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In this thesis we study the additive model of errors in variables, which is also known as the deconvolution problem. The objective consists particularly in the reconstruction of the distribution F associated with a random variable X, which is observable only through a sample of a blurred variable Y, due to an additive random error ε with known distribution H. Our initial considerations yield an unbiased estimator for F for various discrete and some continuous distributions. A more general approach then leads us to the symmetrized model of errors in variables. It is obtained by an additional convolution of G with the conjugate error distribution of H, thereby resulting in an error distribution of symmetric type. As a consequence the characteristic function of X can be represented as the limit of a geometric series. By truncation of this series we deduce an approximation of F, which is valid for arbitrary error distributions. This approximation, termed the deconvolution function, converges to F in many cases. To determine the corresponding rates of convergence, techniques from complex calculus and particularly Mellin-Barnes integrals turn out to be appropriate. The latter describe a special class of integrals that can be evaluated by residue analysis. The results are established in a more general setting, which makes them applicable to other Laplace-type integrals. With the aid of the deconvolution function we also construct an estimator for F. The asymptotic properties of its variance, a peculiar integral of dimension two, can be specified by virtue of our findings from the concluding chapter. These results rely on iterated Mellin-Barnes integrals.

In this note, it is proved the existence of a $$\mathfrak {c}$$ c -dimensional vector space of real-entire functions all of whose nonzero members are non-integrable in the sense of Lebesgue but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space can be chosen to be dense in the space of all real $$C^\infty $$ C ∞ -functions on the plane endowed with the topology of uniform convergence on compacta for all derivatives of all orders. If the condition of being entire is dropped, then a closed infinite dimensional subspace satisfying the same properties can be obtained.

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.

The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ \zeta(1,3,\ldots,1,3) $ and summing, one obtains an explicit rational multiple of a power of $ \pi $. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 \zeta(3,3,\{2\}^m) - \zeta(3,\{2\}^m,(1,2)) $. In this paper we introduce the 'generalised cyclic insertion conjecture', which we describe using a new combinatorial structure on iterated integrals -- the so-called alternating block decomposition. We see that both the original BBBL cyclic insertion conjecture, and Hoffman's conjectural identity, are special cases of this 'generalised' cyclic insertion conjecture. By using Brown's motivic MZV framework, we establish that some symmetrised version of the generalised cyclic insertion conjecture always holds, up to a rational; this provides some evidence for the generalised conjecture., Comment: 40 pages, 1 figure created with Inkscape. Added an observation due to Panzer about the structure of D_odd cycle I(\ell_1, \ldots, \ell_n), in the odd weight case. Added a reference to the recent paper from Hirose and Sato, which proves Hoffman's conjectural identity exactly

This article is a research announcement of my upcoming joint paper with Minoru Hirose on a certain class of Q-linear relations among the multiple zeta values (MZVs), which we call confluence relations. As is well known, MZVs are iterated integrals of meromorphic one-forms dt/t and dt/t-1 on a projective line. Here we consider more general iterated integrals of three different one-forms dt/t, dt/t-1 and dt/t-z and define the confluence relations as limits as z → 1 of Q-linear relations among these iterated integrals. At first, we define standard relations among the iterated integrals which naturally arise by regarding them as functions of z and thus using their differential structure with respect to z, and then we consider their limits as z → 1. The confluence relations seem to give a very rich family of Q-linear relations among MZVs and we even propose a conjecture that they exhaust all the Q-linear relations among MZVs. As a good reason for our conjecture, we prove that the confluence relations imply the extended double shuffle relations (also the duality realtion). A small table up to weight 4 of the confluence relations is given at the end., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

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.

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.

For an elliptic curve E defined over a field k⊂C, we study iterated path integrals of logarithmic differential forms on Et, the universal vectorial extension of E. These are generalizations of the classical periods and quasi-periods of E, and are closely related to multiple elliptic polylogarithms and elliptic multiple zeta values. Moreover, if k is a finite extension of Q, then these iterated integrals along paths between k-rational points are periods in the sense of Kontsevich-Zagier.

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.

