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In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura $$\eta $$ -values. At the end of the paper, we study a variant of Kaneko–Tsumura $$\eta $$ -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.

In this paper we consider iterated integral representations of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst [3] , and give some general results.

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.

We introduce the notion of a matching Rota-Baxter algebra motivated by the recent work on multiple pre-Lie algebras arising from the study of algebraic renormalization of regularity structures [10] , [19] . This notion is also related to iterated integrals with multiple kernels and solutions of the associative polarized Yang-Baxter equation. Generalizing the natural connection of Rota-Baxter algebras with dendriform algebras to matching Rota-Baxter algebras, we obtain the notion of matching dendriform algebras. As in the classical case of one operation, matching Rota-Baxter algebras and matching dendriform algebras are related to matching pre-Lie algebras which coincide with the aforementioned multiple pre-Lie algebras. More general notions and results on matching tridendriform algebras and matching PostLie algebras are also obtained.

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.

We generalize the definition of overconvergent $p$-adic multiple polylogarithms and of $p$-adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized $p$-adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent $p$-adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on $p$-adic cyclotomic multiple zeta values., 25 pages. This is Part I-1 of "$p$-adic cyclotomic multiple zeta values and $p$-adic pro-unipotent harmonic actions"

Let F ∈ ℂ [ x , y ] be a polynomial, γ ( z ) ∈ π 1 ( F − 1 ( z ) ) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation d F + e ω = 0 of the integrable foliation given by F . We study different invariants: the orbit depth k , the nilpotence class n , the derivative length d associated with the couple ( F , γ ) . These invariants bind the length l of the first nonzero Melnikov function of the deformation d F + e ω along γ . We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this behavior with the length of the corresponding Godbillon–Vey sequence. We formulate a conjecture motivated by the study of this example.

In this paper we consider iterated integrals on P 1 ∖ { 0 , 1 , ∞ , z } and define a class of Q -linear relations among them, which arises from the differential structure of the iterated integrals with respect to z. We then define a new class of Q -linear relations among the multiple zeta values by taking their limits of z → 1 , which we call confluence relations (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values. As a good reason for this, we show that confluence relations imply both the regularized double shuffle relations and the duality relations.

In this paper, we prove a family of identities among multiple zeta values, which contains as a special case a conjectural identity of Hoffman. We use the iterated integrals on [Formula: see text] for our proof.

We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points $0,1,z$ and $\infty$. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals are given., Comment: 11 pages

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 prove a formula among finite multiple zeta values with four parameters. This formula is proved by using the iterated integral expression of the multiple polylogarithms. By specializing these parameters, we give some weighted sum formulas for finite multiple zeta values.

It has recently been shown that two-loop kite-type diagrams can be computed analytically in terms of iterated integrals with algebraic kernels. This result was obtained using a new integral representation for two-loop sunset subgraphs. In this paper, we have developed a similar representation for a three-loop banana integral in $d = 2-2\varepsilon$ dimensions. This answer can be generalized up to any given order in the $\varepsilon$-expansion and can be calculated numerically both below and above the threshold. We also demonstrate how this result can be used to compute more complex three-loop integrals containing the three-loop banana as a subgraph., 17 pages, 9 figures, references added

We introduce eight “flavors” of cyclic homology of a mixed complex and study their properties. In particular, we determine their behavior with respect to Chen’s iterated integrals.

We give an overview of some work on elliptic multiple zeta values. First defined by Enriquez as the coefficients of the elliptic KZB associator, elliptic multiple zeta values are also special values of multiple elliptic polylogarithms in the sense of Brown and Levin. Common to both approaches to elliptic multiple zeta values is their representation as iterated integrals on a once-punctured elliptic curve. Having compared the two approaches, we survey various recent results about the algebraic structure of elliptic multiple zeta values, as well as indicating their relation to iterated integrals of Eisenstein series, and to a special algebra of derivations.

We describe some particular finite sums of multiple zeta values which arise from J. Ecalle's "arborification", a process which can be described as a surjective Hopf algebra morphism from the Hopf algebra of decorated rooted forests onto a Hopf algebra of shuffles or quasi-shuffles. This formalism holds for both the iterated sum picture and the iterated integral picture. It involves a decoration of the forests by the positive integers in the first case, by only two colours in the second case., From a talk at the ESF exploratory workshop "New approaches to Multiple Zeta Values", ICMAT, Madrid, Sept-Oct. 2013

We study the zero location and the asymptotic behavior of iterated integrals of polynomials. Borwein–Chen–Dilcher’s polynomials play an important role in this issue. For these polynomials we find their strong asymptotics and give the limit measure of their zero distribution. We apply these results to describe the zero asymptotic distribution of iterated integrals of ultraspherical polynomials with parameters $$(2\alpha +1)/2$$ , $$\alpha \in \mathbb {Z}_{+}$$ .

