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The Poland–Scheraga model, introduced in the 1970s, is a reference model to describe the denaturation transition of DNA. More recently, it has been generalized in order to allow for asymmetry in the strands lengths and in the formation of loops: the mathematical representation is based on a bivariate renewal process, that describes the pairs of bases that bond together. Here, we consider a disordered version of the model, in which the two strands interact via a potential β V (ω ^ i , ω ¯ j) + h when the ith monomer of the first strand and the jth monomer of the second strand meet. Here, h ∈ R is a homogeneous pinning parameter, (ω i ^) i ≥ 1 and ( ω ¯ j) j ≥ 1 are two sequences of i.i.d. random variables attached to each DNA strand, V (· , ·) is an interaction function and β > 0 is the disorder intensity. In our main result, we find some condition on the underlying bivariate renewal so that, if one takes β , h ↓ 0 at some appropriate (explicit) rate as the length of the strands go to infinity, the partition function of the model admits a non-trivial, disordered, scaling limit. This is known as an intermediate disorder regime and is linked to the question of disorder relevance for the denaturation transition. Interestingly, and unlike any other model of our knowledge, the rate at which one has to take β ↓ 0 depends on the interaction function V (· , ·) and on the distribution of (ω i ^) i ≥ 1 , ( ω ¯ j) j ≥ 1 . On the other hand, the intermediate disorder limit of the partition function, when it exists, is universal: it is expressed as a chaos expansion of iterated integrals against a Gaussian process M , which arises as the scaling limit of the field (e β V (ω ^ i , ω ¯ j)) i , j ≥ 0 and exhibits correlations on lines and columns. [ABSTRACT FROM AUTHOR]

In this article we continue our proofs on integrals of continuous functions of three variables in Mizar. In fact, we use similar techniques as in the case of two variables: we deal with projections of continuous function, the continuity of three variable functions in general, aiming at pure real-valued functions (not necessarily extended real-valued functions), concluding with integrability and iterated integrals of continuous functions of three variables. [ABSTRACT FROM AUTHOR]

First we survey and explain the strategy of some recent results that construct holomorphic sl (2 , ℂ) -differential systems over some Riemann surfaces Σ g of genus g ≥ 2 , satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, Fuchsian sl (2 , ℂ) -systems of compact Riemann surfaces [with an appendix by Takuro Mochizuki], preprint, arXiv:org/abs/2104.04818] or some cocompact Kleinian subgroup Γ ⊂ SL (2 , ℂ) as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from Σ g to the quotient space SL (2 , ℂ) / Γ , where Γ ⊂ SL (2 , ℂ) is a cocompact lattice, that do not factor through any elliptic curve [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. This answers positively a question of Ghys in [E. Ghys, Déformations des structures complexes sur les espaces homogènes de SL (2 , ℂ) , J. Reine Angew. Math. 468 (1995) 113–138]; the question was also raised by Huckleberry and Winkelmann in [A. H. Huckleberry and J. Winkelmann, Subvarieties of parallelizable manifolds, Math. Ann. 295 (1993) 469–483]. Then we prove that when M is a Riemann surface, a Torelli-type theorem holds for the affine group scheme over ℂ obtained from the category of holomorphic connections on étale trivial holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact Kähler manifold M , we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over M. If ϖ (respectively, Θ) stands for the affine group scheme over ℂ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism v : (Θ) → (ϖ) of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals. [ABSTRACT FROM AUTHOR]

