70 results on '"Iterated Integrals"'
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2. Construction of Homotopic Invariants of Maps from Spheres to Compact Closed Manifolds
- Author
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I. S. Zubov
- Subjects
Statistics and Probability ,Hopf invariant ,Pure mathematics ,Iterated integrals ,Applied Mathematics ,General Mathematics ,Bibliography ,SPHERES ,Mathematics::Geometric Topology ,Mathematics - Abstract
We study the homotopic classifications of maps from circles and spheres to manifolds and compare the classical approach to define the Hopf invariant with the approach based on Chen’s iterated integrals. Bibliography: 5 titles.
- Published
- 2020
3. The polylog quotient and the Goncharov quotient in computational Chabauty–Kim theory II
- Author
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David Corwin and Ishai Dan-Cohen
- Subjects
Discrete mathematics ,Conjecture ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Galois group ,Boundary (topology) ,01 natural sciences ,Iterated integrals ,Line (geometry) ,0101 mathematics ,Quotient ,Mathematics - Abstract
This is the second installment in a multi-part series starting with Corwin–Dan-Cohen [arXiv:1812.05707v3]. Building on previous work by Dan-Cohen–Wewers, Dan-Cohen, and F. Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of Spec Z \operatorname {Spec}\mathbb {Z} . To do so, we develop a refined version of the algorithm of Dan-Cohen–Wewers tailored specifically to this case. We also commit ourselves fully to working with the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth- 1 1 part of the mixed Tate Galois group studied extensively by Goncharov. An application was given in Corwin–Dan-Cohen [arXiv:1812.05707v3], where we verified Kim’s conjecture in an interesting new case.
- Published
- 2020
4. Modular iterated integrals associated with cusp forms
- Author
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Nikolaos Diamantis
- Subjects
Cusp (singularity) ,Pure mathematics ,Mathematics - Number Theory ,business.industry ,Applied Mathematics ,General Mathematics ,Modular form ,Extension (predicate logic) ,Construct (python library) ,Modular design ,Iterated integrals ,11F37 ,FOS: Mathematics ,Number Theory (math.NT) ,Invariant (mathematics) ,business ,Mathematics - Abstract
We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.
- Published
- 2022
5. Real-Analytic Non-Integrable Functions on the Plane with Equal Iterated Integrals
- Author
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Luis Bernal-González, María del Carmen Calderón-Moreno, Andreas Jung, Universidad de Sevilla. Departamento de Análisis matemático, and Universidad de Sevilla. FQM127: Análisis Funcional no Lineal
- Subjects
Mathematics (miscellaneous) ,Applied Mathematics ,Fubini’s theorem ,Real analytic functions ,Iterated integrals - Abstract
In this note, a vector space of real-analytic functions on the plane is explicitly constructed such that all its nonzero functions are non-integrable but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space is dense in the space of all real continuous functions on the plane endowed with the compact-open topology.
- Published
- 2021
6. Alternative expectation formulas for real-valued random vectors
- Author
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Haruhiko Ogasawara
- Subjects
Statistics and Probability ,021103 operations research ,Distribution (number theory) ,Multivariate random variable ,0211 other engineering and technologies ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,Hoeffding's lemma ,Survival function ,Iterated integrals ,Product (mathematics) ,Applied mathematics ,0101 mathematics ,Random variable ,Mathematics - Abstract
When the elements of a random vector take any real values, formulas of product moments are obtained for continuous and discrete random variables using distribution/survival functions. The r...
- Published
- 2019
7. Fubini’s Theorem
- Author
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Noboru Endou
- Subjects
multiple integral ,Applied Mathematics ,Multiple integral ,68t99 ,product measure ,Algebra ,Computational Mathematics ,03b35 ,Iterated integrals ,Fubini's theorem ,fubini’s theorem ,QA1-939 ,Product measure ,iterated integral ,28a35 ,Mathematics - Abstract
Summary Fubini theorem is an essential tool for the analysis of high-dimensional space [8], [2], [3], a theorem about the multiple integral and iterated integral. The author has been working on formalizing Fubini’s theorem over the past few years [4], [6] in the Mizar system [7], [1]. As a result, Fubini’s theorem (30) was proved in complete form by this article.
- Published
- 2019
8. An extension of the Hermite–Hadamard inequality for convex and s-convex functions
- Author
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Péter Kórus
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Iterated integrals ,Hermite–Hadamard inequality ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Convex function ,Mathematics - Abstract
The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.