Criteria for the boundedness of sublinear integral two-kernel operators of iterated type on cones of monotone functions in Lebesgue spaces on the real semiaxis are given.

This paper explains a method to calculate the coefficients of the Alekseev–Torossian associator as linear combinations of iterated integrals of Kontsevich weight forms of Lie graphs.

We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Then, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Moreover, we investigate the quasi-symmetric functions corresponding to the Schur multiple zeta functions. We obtain the similar results as above for them and, furthermore, describe the images of them by the antipode of the Hopf algebra of quasi-symmetric functions explicitly. Finally, we establish iterated integral representations of the Schur multiple zeta values of ribbon type, which yield a duality for them in some cases., Comment: 42 pages, 2 figures

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 study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.

This paper contains examples of shuffle relations among multiple Dedekind zeta values. Dedekind zeta values were defined by the author in his paper "Multiple Dedekind zeta functions". Here we concentrate on the cases of real or imaginary quadratic fields with up to double iteration. We give examples of integral shuffle relation in terms of iterated integrals over membranes and of infinite sum shuffle relation, sometimes called stuffle relation. Using both types of shuffles for the product $\zeta_{K,C}(2)\zeta_{K,C}(2)$, we find relations among multiple Dedekind zeta values for real and for imaginary quadratic fields., Comment: Defines more general multiple Dedekind zeta values; contains 43 diagrams; 25 pages

In this talk we discuss a class of Feynman integrals, which can be expressed to all orders in the dimensional regularisation parameter as iterated integrals of modular forms. We review the mathematical prerequisites related to elliptic curves and modular forms. Feynman integrals, which evaluate to iterated integrals of modular forms go beyond the class of multiple polylogarithms. Nevertheless, we may bring for all examples considered the associated system of differential equations by a non-algebraic transformation to an \(\varepsilon \)-form, which makes a solution in terms of iterated integrals immediate.

We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among multiple zeta values, which uses iterated log-sine integrals, and give alternative proofs of several known results related to multiple zeta values and log-sine integrals., Comment: 18 pages

In this paper we show that the iterated integrals on products of one variable multiple polylogarithms from 0 to 1 are actually multiple zeta values if they are convergent. In the divergent case, we define regularized iterated integrals from 0 to 1. By the same method, we show that the regularized iterated integrals are also multiple zeta values. As an application, we give new series representations for multiple zeta values and calculate some interesting examples of iterated integrals., Comment: 18 pages

We present the results for two-loop massive kite master integrals with elliptics in terms of iterated integrals with algebraic kernels. The key ingredients are new integral representations for sunset subgraphs in $d=4-2\epsilon$ and $d=2-2\epsilon$ dimensions together with differential equations for considered kite master integrals in $A+B\epsilon$ form. The obtained results can be easily generalized to all orders in $\epsilon$-expansion and show that the class of functions defined as iterated integrals with algebraic kernels may be large enough for writing down results for a large class of massive Feynman diagrams., Comment: 21 page, 4 figures

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

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for ($q$-)multiple zeta values., 5 pages, to appear in Int. J. Number Theory

In this paper, we prove certain algebraic identities, which correspond to differentiation of the shuffle relation, the stuffle relation, and the relations which arise from Mobius transformations of iterated integrals. These formulas provide fundamental and useful tools in the study of iterated integrals on a punctured projective line.

We present a general expression for any term of the Magnus series as an iterated integral of a linear combination of independent right-nested commutators with given coefficients. The relation with the Malvenuto--Reutenauer Hopf algebra of permutations is also discussed., 16 pages

In Sect. 11.3 we have shown how to calculate the volume of solids of revolution. If there is no rotational symmetry, however, one needs an extension of integral calculus to functions of two variables. This arises, for example, if one wants to find the volume of a solid that lies between a domain D in the (x, y)-plane and the graph of a non-negative function \(z = f(x, y)\). In this section we will extend the notion of Riemann integrals from Chap. 11 to double integrals of functions of two variables. Important tools for the computation of double integrals are their representation as iterated integrals and the transformation formula (change of coordinates). The integration of functions of several variables occurs in numerous applications, a few of which we will discuss.

We give new instances where Chabauty–Kim sets can be proved to be finite, by developing a notion of “generalised height functions” on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the 1st explicit nonabelian Chabauty result for a curve $X/\mathbb{Q}$ whose Jacobian has Mordell–Weil rank larger than its genus.

