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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.

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.

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.

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.

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...

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.

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.

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

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

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 .

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.

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.

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

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.

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.

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

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.

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.

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.

We present a survey of some results and open problems related to the center problem for ODEs \({\frac{dv}{dx}=\sum_{i=1}^{\infty}a_{i}(x)\,v^{i+1}}\).

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”.

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

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.

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.

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.

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.

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.

For $ \alpha>0$, $ 0\le \beta \beta$ for $ z$ in the unit disc $ D=\{z:|z

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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

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.

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. >

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.

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. >

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.