- Published
- 2019
9. Invariants of Multidimensional Time Series Based on Their Iterated-Integral Signature
- Author
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Jeremy Reizenstein and Joscha Diehl
- Subjects
FOS: Computer and information sciences ,Pure mathematics ,Partial differential equation ,biology ,Computer Vision and Pattern Recognition (cs.CV) ,Applied Mathematics ,010102 general mathematics ,Computer Science - Computer Vision and Pattern Recognition ,General linear group ,biology.organism_classification ,01 natural sciences ,Ambient space ,010101 applied mathematics ,Chen ,Iterated integrals ,FOS: Mathematics ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Representation Theory ,Mathematics - Abstract
We introduce a novel class of features for multidimensional time series, that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen's iterated-integral signature., Comment: complete rewrite of Section 3.3
- Published
- 2018
10. Perturbation theory of the quadratic Lotka-Volterra double center
- Author
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Lubomir Gavrilov, Jean-Pierre Françoise, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
limit cycles ,Applied Mathematics ,General Mathematics ,Double centers ,Dynamical Systems (math.DS) ,Center (group theory) ,Quadratic system ,Bifurcation function ,Quadratic equation ,Bifurcation theory ,Iterated integrals ,Bautin ideal ,FOS: Mathematics ,iterated integrals ,Perturbation theory ,[MATH]Mathematics [math] ,Mathematics - Dynamical Systems ,34C05, 37F75, 34M35 ,Mathematical physics ,Mathematics - Abstract
We revisit the bifurcation theory of the Lotka-Volterra quadratic system \begin{eqnarray} X_0 :\left\{\begin{aligned} \dot{x}=& - y -x^2+y^2 ,\\ \dot{y}= &\;\;\;\;x - 2xy \end{aligned} \right. \end{eqnarray} with respect to arbitrary quadratic deformations. The system $X_0$ has a double center, which is moreover isochronous. We show that the deformed system $X_0$ can have at most two limit cycles on the finite plane, with possible distribution $(i,j)$, where $i+j\leq2$. Our approach is based on the study of pairs of bifurcation functions associated to the centers, expressed in terms of iterated path integrals of length two., 40 pages
- Published
- 2020
11. Infinite orbit depth and length of Melnikov functions
- Author
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L. Ortiz-Bobadilla, Jessie Pontigo-Herrera, Dmitry Novikov, Pavao Mardešić, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Weizmann Institute of Science [Rehovot, Israël], Instituto de Matematicas (UNAM), Universidad Nacional Autónoma de México (UNAM), and Israel Science Foundation1167/17UNAM PREI Dgapa Unidad Mixta Internacional Laboratorio Solomon Lefschetz (LASOL) FONCICYT Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica (PAPIIT)Universidad Nacional Autonoma de MexicoIN106217ECOS Nord-Conacyt 249542Consejo Nacional de Ciencia y Tecnologia (CONACyT)291231
- Subjects
Polynomial ,Dynamical Systems (math.DS) ,Iterated integrals ,01 natural sciences ,Hamiltonian system ,03 medical and health sciences ,0302 clinical medicine ,FOS: Mathematics ,Center problem ,030212 general & internal medicine ,0101 mathematics ,Mathematics - Dynamical Systems ,[MATH]Mathematics [math] ,Mathematical Physics ,Mathematical physics ,Poincaré map ,Physics ,Conjecture ,Plane (geometry) ,Applied Mathematics ,010102 general mathematics ,MSC : primary, 34C07 ,secondary, 34C05 ,34C08 ,Loop (topology) ,Bounded function ,MAP ,Orbit (control theory) ,Analysis ,34C07, 34C05, 34C08 - Abstract
In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along γ. We construct deformations d F + ϵ ω = 0 whose first nonzero Melnikov function M μ is of length three and explain the difficulties in constructing deformations having high length first nonzero Melnikov functions M μ .
- Published
- 2019
12. Explicit relations between multiple zeta values and related variants
- Author
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Ce Xu
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Logarithm ,Applied Mathematics ,Computation ,010102 general mathematics ,Harmonic (mathematics) ,Star (graph theory) ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Iterated integrals ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Binomial coefficient ,Mathematics - Abstract
In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Apery-type variant of MZSVs ζ B ⋆ ( k ) (called multiple zeta B-star values, abbr. MZBSVs) which involve MHSSs and central binomial coefficients, and establish some explicit connections among MZVs, alternating MZVs and MZBSVs by using the method of iterated integrals. Finally, some interesting consequences and illustrative examples are presented.
- Published
- 2021
13. Period functions associated to real-analytic modular forms
- Author
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Nikolaos Diamantis and Joshua Drewitt
- Subjects
Class (set theory) ,Polynomial ,Mathematics - Number Theory ,business.industry ,Applied Mathematics ,Modular form ,11M99, 11F67 ,Construct (python library) ,Modular design ,Theoretical Computer Science ,Algebra ,Computational Mathematics ,Mathematics (miscellaneous) ,Iterated integrals ,FOS: Mathematics ,Number Theory (math.NT) ,business ,Period (music) ,Mathematics - Abstract
We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals of length one.