Elliptic polylogarithms can be defined as iterated integrals on a genus-one Riemann surface of a set of integration kernels whose generating series was already considered by Kronecker in the 19th century. In this article, we employ the Schottky parametrization of a Riemann surface to construct higher-genus analogues of Kronecker's generating series, which we refer to as Schottky–Kronecker forms. Our explicit construction generalizes ideas from Bernard's higher-genus construction of the Knizhnik-Zamolodchikov connection. Integration kernels generated from the Schottky–Kronecker forms are defined as Poincaré series. Under technical assumptions, related to the convergence of these Poincaré series on the underlying Riemann surface, we argue that these integration kernels coincide with a set of differentials defined by Enriquez, whose iterated integrals constitute higher-genus analogues of polylogarithms. Enriquez' original definition is not well-suited for numerical evaluation of higher-genus polylogarithms. In contrast, the Poincaré series defining our integration kernels can be evaluated numerically for real hyperelliptic curves, for which the above-mentioned convergence assumptions can be verified. We numerically evaluate several examples of genus-two polylogarithms, thereby paving the way for numerical evaluation of hyperelliptic analogues of polylogarithms. [ABSTRACT FROM AUTHOR]

We propose a modification of the algorithm aimed at the construction of analytic solutions of plane axisymmetric problems of elasticity and thermoelasticity in stresses for solid cylinders formed by radially inhomogeneous coaxial layers. Unlike the original approach based on the application of the method of direct integration and the procedure of generalized differentiation, the proposed modification makes it possible to formulate and solve the governing equations of problems of this kind in terms of the classical derivatives with respect to the radial coordinate. In this way, we simplify the procedure of construction of iterated integral kernels of the resolvent solution of the governing integral equation of continuity to which the solution of the analyzed problems is reduced by the method of direct integration. [ABSTRACT FROM AUTHOR]

The signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterizes the path up to a generalized form of reparameterization. It is a classical result of Chen that the log‐signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log‐signature is polynomial if and only if the path is a straight line up to reparameterization. Consequently, the log‐signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses rough path theory, in particular that the signature characterizes a rough path up to reparameterization. [ABSTRACT FROM AUTHOR]

We determine the asymptotic distribution of Manin's iterated integrals of length at most 2. For all lengths, we compute all the asymptotic moments. We show that if the length is at least 3, these moments do in general not determine a unique distribution. [ABSTRACT FROM AUTHOR]

The author apples the Atiyah-Bott-Berline-Vergne formula to the equivariant tautological integrals over flag varieties of types A, B, C, D, and recovers the formulas expressing the integrals as iterated residues at infinity, which were first obtained by Zielenkiewicz using symplectic reduction. [ABSTRACT FROM AUTHOR]

This paper introduces a cascade one proportional derivative incorporating filter (1PDf)-proportional integral (PI) controller abbreviated as c-1PDf-PI to deal effectively with the speed control issue of brushless DC (BLDC) motors. Two problems exist with implementing this controller such as iterated integral overflow and derivation-based chattering owing to the noise. The former is resolved by using an equivalent expression for the integral operation, while the latter is addressed by putting a first-order filter on the derivative term. To achieve the best performance from the controller, snake optimizer (SO) is fruitfully employed for optimizing the controller parameters without need for expert knowledge/interpretation. Here, a more reasonable cost function to assess the candidate solutions is also described. Simulations and laboratory experiments using DSP of TI TMS320F28335 are performed and the results are presented which show that the reference tracking performance, torque disturbance capability and robustness of the c-1PDf-PI controller have potential. These results are also contrasted by those offered by PI and 1PDf speed control schemes individually, affirming the superior performance of our proposal. As per the results, discussion and observation of this research, we stress that good performance and simplicity are salient advantages of the c-1PDf-PI controller, rendering it a good alternative over the complicated controller designs. [ABSTRACT FROM AUTHOR]

We analyse the family of Calabi-Yau varieties attached to four-point fishnet integrals in two dimensions. We find that the Picard-Fuchs operators for fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder integrals. This implies that the periods of the Calabi-Yau varieties for fishnet integrals can be written as determinants of periods for ladder integrals. The representation theory of the geometric monodromy group plays an important role in this context. We then show how the determinant form of the periods immediately leads to the well-known Basso-Dixon formula for four-point fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau geometry implies that the volume is also expressible via a determinant formula of Basso-Dixon type. Finally, we show how the fishnet integrals can be written in terms of iterated integrals naturally attached to the Calabi-Yau varieties. [ABSTRACT FROM AUTHOR]