Dimensionally-regulated Feynman integrals are a cornerstone of all perturbative computations in quantum field theory. They are known to exhibit a rich mathematical structure, which has led to the development of powerful new techniques for their computation. We review some of the most recent advances in our understanding of the analytic structure of multiloop Feynman integrals in dimensional regularisation. In particular, we give an overview of modern approaches to computing Feynman integrals using differential equations, and we discuss some of the properties of the functions that appear in the solutions. We then review how dimensional regularisation has a natural mathematical interpretation in terms of the theory of twisted cohomology groups, and how many of the well-known ideas about Feynman integrals arise naturally in this context. This is Chapter 3 of a series of review articles on scattering amplitudes, of which Chapter 0 [arXiv:2203.13011] presents an overview and Chapter 4 [arXiv:2203.13015] contains closely related topics., Comment: 62 pages, see also the overview article arXiv:2203.13011. v3: journal version

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"

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.

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.

We consider the genus-one curves which arise in the cuts of the sunrise and in the elliptic double-box Feynman integrals. We compute and compare invariants of these curves in a number of ways, including Feynman parametrization, lightcone and Baikov (in full and loop-by-loop variants). We find that the same geometry for the genus-one curves arises in all cases, which lends support to the idea that there exists an invariant notion of genus-one geometry, independent on the way it is computed. We further indicate how to interpret some previous results which found that these curves are related by isogenies instead., Comment: 24 pages, 2 figures

Let $$P_n^{(\alpha ,\beta )}$$ be the n-th monic Jacobi polynomial with $$\alpha ,\beta >-1$$. Given m numbers $$\omega _1, \ldots , \omega _m \in \mathbb {C} \setminus [-1,1]$$, let $$\varOmega _m=( \omega _1, \ldots , \omega _m)$$ and $$\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}$$ be the m-th iterated integral of $$\frac{(n+m)!}{n!}\;P^{(\alpha ,\beta )}_{n}$$ normalized by the conditions $$\begin{aligned} \frac{\mathrm{d}^k\,\mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}}{\mathrm{d}z^k}(\omega _{m-k})=0, \; \text { for } \; k=0,1,\ldots , m-1. \end{aligned}$$The main purpose of the paper is to study the algebraic and asymptotic properties of the sequence of monic polynomials $$\{ \mathscr {P}_{n,m,\varOmega _m}^{(\alpha , \beta )}\}_n$$. In particular, we obtain the relative asymptotic for the ratio of the sequences $$\{ \mathscr {P}^{(\alpha ,\beta )}_{n,m,\varOmega _m}\}_n$$ and $$\{P_n^{(\alpha ,\beta )}\}_n$$. We prove that the zeros of these polynomials accumulate on a suitable ellipse.

We study the algebra of certain q-series, called bi-brackets, whose coefficients are given by weighted sums over partitions. These series incorporate the theory of modular forms for the full modular group as well as the theory of multiple zeta values (MZV) due to their appearance in the Fourier expansion of regularized multiple Eisenstein series. Using the conjugation of partitions we obtain linear relations between bi-brackets, called the partition relations, which yield naturally two different ways of expressing the product of two bi-brackets similar to the stuffle and shuffle product of multiple zeta values. In a recent work, the author and K. Tasaka defined (shuffle) regularized multiple Eisenstein series G ⧢ , by using an explicit connection to the coproduct on formal iterated integrals introduced by Goncharov. These satisfy the shuffle product formula. Applying the same concept for the coproduct on quasi-shuffle algebras enables us to define (stuffle) regularized multiple Eisenstein series G ⁎ satisfying the stuffle product formula. We show that both G ⧢ and G ⁎ are given by linear combinations of products of MZV and bi-brackets.

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.

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.

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

This research paper concentrates on the Kakeya problem. After the introduction of historical issue, we provide a thorough presentation of the results of Kakeya problem with some examples of the early solutions as well as the proof of the final outcome of this problem, the solution of which is known as Besicovitch Set. We give 3 different construction of Besicovitch set as well as the intuition of construction, which is related to iterated integral of 2-variable real function. We also give the Cunningham construction in which the area of a simply connected Kakeya set can also tend to 0. Furthermore, we generalize the process of generating a Kakeya set into a Kakeya dynamic. The definition of multiplicity enables us to estimate the area of a Kakeya set. In following discussion we provided a conjecture related to the solution in particular range. Finally, the derivation of the Kakeya problem is presented.

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 extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrands and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a closed form solution in full generality, we may succeed in computing the first coefficients of the Laurent series expansions of such integrals in terms of indefinite nested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate, can be considered as an enhanced implementation of the (continuous) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals. Together with the summation package Sigma and the package HarmonicSums our package provides methods to compute closed form representations (or coefficients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums or iterated integrals., Comment: 25 pages

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

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 $$(2\alpha +1)/2$$ , $$\alpha \in \mathbb {Z}_{+}$$ .