- Published
- 2019
14. ITERATED INTEGRAL TRANSFORMS OF ACTIVATED SIGMOID FUNCTION AS RELATED TO CARATHÉODORY FAMILY
- Author
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A. T. Oladipo and O. A. Fadipe-Joseph
- Subjects
Iterated integrals ,Calculus ,Applied mathematics ,Sigmoid function ,Mathematics - Published
- 2016
15. Estimation of Unknown Function of a Class of Retarded Iterated Integral Inequalities
- Author
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Wu-Sheng Wang and Ricai Luo
- Subjects
Estimation ,Class (set theory) ,Inequality ,Iterated integrals ,media_common.quotation_subject ,Calculus ,Applied mathematics ,Function (mathematics) ,Mathematics ,media_common - Published
- 2016
16. Decay rate of iterated integrals of branched rough paths
- Author
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Horatio Boedihardjo
- Subjects
Factorial ,Pure mathematics ,Differential equation ,Applied Mathematics ,010102 general mathematics ,Probability (math.PR) ,Extension (predicate logic) ,01 natural sciences ,010101 applied mathematics ,Mathematics - Classical Analysis and ODEs ,Iterated integrals ,Iterated function ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,High Energy Physics::Experiment ,0101 mathematics ,QA ,Mathematics - Probability ,Mathematical Physics ,Analysis ,Counterexample ,Mathematics - Abstract
Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals., Comment: 26 pages, 2 figures. Accepted version
- Published
- 2018
17. Method of Iterated Integrals for Solution of Stochastic Integrals with Applications to Monte Carlo Simulation of Stochastic Differential Equations
- Author
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Ahsan Amin
- Subjects
Stochastic differential equation ,Hermite polynomials ,Terminal time ,Computer simulation ,Iterated integrals ,Monte Carlo method ,Time derivative ,Applied mathematics ,Initial value problem ,Mathematics - Abstract
In this article we suggest a new method for solutions of stochastic integrals where the dynamics of the variables in integrand are given by some stochastic differential equation. We also propose numerical simulation of stochastic differential equations which is based on iterated integrals method employing Ito change of variable expansion of the drift and volatility function of the SDE. If a function has to be evaluated along time, it would be possible to calculate its value in time using the initial value of the function and an integral of the time derivative of the function till terminal time. Method of iterated integrals is a method of solution of integrals in which we repeatedly expand different time dependent terms in the integral as an initial value and integral of its derivative along time so that their sum would equal the original term in the integrand that was expanded. We continue to repeatedly expand the integrand in time dependent terms into initial value terms and integral of further time dependent higher derivative and continue until we have enough accuracy in the terms evaluated at initial values so as to be able to neglect the time dependent highest derivative terms. We evaluate all these iterated integrals terms at the initial time of the simulation along with the solution of the stochastic integrals which is found in terms of Hermite polynomials and variance of the integrals. We apply the method of iterated integrals to simulation of various stochastic differential equations. We find that the accuracy of simulation sharply increases using the method of iterated integral as compared to naive and simple monte carlo simulation methods.
- Published
- 2018
18. Frontmatter.
- Subjects
- *
TAUBERIAN theorems , *APPLIED mathematics , *ITERATED integrals , *MATHEMATICAL physics , *EISENSTEIN series , *LORENTZ spaces , *QUADRICS - Published
- 2021
- Full Text
- View/download PDF
19. Identities for the multiple zeta (star) values
- Author
-
Ce Xu
- Subjects
Logarithm ,Series (mathematics) ,Mathematics - Number Theory ,Bar (music) ,Applied Mathematics ,Computation ,Star (game theory) ,Mathematics::Number Theory ,010102 general mathematics ,0102 computer and information sciences ,Function (mathematics) ,01 natural sciences ,Combinatorics ,Mathematics (miscellaneous) ,010201 computation theory & mathematics ,Iterated integrals ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By applying the formulas obtained, we prove that the multiple zeta star values whose indices are the sequences $$(\bar{1},\{1\}_m,\bar{1})$$ and $$(2,\{1\}_m,\bar{1})$$ can be expressed polynomially in terms of zeta values, polylogarithms and $$\ln (2)$$ . We also evaluate several restricted sums involving multiple zeta values.