We present the analytic and compact two-loop helicity amplitudes for QCD and QED corrections to the light-by-light scattering process with massive internal fermions. We express the master integrals either in terms of multiple polylogarithms or in terms of iterated integrals with dlog one-forms. We also elaborate on optimising the analytic results for each phase-space region. This makes the numerical evaluation of the scattering amplitudes fast, stable and suitable for phenomenological applications. [ABSTRACT FROM AUTHOR]

Brain-computer interfaces (BCIs) allow direct communication between one's central nervous system and a computer without any muscle movement hence by-passing the peripheral nervous system. They can restore disabled people's ability to interact with their environment, e.g. communication and wheelchair control. However, to this day their performance is still hindered by the non-stationarity of electroencephalography (EEG) signals, as well as their susceptibility to noise from the users' environment and from their own physiological activity. Moreover, a non-negligible amount of users struggle to use BCI systems based on motor imagery. In this paper, a new method based on the path signature is introduced to tackle this problem by using features which are different from the usual power-based ones. The path signature is a series of iterated integrals computed from a multidimensional path. It is invariant under translation and time reparametrization, which makes it a robust feature for multichannel EEG time series. The performance can be further boosted by combining the path signature with the gold standard Riemannian classifier in the BCI field exploiting the geometric structure of symmetric positive definite (SPD) matrices. The results obtained on publicly available datasets show that the signature method is more robust to inter-user variability than classical ones, especially on noisy and low-quality data. Hence, this study paves the way towards the use of mathematical tools that until now have been neglected, in order to tackle the EEG-based BCI variability issue. It also sheds light on the lead-lag relationship captured by path signature which seems relevant to assess the underlying neural mechanisms. [ABSTRACT FROM AUTHOR]

Given cusp forms f and g of integral weight k ≽ 2, the depth two holomorphic iterated Eichler-Shimura integral If,g is defined by ∫i∞τ ƒ(z)(X - z)k-2 Ig (z; Y) dz, where Ig is the Eichler integral of g and X,Y are formal variables. We provide an explicit vector-valued modular form whose top components are given by If,g. We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by εf,g, that can be seen as a higher depth generalization of the scalar-valued Eichler integral εf of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol-Popa. We show that εf,g can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form Δ. This allows for effective computation of εf,g. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the normalized maxima of real-valued diffusive particles with mean-field drift interaction. Our main result establishes propagation of chaos: in the large population limit, the normalized maxima behave as those arising in an i.i.d. system where each particle follows the associated McKean–Vlasov limiting dynamics. Because the maximum depends on all particles, our result does not follow from classical propagation of chaos, where convergence to an i.i.d. limit holds for any fixed number of particles but not all particles simultaneously. The proof uses a change of measure argument that depends on a delicate combinatorial analysis of the iterated stochastic integrals appearing in the chaos expansion of the Radon–Nikodym density. [ABSTRACT FROM AUTHOR]

We define Chen's iterated integrals of forms with values in an Azumaya algebra and show that in the flat case, these iterated integrals provide chain maps, and more generally, the curved structure is preserved. The appropriate trace maps are then defined and their corresponding geometric meanings are studied. We also define and study the wedge product of forms and the concatenation product of paths for these iterated integrals. [ABSTRACT FROM AUTHOR]

We develop an explicit algebraic de Rham theory for the relative completion of {\mathrm {SL}}_2(\mathbb {Z}). This allows the construction of iterated integrals involving modular forms of the second kind, generalizing iterated integrals of holomorphic modular forms that were previously studied by Brown and Manin. These newly constructed iterated integrals provide all 'multiple modular values' defined by Brown. [ABSTRACT FROM AUTHOR]

It is known that multiple zeta values can be written in terms of certain iterated log-sine integrals. Conversely, we evaluate iterated log-sine integrals in terms of multiple zeta values and multiple polylogarithms in this paper. We also suggest some conjectures on multiple zeta values, multiple Clausen values, multiple Glaisher values and iterated log-sine integrals. [ABSTRACT FROM AUTHOR]

