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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]

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]

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 signature transform, defined by the formal tensor series of global iterated path integrals, is a homomorphism between the path space and the tensor algebra that has been studied in geometry, control theory, number theory as well as stochastic analysis. An elegant isometry conjecture states that the length of a bounded variation path can be recovered from the asymptotics of its normalised signature: This property depends on a key topological non-degeneracy notion known as treereducedness (namely, with no tree-like pieces). Existing arguments have relied crucially on having a continuous derivative under the unit speed parametrisation. In this article, we prove the above isometry conjecture for planar paths by assuming only local bounds on the angle of 0 (which ensures the absence of tree-like pieces). Our technique is based on lifting the path onto the special linear group SL2.R/and analysing the behaviour of the associated angle dynamics at a microscopic level. [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]

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 study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let (Fj)j≥1 be a sequence of normalized homeomorphic solutions to the planar Beltrami equation ∂z-Fj(z) = µj(z,ω) ∂zFj(z), where the random dilatation satisfies ∣µj∣ ≤ k < 1 and has locally periodic statistics, for example of the type (0.1) ... where g(z,ω) decays rapidly in z, the random variables Xn are i.i.d., and ϕ ∈ C0∞. We establish the almost sure and local uniform convergence as j → ∞ of the maps Fj to a deterministic quasiconformal limit Fj. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let T1, ..., Tm be translation and dilation invariant singular integrals on Rd, and consider a d-dimensional version of µj, e.g., as defined above or within a more general setting, see Definition 3.4 below. We then prove that there is a deterministic function f such that almost surely, µj Tmµj ... T1µj → f as j →∞ weakly in Lp, for 1 < p < ∞. [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]

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]

Fractional differential equations with constant coefficients can be readily handled by a number of methods, but those with variable coefficients are much more challenging. Recently, a method has appeared in the literature for solving fractional differential equations with variable coefficients, the solution being in the form of an infinite series of iterated fractional integrals. In the current work, we consider fractional differential equations with Atangana–Baleanu integro-differential operators and continuous variable coefficients, and establish analytical solutions for such equations. The representation of the solution is given by a uniformly convergent infinite series involving Atangana–Baleanu operators. To the best of our knowledge, this is the first time that explicit analytical solutions have been given for such general Atangana–Baleanu differential equations with variable coefficients. The corresponding results for fractional differential equations with constant coefficients are also given. [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 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]

Non-local properties of non-linear systems, linear with respect to control function, are studied. We consider a class of such systems that are homogeneous at the origin. At a point different from the origin the system may be non-homogeneous, so its homogeneous approximation is of interest. However, at such points, the system may become less singular than at the origin. We describe a set of points where the system is as singular as at the origin. It is shown that such a set forms a linear subspace. The main tool used in the paper is a free algebra approach proposed in our previous papers. [ABSTRACT FROM AUTHOR]

The unpolarized and polarized massive operator matrix elements A Q g (3) and Δ A Q g (3) contain first–order factorizable and non–first–order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first–order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color– ζ factors for the cases in which also non–first–order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin N –space, and correspondingly, on Kummer–Poincaré and square–root valued alphabets in Bjorken– x space. We present a complete discussion of the possibilities of solving the present problem in N –space analytically and we also discuss the limitations in the present case to analytically continue the given N –space expressions to N ∈ C by strict methods. The representation through generating functions allows a well synchronized representation of the first–order factorizable results over a 17–letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in x –space, also containing up to weight w = 5 special constants, which can be rationalized to Kummer–Poincaré iterated integrals at special arguments. The analytic x –space representation requires separate analyses for the intervals x ∈ [ 0 , 1 / 4 ] , [ 1 / 4 , 1 / 2 ] , [ 1 / 2 , 1 ] and x > 1. We also derive the small and large x limits of the first–order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman diagrams, and an independent high–precision numerical solution of the problems. [ABSTRACT FROM AUTHOR]

In this paper, we study new classes of Apéry-type series involving the central binomial coefficients and the multiple t -harmonic sums by combining the methods of iterated integrals and Fourier–Legendre series expansions, where the multiple t -harmonic sums are a variation of multiple harmonic sums in which all the summation indices are restricted to odd numbers only. Our approach also enables us to generalize some old classes of Apéry-type series involving harmonic sums to those with products of multiple harmonic sums and multiple t -harmonic sums. We show that these series can be expressed as either the real or the imaginary part of a Q -linear combination of colored multiple zeta values of level 4. Hopefully, these relations will shed some new lights on their properties which may lead to novel approaches to irrationality questions on the Riemmann zeta values, or more generally, the multiple zeta values. [ABSTRACT FROM AUTHOR]