- Published
- 2017
- Full Text
- View/download PDF
20. Multiple Dedekind zeta functions
- Author
-
Ivan Horozov
- Subjects
Pure mathematics ,11G55, 11M32 ,Integral representation ,Mathematics - Number Theory ,Mathematics::General Mathematics ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Algebraic Geometry ,010104 statistics & probability ,Continuation ,Iterated integrals ,FOS: Mathematics ,Dedekind cut ,Number Theory (math.NT) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Dedekind zeta function ,Mathematics ,Meromorphic function - Abstract
In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1,1,1). Based on such computations, we state two conjectures about MDZV., Comment: This version has substantial improvements in the content and the style. There are more details about the analytic continuation together with new examples of multiple residues. 43 pages
- Published
- 2014
21. COMPUTING INTEGRALS OVER POLYNOMIALLY DEFINED REGIONS AND THEIR BOUNDARIES IN 2 AND 3 DIMENSIONS
- Author
-
Yuzita Yaacob, Stanly Steinberg, and Michael J. Wester
- Subjects
Numerical Analysis ,Correctness ,General Computer Science ,Applied Mathematics ,Surface integral ,d: area integral ,line integral ,Divergence theorem ,General Medicine ,Information technology ,Symbolic computation ,T58.5-58.64 ,Theoretical Computer Science ,Cylindrical algebraic decomposition ,Volume integral ,Algebra ,Order of integration (calculus) ,Modeling and Simulation ,iterated integrals ,cylindrical algebraic decomposition (cad) ,Vector calculus ,volume integral ,Mathematics - Abstract
This study uses the cylindrical algebraic decomposition algorithms implemented in Mathematica to produce procedures to analytically compute integrals over polynomially defined regions and their boundaries in two and three dimensions. Using these results, we can implement the divergence theorem in three dimensions or the Green’s theorems in two dimensions. These theorems are of central importance in the applications of multidimensional integration. They also provide a strong correctness test for the implementation of our results in a computer algebra system. The resulting software can solve many of the two and some of the three dimension a lintegration problems in vector calculu stextbooks. The three dimensional results are being extended. The results in this paper are being included inanautomated student assistant for vector calculus.
- Published
- 2012
22. Parameter Dependent and Iterated Integrals
- Author
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Niels Jacob and Kristian P Evans
- Subjects
Order of integration (calculus) ,Mathematical optimization ,Iterated integrals ,Applied mathematics ,Parameter dependent ,Mathematics - Published
- 2016
23. On Improvement and Deterioration of A Repairable System Under Generalized Stochastic Orders
- Author
-
Amarjit Kundu and Asok K. Nanda
- Subjects
Repairable systems ,Computer science ,Iterated integrals ,Applied mathematics ,General Medicine - Abstract
Improvement (deterioration) of a repairable system in the sense of several generalized stochastic orders has been discussed in this paper. We have also proved some general results for minimal repairable systems.
- Published
- 2011
24. On some Gronwall type inequalities involving iterated integrals
- Author
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Sever S Dragomir, Yeol Je Cho, and Young-Ho Kim
- Subjects
Pure mathematics ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Function (mathematics) ,Type (model theory) ,Order of integration (calculus) ,Iterated integrals ,Simple (abstract algebra) ,Integro-differential equation ,media_common ,Mathematics - Abstract
In this paper we consider simple inequalities involving iterated integrals in the inequality (1.1) for functions when the function u in the both side of the inequality (1.1) are replaced by the function w(u) and ϕ (u) for some functions w,ϕ and provide some retarded integral inequalities involving iterated integrals. Some applications are also given to illustrate the usefulness of our results.
- Published
- 2011
25. Some Algebraic Aspects of the Center Problem for Ordinary Differential Equations
- Author
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Alexander Brudnyi
- Subjects
Discrete mathematics ,Iterated integrals ,Applied Mathematics ,Ordinary differential equation ,Center (category theory) ,Ode ,Discrete Mathematics and Combinatorics ,Algebraic number ,Mathematics - Abstract
We present a survey of some results and open problems related to the center problem for ODEs \({\frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x)\,v^{i+1}}\).
- Published
- 2010
26. Conditional implicit mean and the law of iterated integrals
- Author
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Hiroyuki Ozaki
- Subjects
Discrete mathematics ,Economics and Econometrics ,Law of total expectation ,Implicit function ,Iterated integrals ,Applied Mathematics ,Decision theory ,Functional equation ,Mean value ,Applied mathematics ,Conditional expectation ,Mathematics - Abstract
This paper presents a new framework which generalizes the concept of conditional expectation to mean values which are implicitly defined as unique solutions to some functional equation. We call such a mean value an implicit mean . The implicit mean and its very special example, the quasi-linear mean , have been extensively applied to economics and decision theory. This paper provides a procedure of defining the conditional implicit mean and then analyzes its properties. In particular, we show that the conditional implicit mean is in general “biased” in the sense that an analogue of the law of iterated expectations does not hold and we characterize the quasi-linear mean as the only implicit mean which is “unbiased”.
- Published
- 2009
27. Fredholm equations for non-symmetric kernels with applications to iterated integral operators
- Author
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Christopher S. Withers and Saralees Nadarajah
- Subjects
Pure mathematics ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Fredholm integral equation ,Integral transform ,Fredholm theory ,Mathematics - Spectral Theory ,Computational Mathematics ,symbols.namesake ,Operator (computer programming) ,Iterated integrals ,Singular value decomposition ,FOS: Mathematics ,symbols ,Spectral Theory (math.SP) ,Kernel (category theory) ,Mathematics - Abstract
We give the Jordan form and the Singular Value Decomposition for an integral operator ${\cal N}$ with a non-symmetric kernel $N(y,z)$. This is used to give solutions of Fredholm equations for non-symmetric kernels, and to determine the behaviour of ${\cal N}^n$ and $({\cal N}{\cal N^*})^n$ for large $n$., 12 A4 pages
- Published
- 2008
28. Iterated integral transforms of Caratheodory functions and their applications to analytic and univalent functions
- Author
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T. O. Opoola and K. O. Babalola
- Subjects
Discrete mathematics ,Algebra ,Iterated integrals ,Applied Mathematics ,General Mathematics ,Mathematical proof ,Integral transform ,Mathematics - Abstract
In this paper we develop and study some integral transforms of Caratheodory functions. We apply the transforms to study certain other classes of analytic and univalent functions both to obtain new results and provide new proofs of some known ones.