Spectral representations of iterated Itô and Stratonovich stochastic integrals of arbitrary multiplicity, including integrals from Taylor–Itô and Taylor–Stratonovich expansions, are obtained by the spectral method. They are required for the implementation of numerical methods for solving Itô and Stratonovich stochastic differential equations with high orders of mean-square and strong convergence. The purpose of such numerical methods is the modeling of nonlinear stochastic dynamics in many fields. This paper contains necessary theoretical results, as well as the results of numerical experiments. [ABSTRACT FROM AUTHOR]

The paper considers a Lamb's initial-boundary value problem for an elastic half-space in the case when Poisson's ratio takes the limiting value of 1/2. The existence of a classical solution in the form of an iterated improper integral in the case of axial symmetry is proved. [ABSTRACT FROM AUTHOR]

Classical multiple zeta values can be viewed as iterated integrals of the differentials d t t , d t 1 − t from 0 to 1. In this paper, we reprove Brown's theorem: For a i , b i , c i j ∈ ℤ , the iterated integral of the form ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i t i a i (1 − t i) b i ∏ i < j (t j − t i) c i j d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values of weight ≤ N if convergent. What is more, we show that if p i (t) , 1 ≤ i ≤ N , are in a ℚ [ t , 1 / t , 1 / (1 − t) ] -algebra generated by multiple polylogarithms and their dual, and if q i j (t) , 1 ≤ i < j ≤ N , are in a ℚ [ t , 1 / t ] -algebra generated by logarithm, then the iterated integral ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 ∏ i p i (t i) ∏ i < j q i j (t j − t i) d t 1 ⋯ d t N is a ℚ -linear combination of multiple zeta values. As an application of our main results, we show that the coefficients of the Taylor expansions of the Selberg integrals ∫ ⋯ ∫ 0 < t 1 < ⋯ < t N < 1 f ∏ i t i α i (1 − t i) β i ∏ i < j (t j − t i) γ i j d t 1 ⋯ d t N (with respect to α i , β i , γ i j ) at the integral points in some product of right half complex plane are ℚ -linear combinations of multiple zeta values for any f ∈ ℚ [ t i , t i − 1 , (t i − t j) − 1 | 1 ≤ i ≤ N , 1 ≤ i < j ≤ N ]. This statement generalizes Terasoma's original result. [ABSTRACT FROM AUTHOR]

The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ⁢ (X T ε , X ¯ T ε , Mil , (n) ) ≤ C ⁢ ε 2 / n and d TV ⁢ (X T ε , X ¯ T ε , EM , (n) ) ≤ C ⁢ ε / n , where d TV is the total variation distance, X ε is a solution of a stochastic differential equation with a small parameter 휀, and X ¯ ε , Mil , (n) and X ¯ ε , EM , (n) are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method. [ABSTRACT FROM AUTHOR]

We present a method to calculate the x-space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index N, directly, without computing the Mellin-space expressions in explicit form analytically. Here N belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non-iterative integrals, generalizing the former ones. The x-space expressions are needed to derive the small-x behaviour of the respective quantities, which usually cannot be accessed in N-space. We illustrate the method for different (iterated) alphabets, including non-iterative 2F1 and elliptic structures, as examples. These structures occur in different massless and massive three-loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first-order factors. [ABSTRACT FROM AUTHOR]

For the approximation and simulation of twofold iterated stochastic integrals and the corresponding Lévy areas w.r.t. a multi-dimensional Wiener process, we review four algorithms based on a Fourier series approach. Especially, the very efficient algorithm due to Wiktorsson and a newly proposed algorithm due to Mrongowius and Rößler are considered. To put recent advances into context, we analyse the four Fourier-based algorithms in a unified framework to highlight differences and similarities in their derivation. A comparison of theoretical properties is complemented by a numerical simulation that reveals the order of convergence for each algorithm. Further, concrete instructions for the choice of the optimal algorithm and parameters for the simulation of solutions for stochastic (partial) differential equations are given. Additionally, we provide advice for an efficient implementation of the considered algorithms and incorporated these insights into an open source toolbox that is freely available for both Julia and Matlab programming languages. The performance of this toolbox is analysed by comparing it to some existing implementations, where we observe a significant speed-up. [ABSTRACT FROM AUTHOR]

