Dynamics, ISSN 0924-090X, 7/2008, Volume 53, Issue 1, pp. 67 - 74

Fractional mechanics describe both conservative and nonconservative systems. The fractional variational principles gained importance in studying the fractional...

Automotive and Aerospace Engineering, Traffic | Fractional Riemann–Liouville derivative | Faà di Bruno formula | Engineering | Vibration, Dynamical Systems, Control | Fractional Euler–Lagrange equations | Mechanics | Fractional Lagrangians | Mechanical Engineering | Fractional calculus | Fractional Euler-Lagrange equations | Fractional Riemann-Liouville derivative | CLASSICAL FIELDS | fractional calculus | SEQUENTIAL MECHANICS | Faa di Bruno formula | CALCULUS | fractional Lagrangians | FORMULATION | ENGINEERING, MECHANICAL | fractional Riemann-Liouville derivative | fractional Euler-Lagrange equations | MECHANICS | LINEAR VELOCITIES | SYSTEMS | VARIATIONAL-PROBLEMS | FORMALISM | Mechanics (physics) | Mathematical analysis | Classical mechanics | Euler-Lagrange equation | Equations of motion | Variational principles

Automotive and Aerospace Engineering, Traffic | Fractional Riemann–Liouville derivative | Faà di Bruno formula | Engineering | Vibration, Dynamical Systems, Control | Fractional Euler–Lagrange equations | Mechanics | Fractional Lagrangians | Mechanical Engineering | Fractional calculus | Fractional Euler-Lagrange equations | Fractional Riemann-Liouville derivative | CLASSICAL FIELDS | fractional calculus | SEQUENTIAL MECHANICS | Faa di Bruno formula | CALCULUS | fractional Lagrangians | FORMULATION | ENGINEERING, MECHANICAL | fractional Riemann-Liouville derivative | fractional Euler-Lagrange equations | MECHANICS | LINEAR VELOCITIES | SYSTEMS | VARIATIONAL-PROBLEMS | FORMALISM | Mechanics (physics) | Mathematical analysis | Classical mechanics | Euler-Lagrange equation | Equations of motion | Variational principles

Journal Article

Discrete Mathematics, ISSN 0012-365X, 2011, Volume 311, Issue 6, pp. 387 - 392

Faà di Bruno’s formula is the higher chain rule for differentiation. By means of Gessel’s q -composition we derive a q -analogue of Faà di Bruno’s determinant...

Determinant | Complete Bell polynomial | Faà di Bruno’s formula | [formula omitted]-analogue

Determinant | Complete Bell polynomial | Faà di Bruno’s formula | [formula omitted]-analogue

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 6/2018, Volume 46, Issue 2, pp. 323 - 344

The commutative ring $$R(P(t))=\mathbb C[t^{\pm 1},u \,|\, u^2=P(t)]$$ R(P(t))=C[t±1,u|u2=P(t)] , where $$P(t)=\sum _{i=0}^na_it^i=\prod _{k=1}^n(t-\alpha...

DJKM algebras | Functions of a Complex Variable | Universal central extensions | Superelliptic Lie algebras | Superelliptic curves | Field Theory and Polynomials | Mathematics | Fáa di Bruno’s formula | 14H55 | Automorphism groups | 42C10 | Fourier Analysis | Pell’s equation | Bell polynomials | 17B65 | Krichever–Novikov algebras | Associated Legendre polynomials | Number Theory | Combinatorics | 05E35 | CENTRAL EXTENSIONS | Faa di Bruno's formula | KRICHEVER-NOVIKOV-TYPE | MATHEMATICS | Pell's equation | Krichever-Novikov algebras | Algebra

DJKM algebras | Functions of a Complex Variable | Universal central extensions | Superelliptic Lie algebras | Superelliptic curves | Field Theory and Polynomials | Mathematics | Fáa di Bruno’s formula | 14H55 | Automorphism groups | 42C10 | Fourier Analysis | Pell’s equation | Bell polynomials | 17B65 | Krichever–Novikov algebras | Associated Legendre polynomials | Number Theory | Combinatorics | 05E35 | CENTRAL EXTENSIONS | Faa di Bruno's formula | KRICHEVER-NOVIKOV-TYPE | MATHEMATICS | Pell's equation | Krichever-Novikov algebras | Algebra

Journal Article

The Journal of Geometric Analysis, ISSN 1050-6926, 7/2018, Volume 28, Issue 3, pp. 2602 - 2608

We prove that a certain positivity condition, considerably more general than pseudoconvexity, enables one to conclude that the regular and singular orders of...

Faa di Bruno formula | Positivity property | 32F18 | Mathematics | Real hypersurface germ | Abstract Harmonic Analysis | Fourier Analysis | Finite type conditions | Convex and Discrete Geometry | Global Analysis and Analysis on Manifolds | Plurisubharmonic function | Differential Geometry | Dynamical Systems and Ergodic Theory | 32T27 | 32U05 | 32T25 | 32V35 | MATHEMATICS | PSEUDOCONVEX DOMAINS | NEUMANN PROBLEM | CONVEX DOMAINS

Faa di Bruno formula | Positivity property | 32F18 | Mathematics | Real hypersurface germ | Abstract Harmonic Analysis | Fourier Analysis | Finite type conditions | Convex and Discrete Geometry | Global Analysis and Analysis on Manifolds | Plurisubharmonic function | Differential Geometry | Dynamical Systems and Ergodic Theory | 32T27 | 32U05 | 32T25 | 32V35 | MATHEMATICS | PSEUDOCONVEX DOMAINS | NEUMANN PROBLEM | CONVEX DOMAINS

Journal Article

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, ISSN 0191-2216, 12/2009, pp. 3155 - 3161

This paper gives a bound on the continuity of solutions to nonlinear ordinary differential equations. Continuity is measured with respect to an arbitrary...

Sensitivity analysis | Stability | Veins | Differential equations | FAA | Lyapunov method | Aerospace materials | Polynomials | Functional programming | Aerospace engineering

Sensitivity analysis | Stability | Veins | Differential equations | FAA | Lyapunov method | Aerospace materials | Polynomials | Functional programming | Aerospace engineering

Conference Proceeding

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