- Published
- 2006
29. ON SOME GRONWALL TYPE INEQUALITIES FOR A SYSTEM INTEGRAL EQUATION
- Author
-
Byung-Il Kim
- Subjects
Inequality ,Iterated integrals ,Integro-differential equation ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Applied mathematics ,Function (mathematics) ,Type (model theory) ,Integral equation ,Mathematics ,media_common - Abstract
In this paper we consider analogous of Gronwall-type inequalities involving iterated integrals in the inequality (1.2) for functions when the function u in the right-hand side of the in- equality (1.2) is replaced by the function u p for some p: These inequalities are efiective tools in the study of a system of an inte- gral equation. We also provide some integral inequalities involving iterated integrals.
- Published
- 2005
30. Growth estimates on certain integral inequalities in two variables involving iterated integrals
- Author
-
B. G. Pachpatte
- Subjects
Variables ,Inequality ,Applied Mathematics ,General Mathematics ,Multiple integral ,media_common.quotation_subject ,Mathematical analysis ,Integral equation ,Order of integration (calculus) ,Iterated function ,Iterated integrals ,Applied mathematics ,Mathematics ,media_common - Abstract
The aim of the present paper is to establish growth estimates on some new integral inequalities in two independent variables involving iterated double integrals, which can be used to study the qualitative behavior of solutions of certain partial integrodifferential and integral equations. Applications are also given to illustrate the usefulness of one of our results.
- Published
- 2005
31. THEORY OF DISTRIBUTION IN THE SENSE OF CONNES–HIDA AND FEYNMAN PATH INTEGRAL ON A MANIFOLD
- Author
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Rémi Léandre
- Subjects
Statistics and Probability ,Algebra ,Current (mathematics) ,Iterated integrals ,Applied Mathematics ,Path integral formulation ,Statistical and Nonlinear Physics ,Mathematical Physics ,Manifold ,Distribution (differential geometry) ,Mathematical physics ,Mathematics - Abstract
On p. 1096 of Ref. 31, we remark that the Witten current defined in Ref. 13 is an analogue of Hida distribution, Wiener chaos being replaced by Chen iterated integrals. We clarify this remark in the case of a Feynman path integral on a manifold.
- Published
- 2003
32. Classroom notes: Iterated integrals: an algebraic approach
- Author
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Leo G Ii Chouinard
- Subjects
Algebra ,Order of integration (calculus) ,Mathematics (miscellaneous) ,Iterated integrals ,Applied Mathematics ,Calculus ,Graphics ,Algebraic number ,Education ,Mathematics - Abstract
Setting up iterated integrals is only approached graphically in standard calculus textbooks. However, for three-dimensional regions of integration in particular, the graphics shown in textbooks are often not capable of reproduction by students. This article explains an algebraic approach to setting up iterated integrals that has been used by the author for about 15 years to bypass or simplify graphing difficulties in these problems.
- Published
- 2003
33. On a subclass of Bazileviv c functions
- Author
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S. A. Halim
- Subjects
Combinatorics ,Iterated integrals ,Applied Mathematics ,General Mathematics ,education ,Calculus ,Alpha (ethology) ,Beta (velocity) ,Unit (ring theory) ,Subclass ,Mathematics - Abstract
For $ \alpha>0$, $ 0\le \beta \beta$ for $ z$ in the unit disc $ D=\{z:|z
- Published
- 2002
34. Synthetic Differential Geometry of Chen's Iterated Integrals
- Author
-
Hirokazu Nishimura
- Subjects
Path (topology) ,Mathematics - Differential Geometry ,Pure mathematics ,biology ,Applied Mathematics ,Synthetic differential geometry ,biology.organism_classification ,Mathematics::Algebraic Geometry ,Mathematics (miscellaneous) ,Chen ,Differential Geometry (math.DG) ,Iterated integrals ,FOS: Mathematics ,Path space ,Mathematics - Abstract
Chen's iterated integrals are treated within synthetic dierential geome- try. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.
- Published
- 2014
35. Eigenvalues of Sturm-Liouville Problems Using Fliess Series
- Author
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B. Chanane
- Subjects
Series (mathematics) ,Applied Mathematics ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Boundary (topology) ,Sturm–Liouville theory ,Function (mathematics) ,Mathematics::Spectral Theory ,Systems theory ,Iterated integrals ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Applied mathematics ,Representation (mathematics) ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper deals with the approximation of, and error bounds on, the eigenvalues of regular Sturm-Liouville problems. Concepts from systems theory are used to derive a novel representation for the boundary function.