In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher levels are yielded from the inverse relationship with an s-step function of the exponential function. We show such a function with recurrence relations and give the expressions of poly-Cauchy numbers with higher levels. Poly-Cauchy numbers with higher levels can be also expressed in terms of iterated integrals and a combinatorial summation. Poly-Cauchy numbers with higher levels for negative indices have a double summation formula. In addition, Cauchy numbers with higher levels can be also expressed in terms of determinants. [ABSTRACT FROM AUTHOR]

In this paper, we provide variation of constants formulae for linear (forward) stochastic Volterra integral equations (SVIEs, for short) and linear Type-II backward stochastic Volterra integral equations (BSVIEs, for short) in the usual Itô's framework. For these purposes, we define suitable classes of stochastic Volterra kernels and introduce new notions of the products of adapted L 2 -processes. Observing the algebraic properties of the products, we obtain the variation of constants formulae by means of the corresponding resolvent. Our framework includes SVIEs with singular kernels such as fractional stochastic differential equations. Also, our results can be applied to general classes of SVIEs and BSVIEs with infinitely many iterated stochastic integrals. The duality principle between generalized SVIEs and generalized BSVIEs is also proved. [ABSTRACT FROM AUTHOR]

We prove the existence of a maximal dimensional vector space of real functions on the real plane all of whose nonzero members are bounded, entire, non-Lebesgue-integrable, and satisfy the equality of the two iterated integrals given in the conclusion of the Fubini theorem, with the additional property that these iterated integrals exist in the Riemann sense, but not in the Lebesgue one. Moreover, this vector space is dense in the space of smooth functions on the plane under its natural topology. [ABSTRACT FROM AUTHOR]

In this paper, for the mapping of a sphere into a compact orientable manifold Sn → M, n ≥ 1, we solve the problem of determining whether it represents a nontrivial element in the homotopy group of the manifold πn(M). For this purpose, we consistently use the theory of iterated integrals developed by Chen. It should be noted that the iterated integrals as repeated integration were previously meaningfully used by Lappo-Danilevsky to represent solutions of systems of linear differential equations and by Whitehead for the analytical description of the Hopf invariant for mappings f : S2n−1 → Sn, n ≥ 2. We give a brief description of Chen's theory representing Whitehead's and Haefliger's formulas for the Hopf invariant and generalized Hopf invariant. Examples of calculating these invariants using the technique of iterated integrals are given. Further, it is shown how one can detect any element of the fundamental group of a Riemann surface using iterated integrals of holomorphic forms. This required to prove that the intersection of the terms of the lower central series of the fundamental group of a Riemann surface is a unit group. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce and study a new class of nonlinear coupled Hilfer differential equations with nonlocal boundary conditions involving Riemann–Liouville and Hadamard-type iterated fractional integral operators. By applying the Leray–Schauder alternative and Krasnosel'skiĭ's fixed point theorem, two results presenting different criteria for the existence of solutions to the given problem are proven. The third result provides a sufficient criterion for the existence of a unique solution to the problem at hand. Numerical examples are constructed to demonstrate the application of the results obtained. Two graphs show asymmetric solutions when a Hilfer parameter is varied. The work presented in this paper is novel and significantly enriches the literature on the topic. [ABSTRACT FROM AUTHOR]

We compute the master integrals relevant for the two-loop corrections to pseudo-scalar quarkonium and leptonium production and decay. We present both analytic and high-precision numerical results. The analytic expressions are given in terms of multiple polylogarithms (MPLs), elliptic multiple polylogarithms (eMPLs) and iterated integrals of Eisenstein series. As an application of our results, we obtain for the first time an analytic expression for the two-loop amplitude for para-positronium decay to two photons at two loops. [ABSTRACT FROM AUTHOR]