In this paper we show that certain Feynman integrals can be expressed as linear combinations of iterated integrals of modular forms to all orders in the dimensional regularisation parameter $\varepsilon$ . We discuss explicitly the equal mass sunrise integral and the kite integral. For both cases we give the alphabet of letters occurring in the iterated integrals. For the sunrise integral we present a compact formula, expressing this integral to all orders in $\varepsilon$ as iterated integrals of modular forms., 53 pages, v2: section on relation to Gamma_1(6) added, v3: section on the choice of the periods added

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

We construct an A ∞ -quasi-equivalence of dg-categories between P A — the category of prefect A 0 -modules with flat Z -connection, associated to the de Rham dga A • of a compact manifold M — and the dg-category of infinity-local systems on M — homotopy-coherent representations of the smooth singular simplicial set of M. We understand this as a generalization of the classical Riemann–Hilbert correspondence to Z -connections ( Z -graded superconnections in some circles). This theory makes crucial use of Igusaʼs notion of higher holonomy transport for Z -connections which is a derivative of Chenʼs idea of generalized holonomy.

We use higher parallel transport—more precisely, the integration \(\mathsf{A }_\infty \)-functor constructed in Arias Abad and Schatz (The \(\mathsf{A}_\infty \) de Rham theorem and the integration of representations up to homotopy. Int Math Res Not, 2013) and Block and Smith (A Riemann–Hilbert correspondence for infinity local systems. arXiv:0908.2843, 2012)—to define Reidemeister torsion for flat superconnections. We conjecture a version of the Cheeger–Muller theorem, namely that the combinatorial Reidemeister torsion coincides with the analytic torsion defined by Mathai and Wu (Contemp Math 546, 199–212, 2011).

A multiple zeta value can always be represented by its Drinfel’d integral. If we add some factors appeared in the integrand of the integral representation of the multiple zeta value, it would still represent a linear combination of multiple zeta values, but the depths and weights may decrease. In this paper, we shall investigate some of multiple zeta values obtained from Drinfel’d integral with additional factors aforementioned and study a class of deformation of multiple zeta values. Results are then obtained as analogues or generalizations of the sum formula of multiple zeta values.

The Coleman integral is a p-adic line integral. Double Coleman integrals on elliptic curves appear in Kim’s nonabelian Chabauty method, the first numerical examples of which were given by the author, Kedlaya, and Kim [3]. This paper describes the algorithms used to produce those examples, as well as techniques to compute higher iterated integrals on hyperelliptic curves, building on previous joint work with Bradshaw and Kedlaya [2].

For a general differential system $\dot x(t) = \sum_{d=1}^3 u_d(t)X_d$, where $X_d$ generates the simple Lie algebra of type $\mathfrak{a}_1$, we compute the explicit solution in terms of iterated integrals of products of $u_d$'s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures., Comment: 16 pages. The final publication is available at link.springer.com

In this paper, we characterize the universal centres of the ordinary differential equations , where ai(θ) are trigonometric polynomials, in terms of the composition conditions. These centres are closely related with the classical Poincaré centre problem for planar analytic differential systems. Additionally, we show that the notion of universal centre is not invariant under changes of variables, and we also provide different families of universal centres. Finally, we characterize all the universal centres for the quadratic polynomial differential systems. Universal centres and composition conditions. The first and second authors are partially supported by a MICINN/FEDER grant number MTM2011-22877 and by a Generalitat de Catalunya grant number 2009SGR 381. The third author is partially supported by a MICINN/ FEDER grant number MTM2008-03437, by a Generalitat de Catalunya grant number 2009SGR 410 and by ICREA Academia.

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 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

We study an elliptic analogue of multiple zeta values, the elliptic multiple zeta values of Enriquez, which are the coefficients of the elliptic KZB associator. Originally defined by iterated integrals on a once-punctured complex elliptic curve, it turns out that they can also be expressed as certain linear combinations of indefinite iterated integrals of Eisenstein series and multiple zeta values. In this paper, we prove that the $\mathbb{Q}$-span of these elliptic multiple zeta values forms a $\mathbb{Q}$-algebra, which is naturally filtered by the length and is conjecturally graded by the weight. Our main result is a proof of a formula for the number of $\mathbb{Q}$-linearly independent elliptic multiple zeta values of lengths one and two for arbitrary weight., Comment: 22 pages

t d s d d u g u h s u s u s f u t u 0 0 0 ) ) ) ( ) ( ) ( )( ( ) ( )( ( ) ( ) ( 0 τ ξ ξ ξ τ τ τ (1) To avoid the shortcoming of these results, Medved (2) presented a new method to discuss nonlinear singular integral inequalities of Henry type and their Bihari version as follows