- Published
- 1998
36. Generalized retarded nonlinear integral inequalities involving iterated integrals and an application
- Author
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Wu-Sheng Wang, Xuefang Li, and Deqing Huang
- Subjects
Order of integration (calculus) ,Work (thermodynamics) ,Nonlinear system ,Iterated integrals ,Applied Mathematics ,Multiple integral ,Mathematical analysis ,Discrete Mathematics and Combinatorics ,Analysis ,Mathematics ,Volume integral - Abstract
In this work, some new generalized retarded nonlinear integral inequalities, which include nonlinear composite functions of unknown functions between iterated integrals, are discussed. By adopting novel analysis techniques, the upper bounds of the embedded unknown functions are estimated explicitly. The derived results can be applied in the study of differential-integral equations and some practical problems in engineering.
- Published
- 2013
37. Summations of polylogarithms via evaluation transform
- Author
-
Hoang Ngoc Minh
- Subjects
Numerical Analysis ,General Computer Science ,Applied Mathematics ,TheoryofComputation_GENERAL ,Symbolic computation ,Theoretical Computer Science ,Riemann zeta function ,Algebra ,symbols.namesake ,Nonlinear system ,Combinatorics on words ,Special functions ,Iterated integrals ,Modeling and Simulation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Generating series ,Mathematics ,Meromorphic function - Abstract
In this work, the Evaluation transform is adapted to interpret the polylogarithms as being the Evaluation of the words yxn, for meromorphic kernels and meromorphic inputs. The functional equations on polylogarithms can be obtained, then, via combinatorics on words and the relations between the inputs. Some identities concerning these special functions are also proposed via the products of generating series.
- Published
- 1996
38. Products of quasi-measures
- Author
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D. Grubb
- Subjects
Combinatorics ,Physics ,Mathematics Subject Classification ,Iterated integrals ,Set function ,Applied Mathematics ,General Mathematics ,Fubini's theorem ,Hausdorff space ,Product topology ,Function (mathematics) ,Measure (mathematics) - Abstract
A quasi-state is a positive functional on C(X) that is only assumed to be linear on singly-generated subalgebras. We consider the “iterated integral” of two quasi-states and determine when this gives a quasi-state on the product space. We also provide explicit formulas for the corresponding quasi-measures in case it does. Finally, we show the general failure of Fubini’s Theorem for quasi-states. If X is a compact, Hausdorff space, we let C(X) denote the collection of realvalued continuous functions on X . We let sp f denote the range of f . A quasi-state is a function ρ : C(X)→ R such that: (i) If f ≥ 0, then ρ(f) ≥ 0. (ii) ρ(1) = 1. (iii) If r ∈ R, then ρ(rf) = rρ(f). (iv) If φ,ψ ∈ C(sp f), then ρ(φ ◦ f + ψ ◦ f) = ρ(φ ◦ f) + ρ(ψ ◦ f). In [1], Aarnes answered the question of whether every quasi-state must be linear in the negative. He did this by establishing a correspondence between quasi-states and certain set functions. In particular, a function μ defined for the subsets of X that are either open or closed is a quasi-measure if: a) μ(A) ≥ 0 for all A. b) A ⊆ B implies that μ(A) ≤ μ(B). c) A ∩B = ∅ implies μ(A ∪B) = μ(A) + μ(B). d) If U is open, μ(U) = sup{μ(K) : K ⊆ U,K closed }. e) μ(X) = 1. The primary difference between a quasi-measure and a finitely additive measure is that quasi-measures do not have to be subadditive. The quasi-state ρ corresponds to the quasi-measure μ if μ(U) = sup{ρ(f) : 0 ≤ f ≺ U} for U open and μ(K) = inf{ρ(f) : K ≺ f} for K closed. Here, f ≺ U means that 0 ≤ f ≤ 1 and the support of f is contained in U . Also, K ≺ f means that f ≥ 0 and f ≥ 1 on K. This construction is detailed in [1], where a particular example of a quasi-measure that is not a measure is given. Other examples may be found in [2, 3] and [5]. Received by the editors October 26, 1994 and, in revised form, February 7, 1995. 1991 Mathematics Subject Classification. Primary 28C05. c ©1996 American Mathematical Society
- Published
- 1996
39. Stochastic Dominance on Unidimensional Grids
- Author
-
Irving H. Lavalle and Peter C. Fishburn
- Subjects
Iterated integrals ,Iterated function ,General Mathematics ,Cumulative distribution function ,Calculus ,Applied mathematics ,Probability distribution ,Stochastic dominance ,Management Science and Operations Research ,Grid ,Computer Science Applications ,Mathematics ,Sign (mathematics) - Abstract
Special stochastic-dominance relations for probability distributions on a finite grid of evenly-spaced points are considered. The relations depend solely on iterated partial sums of grid-point probabilities and are very computer efficient. Their corresponding classes of utility functions for expected-utility comparisons consist of functions defined on the grid that mimic in the large the traditional continuous functions whose derivatives alternate in sign. The first-degree and second-degree relations are identical to their traditional counterparts defined from iterated integrals of cumulative distribution functions. The higher-degree relations differ from the traditional relations in interesting and sometimes subtle ways. The paper explores aspects of the partial-sums relations, including effects of grid refinements and extensions, and describes their relationships to the traditional relations.