We provide a comprehensive summary of concepts from Calabi-Yau motives relevant to the computation of multi-loop Feynman integrals. From this we derive several consequences for multi-loop integrals in general, and we illustrate them on the example of multi-loop banana integrals. For example, we show how Griffiths transversality, known from the theory of variation of mixed Hodge structures, leads quite generically to a set of quadratic relations among maximal cut integrals associated to Calabi-Yau motives. These quadratic relations then naturally lead to a compact expression for l-loop banana integrals in D = 2 dimensions in terms of an integral over a period of a Calabi-Yau (l − 1)-fold. This new integral representation generalizes in a natural way the known representations for l ≤ 3 involving logarithms with square root arguments and iterated integrals of Eisenstein series. In a second part, we show how the results obtained by some of the authors in earlier work can be extended to dimensional regularization. We present a method to obtain the differential equations for banana integrals with an arbitrary number of loops in dimensional regularization without the need to solve integration-by-parts relations. We also present a compact formula for the leading asymptotics of banana integrals with an arbitrary number of loops in the large momentum limit. This generalizes the novel Γ ̂ -class introduced by some of the authors to dimensional regularization and provides a convenient boundary condition to solve the differential equations for the banana integrals. As an application, we present for the first time numerical results for equal-mass banana integrals with up to four loops and up to second order in the dimensional regulator. [ABSTRACT FROM AUTHOR]

We show that the differential equation for the three-loop equal-mass banana integral can be cast into an ε-factorised form with entries constructed from (meromorphic) modular forms and one special function, which can be given as an iterated integral of meromorphic modular forms. The ε-factorised form of the differential equation allows for a systematic solution to any order in the dimensional regularisation parameter ε. The alphabet of the iterated integrals contains six letters. [ABSTRACT FROM AUTHOR]

The usual iterated integral map given by Chen produces an equivalence between the two-sided bar complex on differential forms and the de Rham complex on the path space. This map fails to make sense when considering the curved differential graded algebra of bundle-valued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen's iterated integral that incorporates parallel transport and maps an analog of the two-sided bar construction on bundle-valued forms to bundle-valued forms on the path space. This iterated integral is proven to be a homotopy equivalence of curved differential graded algebras, and for real-valued forms it factors through the usual Chen iterated integral. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of representing any polynomial in terms of the degenerate Bernoulli polynomials arising from Volkenborn integral and more generally of the higher‐order degenerate Bernoulli polynomials arising from iterated Volkenborn integral. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples. [ABSTRACT FROM AUTHOR]

In this paper, we prove the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function on a line parallel to the real axis. [ABSTRACT FROM AUTHOR]

We study the Hölderian regularity of Gaussian wavelets series and show that they display, almost surely, three types of points: slow, ordinary and rapid. In particular, this fact holds for the Fractional Brownian Motion. Finally, we remark that the existence of slow points is specific to these functions. [ABSTRACT FROM AUTHOR]

We revisit the bifurcation theory of the Lotka–Volterra quadratic system X 0 : ẋ = − y − x 2 + y 2 , ẏ = x − 2 x y with respect to arbitrary quadratic deformations. The system has a double center, which is moreover isochronous. We show that the deformed system can have at most two limit cycles on the finite plane, with possible distribution (i , j) , where i + j ≤ 2. 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. [ABSTRACT FROM AUTHOR]

Given a stochastic differential equation (SDE) in Rn whose solution is constrained to lie in some manifold M⊂Rn , we identify a class of numerical schemes for the SDE whose iterates remain close to M to high order. These schemes approximate a geometrically invariant scheme, which gives perfect solutions for any SDE that is diffeomorphic to n -dimensional Brownian motion. Unlike projection-based methods, they may be implemented without explicit knowledge of M. They can even be implemented if the solution merely remains close to M , without being exactly confined to it. Our approach does not require simulating any iterated Itô integrals beyond those needed to implement the Euler–Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their geometric advantages in a variety of numerical contexts, including Monte Carlo simulation of the Riemannian Langevin equation. [ABSTRACT FROM AUTHOR]