We consider the on-shell mass and wave function renormalization constants $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters $\sqrt{1-\tau^2}$ and $\sqrt{1-\tau^2}/\tau$ which extends the findings from Ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order ${\cal O}(\epsilon^2)$ and ${\cal O}(\epsilon)$ at two- and three-loop order which are crucial ingrediants for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation., Comment: 14 pages, 3 figures

Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's formula when interpreted in the framework of signatures of paths., Comment: 25 pages

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 (2α+1)/2, α∈Z+. Research partially supported by Ministry of Economy and Competitiveness of Spain, under Grant MTM2014-54043-P. Héctor Pijeira-Cabrera: Research partially supported by Ministry of Science, Innovation and Universities of Spain, under Grant PGC2018-096504-B-C33.

Let X be a smooth projective curve of genus $$\ge 1$$ over $${{\mathbb {C}}}$$ , and $$e,\infty \in X({{\mathbb {C}}})$$ be distinct points. Let $$L_n$$ be the mixed Hodge structure of functions on $$\pi _1(X-\{\infty \},e)$$ given by iterated integrals of length $$\le n$$ (as defined by Hain). Building on a work of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012), we express the mixed Hodge extension $${\mathbb {E}}^\infty _{n,e}$$ given by the weight filtration on $$\frac{L_n}{L_{n-2}}$$ as the Abel–Jacobi image of a null-homologous algebraic cycle on $$X^{2n-1}$$ . This algebraic cycle is constructed using the different embeddings of $$X^{n-1}$$ into $$X^n$$ . As a corollary, we show that the extension $${\mathbb {E}}^\infty _{n,e}$$ determines the point $$\infty \in X-\{e\}$$ . When $$n=2$$ , our main result is a strengthening of a theorem of Darmon et al. (Publications Mathematiques de Besancon 2:19–46, 2012). In the final section we assume that $$X,e,\infty $$ are defined over a subfield K of $${{\mathbb {C}}}$$ . Generalizing a construction in (Darmon et al., Publications Mathematiques de Besancon 2:19–46, 2012), we use the extension $${\mathbb {E}}^\infty _{n,e}$$ to define a family of K-rational points on the Jacobian of X parametrized by $$(n-1)$$ -dimensional algebraic cycles on $$X^{2n-2}$$ defined over K.

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.

We argue that in many physical calculations where the “elliptic” sectors are involved, one can express the results via iterated integrals with almost all weights being rational. Our method is based on the existence of 𝜖-regular basis, which is akin to the 𝜖-finite basis defined in ref. [1]. As a demonstration of our technique, we calculate the photon contribution to the total cross section of the production of twoQ$$ \overline{Q} $$Q¯electron-positron collisions.

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 .

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 report on an implementation within GiNaC to evaluate iterated integrals related to elliptic Feynman integrals numerically to arbitrary precision within the region of convergence of the series expansion of the integrand. The implementation includes iterated integrals of modular forms as well as iterated integrals involving the Kronecker coefficient functions $g^{(k)}(z,\tau)$. For the Kronecker coefficient functions iterated integrals in $d\tau$ and $dz$ are implemented. This includes elliptic multiple polylogarithms., Comment: 37 pages, v2: version to be published

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

This is a review of our study of the crystalline pro-umipotent fundamental groupoid of mathrm{I}mathrm{P}^{1}-({0, infty}cup$mu$_{N}) ([Jn], nin{1, ldots , 12 This review is restricted to the parts I and II of this study ([Jn], nin{1, ldots , 6 and to N=1. The part I ([JI], [J2], [J3]) is an explicit computation of the Frobenius. The part II ([J4], [J5], [J6]) is an explicit algebraic study of p-adic multiple zeta values, which are associated with the Frobenius. The theory is centered around the notion of harmonic Ihara action, defined in [J2], [J3]. The harmonic Ihara action is connected to the usual Ihara action, which is a byproduct of a motivic Galois action ; it has three different incarnations related to each other by "comparison " maps ; it enables to express our explicit formulas for the Frobenius, which relate p-adic multiple zeta values and certain sequences of multiple harmonic sums to each other ; it then also enables to understand jointly the algebraic properties of these two objects. We try to emphasize the ideas and the philosophy of this work. We review its context and its motivations (§1) and, after a few technical preliminaries (§2), our general strategy (S3). We state most of the main results of part I (§4) and of part II (§5), and we also summarize and motivate the methods of the proofs. We conclude on the main messages of the theory (§6).