- Published
- 1995
40. Some new generalized retarded nonlinear integral inequalities with iterated integrals and their applications
- Author
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Wu-Sheng Wang
- Subjects
Order of integration (calculus) ,Nonlinear system ,Integro-differential equation ,Iterated integrals ,Applied Mathematics ,Multiple integral ,Mathematical analysis ,Discrete Mathematics and Combinatorics ,Upper and lower bounds ,Analysis ,Volume integral ,Mathematics - Abstract
Some new generalized retarded nonlinear integral inequalities with iterated integrals are discussed and upper bound estimations of unknown functions are given by analysis technique. These estimations can be used as tools in the study of differential-integral equations with the initial conditions.
- Published
- 2012
41. Lagrangian and Hamiltonian Feynman formulae for some Feller semigroups and their perturbations
- Author
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Yana A. Butko, René L. Schilling, and Oleg G. Smolyanov
- Subjects
Statistics and Probability ,Semigroup ,Applied Mathematics ,Probability (math.PR) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,symbols.namesake ,Mathematics - Analysis of PDEs ,Iterated integrals ,Evolution equation ,symbols ,FOS: Mathematics ,Elementary function ,Feynman diagram ,Hamiltonian (quantum mechanics) ,Mathematics - Probability ,Mathematical Physics ,Lagrangian ,Mathematics ,Probability measure ,Mathematical physics ,Analysis of PDEs (math.AP) - Abstract
A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of n-fold iterated integrals of some elementary functions as n → ∞. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite-dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman–Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.
- Published
- 2012
- Full Text
- View/download PDF
42. Volterra series and permutation groups
- Author
-
Andrei A. Agrachev and Revaz Valerianovich Gamkrelidze
- Subjects
Statistics and Probability ,Discrete mathematics ,Dynamical systems theory ,Algebraic structure ,Iterated integrals ,Applied Mathematics ,General Mathematics ,Volterra series ,Multiplication ,Permutation group ,Asymptotic expansion ,Dynamical system (definition) ,Mathematics - Abstract
Algebraic structures, connected with the asymptotic expansions of perturbations of smooth dynamical systems, are investigated; first of all, the so-called shuffle multiplication for permutations and for iterated integrals.
- Published
- 1994
43. A combinatorial method for calculating the moments of Lévy area
- Author
-
Daniel Levin and Mark Wildon
- Subjects
Combinatorics ,Pure mathematics ,Mathematics::Probability ,Iterated integrals ,Applied Mathematics ,General Mathematics ,Product (mathematics) ,Faculty of Science\Mathematics ,Combinatorial method ,Brownian motion ,Mathematics - Abstract
We present a new way to compute the moments of the Levy area of a two-dimensional Brownian motion. Our approach uses iterated integrals and combinatorial arguments involving the shuffle product.
- Published
- 2009
44. Cumulants as iterated integrals
- Author
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Franz Lehner
- Subjects
Statistics and Probability ,Approximation theory ,60E05, 62E10 ,Applied Mathematics ,Mathematical analysis ,Probability (math.PR) ,Edgeworth series ,Order of integration (calculus) ,Distribution function ,Iterated integrals ,FOS: Mathematics ,Mathematics - Combinatorics ,Order (group theory) ,Applied mathematics ,Combinatorics (math.CO) ,Statistics, Probability and Uncertainty ,Cumulant ,Mathematics - Probability ,Mathematics - Abstract
A formula expressing cumulants in terms of iterated integrals of the distribution function is derived. It generalizes results of Jones and Balakrishnan who computed expressions for cumulants up to order 4., Comment: AMS-LaTeX, 8 pages
- Published
- 2009
- Full Text
- View/download PDF
45. On some new nonlinear retarded integral inequalities with iterated integrals and their applications
- Author
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Qing-Hua Ma and Josip Pečarić
- Subjects
Nonlinear system ,Differential equation ,General Mathematics ,Multiple integral ,Mathematical analysis ,Applied mathematics ,Gronwall-like integral inequalities ,retarded ,iterated integrals ,priori bound ,Functional integration ,Daniell integral ,Type (model theory) ,Integral equation ,Mathematics ,Volume integral - Abstract
Some new nonlinear retarded integral inequalities of Gronwall- like type are established, which mainly generalized some results given by Cho, Dragomir and Kim (J. Korean Math. Soc. 43 (2006), No. 3, pp. 563–578) and can be used in the analysis of various problems in the theory of certain classes of differential equations and integral equations. Applications examples are also indicated.