We give a rigorous construction of the path integral in N = 1 / 2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler–Jones–Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Güneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah–Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaumé, Atiyah, Bismut and Witten. [ABSTRACT FROM AUTHOR]

A new algorithm for the approximation and simulation of twofold iterated stochastic integrals together with the corresponding Lévy areas driven by a multidimensional Brownian motion is proposed. The algorithm is based on a truncated Fourier series approach. However, the approximation of the remainder terms differs from the approach considered by Wiktorsson (2001). As the main advantage, the presented algorithm makes use of a diagonal covariance matrix for the approximation of one part of the remainder term and has a higher accuracy due to an exact approximation of the other part of the remainder. This results in a significant reduction of the computational cost compared to, e.g., the algorithm introduced by Wiktorsson. Convergence in L p (Ω) -norm with p ≥ 2 for the approximations calculated with the new algorithm as well as for approximations calculated by the basic truncated Fourier series algorithm is proved and the efficiency of the new algorithm is analyzed. [ABSTRACT FROM AUTHOR]

In this paper, the Gaussian quadrature formula is used to approximately calculate two-dimensional hypersingular integrals of the form ∬ S f (x , y) r 3 d S on the circular domain. The two-dimensional hypersingular integrals are changed into iterated integrals by polar transformation. The inner layer of the integral has singularity and Gaussian quadrature formula is constructed for this part. The outer layer without singularity is calculated by Gauss–Legendre quadrature formula of Riemann integral. The corresponding theoretical analysis is given. Furthermore, this type of two-dimensional hypersingular integral is extended to the form ∬ S f (x , y) r p + 1 d S (p > 2) and an error estimate is also derived. Several numerical examples are presented to verify the effectiveness of the proposed algorithm, no matter the singularity is located at the center or near the boundary of the circular domain. [ABSTRACT FROM AUTHOR]

We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a category internal to the category of topological spaces. By applying the machinery to a Borel construction, we explicitly determine the mod p cohomology algebra of the free loop space of the real projective space for each odd prime p. This example is emphasized as an important computational case. Moreover, we represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold M with differential forms on the universal cover of M via Chen's iterated integral map. [ABSTRACT FROM AUTHOR]

We consider the problem of obtaining higher order in regularization parameter ε analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct integration of parametric representations in terms of iterated integrals. Taking non-planar elliptic vertex as an example we show that in addition to two mentioned methods one can use analytical solution of differential equations in terms of power series. Moreover, in the last case it is possible to obtain the exact in ε results. [ABSTRACT FROM AUTHOR]

We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of solutions and Ulam's type stabilities including the well‐known Ulam–Hyers stability and the newly extended Ulam–Hyers conformable exponential stability for two classes of fractional differential equations with the conformable fractional derivative and the time delay. The Banach contraction principle, the technique of Picard operator, the Grönwall integral inequalities, and generalized iterated integral inequality in the sense of conformable fractional integral are the main tools for deriving our main results. Finally, several illustrative examples will be presented to demonstrate our work. [ABSTRACT FROM AUTHOR]

We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter x and the dimension d, in the whole kinematic range of x. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the MS ¯ and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy. [ABSTRACT FROM AUTHOR]

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 η -values. At the end of the paper, we study a variant of Kaneko–Tsumura η -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The standard definition for the Atangana–Baleanu fractional derivative involves an integral transform with a Mittag‐Leffler function in the kernel. We show that this integral can be rewritten as a complex contour integral which can be used to provide an analytic continuation of the definition to complex orders of differentiation. We discuss the implications and consequences of this extension, including a more natural formula for the Atangana–Baleanu fractional integral and for iterated Atangana–Baleanu fractional differintegrals. [ABSTRACT FROM AUTHOR]