- Published
- 2008
46. Criteria for the stability of p-linear maps and p-powers
- Author
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I. W. Sandberg
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Multilinear map ,Pure mathematics ,Stability criterion ,Applied Mathematics ,Linear system ,Mathematics::Classical Analysis and ODEs ,Sense (electronics) ,Topology ,Stability (probability) ,Nonlinear system ,Kernel (algebra) ,Operator (computer programming) ,Systems theory ,Iterated integrals ,Applied mathematics ,Uniform boundedness ,Analysis ,Mathematics - Abstract
Iterated integral p-linear maps and p-powers arise in the theory of representations of nonlinear systems. In studies by the author (1989) necessary and sufficient conditions were given for such operators to be stable in a standard sense, and it was shown that a well-known sufficient condition is not necessary for all p>or=2, and corresponding results for discrete-time cases were given. These results are reviewed in the paper. >
- Published
- 1990
47. New Integral Inequalities for Iterated Integrals with Applications
- Author
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Ravi P. Agarwal, Young-Ho Kim, and Cheon Seoung Ryoo
- Subjects
Inequality ,Quantitative Biology::Neurons and Cognition ,media_common.quotation_subject ,Applied Mathematics ,lcsh:Mathematics ,Type (model theory) ,lcsh:QA1-939 ,Nonlinear system ,Iterated integrals ,Calculus ,Discrete Mathematics and Combinatorics ,Daniell integral ,Computer Science::Databases ,Analysis ,media_common ,Mathematics - Abstract
Some new nonlinear retarded integral inequalities of Gronwall type are established. These inequalities can be used as basic tools in the study of certain classes of integrodifferential equations.
- Published
- 2007
48. An algebraic criterion for the onset of chaos in nonlinear dynamical systems
- Author
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Aynur Unal
- Subjects
Power series ,Pure mathematics ,Noncommutative ring ,Dynamical systems theory ,Applied Mathematics ,Mathematical analysis ,Autocorrelation ,Chaotic ,ComputerApplications_COMPUTERSINOTHERSYSTEMS ,Dynamical system ,CHAOS (operating system) ,Nonlinear dynamical systems ,Nonlinear system ,Iterated integrals ,Product (mathematics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Applied mathematics ,Lisp ,Algebraic number ,computer ,Analysis ,Computer Science::Databases ,Mathematics ,computer.programming_language - Abstract
Defines the onset of chaos in a given dynamical system and expresses the necessary and sufficient conditions as a criterion for the onset. The algebra required both in obtaining the generating power series in their shuffle product together with the two limits all can be done on a computer with a symbolic language. The following symbolic languages can be used: PL/I, REDUCE, MACSYMA, and LISP. The criterion, which is computer-algebraic, can be utilized to characterize the physical parameters' ranges for which chaotic regimes will take place and hence can be used in designing nonlinear dynamical system to avoid such regimes. >
- Published
- 2003
49. Iterated path integrals
- Author
-
Kuo-Tsai Chen
- Subjects
Applied Mathematics ,General Mathematics ,58A99 ,53C65 ,53B15 ,55D35 ,55H20 ,Iterated integrals ,Iterated function ,Path integral formulation ,Diffeology ,49F05 ,Applied mathematics ,Functional integration ,Mathematics - Published
- 1977
50. Oscillation properties of perturbed disconjugate equations
- Author
-
William F. Trench
- Subjects
Physics ,Combinatorics ,Nonlinear system ,Iterated integrals ,Oscillation ,Applied Mathematics ,General Mathematics ,Scalar equation ,Multiplicative function ,Mathematical analysis ,Without loss of generality ,Connection (algebraic framework) - Abstract
Oscillation conditions are given for the equation L u + f ( t , u ) = 0 {L_u} + f(t,u) = 0 , where \[ L u = 1 β n d d t 1 β n − 1 ⋯ d d t 1 β 1 d d t u β 0 ( n ⩾ 2 ) , Lu = \frac {1} {{{\beta _n}}}\frac {d} {{dt}}\frac {1} {{{\beta _{n - 1}}}} \cdots \frac {d} {{dt}}\frac {1} {{{\beta _1}}}\frac {d} {{dt}}\frac {u} {{{\beta _0}}}(n \geqslant 2), \] with β 0 , … , β n {\beta _0}, \ldots ,{\beta _n} positive and continuous on ( 0 , ∞ ) , ∫ ∞ β i d t = ∞ ( 1 ⩽ i ⩽ n − 1 ) (0,\infty ),\int {^\infty {\beta _i}dt = \infty (1 \leqslant i \leqslant n - 1)} , and f f subject to conditions which include u f ( t , u ) ⩾ 0 uf(t,u) \geqslant 0 . The results obtained include previously known oscillation conditions for the equation u ( n ) + f ( t , u ) = 0 {u^{(n)}} + f(t,u) = 0 for both linear and nonlinear cases.
- Published
- 1975
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