We calculate the master integrals for bipartite cuts of the three-loop propagator QED diagrams. These master integrals determine the spectral density of the photon self energy. Our results are expressed in terms of the iterated integrals, which, apart from the 4m cut (the cut of 4 massive lines), reduce to Goncharov's polylogarithms. The master integrals for 4m cut have been calculated in our previous paper in terms of the one-fold integrals of harmonic polylogarithms and complete elliptic integrals. We provide the threshold and high-energy asymptotics of the master integrals found, including those for 4m cut. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A profile from the Argo ocean observation array is a sequence of three‐dimensional vectors composed of pressure, salinity, and temperature, appearing as a continuous curve in three‐dimensional space. The shape of this curve is faithfully represented by a path signature, which is a collection of all the iterated integrals. Moreover, the product of two terms of the signature of a path can be expressed as the sum of higher‐order terms. As a result of this algebraic property, a nonlinear function of the profile shape can always be represented by a weighted linear combination of the iterated integrals, which enables machine learning of a complicated function of the profile shape. In this study, we performed supervised learning for existing Argo data with quality control flags by using the signature method and demonstrated the estimation performance by cross validation. Unlike rule‐based approaches, which require several complicated and possibly subjective rules, this method is simple and objective in nature because it relies only on past knowledge regarding the shape of profiles. This technique is critical for realizing automatic quality control for Argo profile data. Key Points: We transformed each vertical sequence of Argo observation into the signatureThen, it becomes easy to express the nonlinear function of the profile shapeWe took advantage of this mathematical property for assigning quality control flags [ABSTRACT FROM AUTHOR]

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated integrals of length one. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

olutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach has a potential to be more suitable for applications. Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given. [ABSTRACT FROM AUTHOR]

• A rigorous construction of Arborified Zeta Values (AZVs) as iterated series and integrals and a proof of their convergence. • An explicit relation between AZVs and Multi Zeta Values (MZVs). • New (to my knowledge) proofs of some of the properties of the MZVs, proofs obtained by purely algebraic means. • The construction of new (to my knowledge) algebraic structures on trees relevant to AZVs. Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees. [ABSTRACT FROM AUTHOR]

We compute fully analytic results for the three-loop diagrams involving two different massive quark flavours contributing to the ρ parameter in the Standard Model. We find that the results involve exactly the same class of functions that appears in the well-known sunrise and banana graphs, namely elliptic polylogarithms and iterated integrals of modular forms. Using recent developments in the understanding of these functions, we analytically continue all the iterated integrals of modular forms to all regions of the parameter space, and in each region we obtain manifestly real and fast-converging series expansions for these functions. [ABSTRACT FROM AUTHOR]

In this paper, we construct extensions of mixed Hodge structure coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth projective pointed curve which correspond to the regulators of certain motivic cohomology cycles on the Jacobian of the curve essentially constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of Colombo (J. Algebr. Geom.11(4) (2002) 761–790) where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve. [ABSTRACT FROM AUTHOR]

We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of 'mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in q,q and log |q| involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series. [ABSTRACT FROM AUTHOR]

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 two Q Q ¯ electron-positron collisions. [ABSTRACT FROM AUTHOR]

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 Poincaré 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. [ABSTRACT FROM AUTHOR]

We generalize Gehrig-Kawski theorem connecting the adjoint of the Eulerian idempotent with the logarithm of identity operator in the convolution product algebra. This has application in dynamical systems, control theory, coordinates of the first kind, generalized BCH-formula, Magnus expansion, etc., and is connected with iterated integrals and the signature of a path. We also show certain algebraic identities, which are meaningful in context of control and path-signature theory. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract Let G be a finite graph, and let G n be the n -th iterated cone over G. We study the structure of the critical group of G n arising in divisor and sandpile theory. [ABSTRACT FROM AUTHOR]

Abstract We calculate the non-singlet, the pure singlet contribution, and their interference term, at O (α 2) due to electron-pair initial state radiation to e + e − annihilation into a neutral vector boson in a direct analytic computation without any approximation. The correction is represented in terms of iterated incomplete elliptic integrals. Performing the limit s ≫ m e 2 we find discrepancies with the earlier results of Ref. [1] and confirm results obtained in Ref. [2] where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in m 2 / s. In this way, we also confirm the validity of the factorization of massive partons in the Drell–Yan process. We also add non-logarithmic terms at O (α 2) which have not been considered in [1]. The corrections are of central importance for precision analyses in e + e − annihilation into γ ⁎ / Z ⁎ at high luminosity. [ABSTRACT FROM AUTHOR]

Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assump- tion on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach has a potential to be more suitable for applications. Rep- resentation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract We investigate linear relations among a class of iterated integrals on the Riemann sphere minus four points 0 , 1 , z and ∞. Generalization of the duality formula and the sum formula for multiple zeta values to the iterated integrals are given. [ABSTRACT FROM AUTHOR]

This is a survey on renormalisation in algebraic locality setup highlighting the role that locality morphisms can play for renormalisation purposes. After describing the general framework to build locality regularisation maps, we illustrate renormalisation by locality algebra homomorphisms on three examples, the renormalisation of conical zeta functions at poles, the definition of branched zeta functions and their evaluation at poles and finally the values of iterated integrals stemming from Kreimer's toy model. [ABSTRACT FROM AUTHOR]

Let G ≅ Z / m 1 Z × … × Z / m r Z be a finite abelian group with 1 < m 1 | … | m r = exp ⁡ ( G ) . The Kemperman Structure Theorem characterizes all subsets A , B ⊆ G satisfying | A + B | < | A | + | B | and has been extended to cover the case when | A + B | ≤ | A | + | B | . Utilizing these results, we provide a precise structural description of all finite subsets A ⊆ G with | n A | ≤ ( | A | + 1 ) n − 3 when n ≥ 3 (also when G is infinite), in which case many of the pathological possibilities from the case n = 2 vanish, particularly for large n ≥ exp ⁡ ( G ) − 1 . The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence S of terms from G having length | S | ≥ 2 | G | − 1 must either have every element of G representable as a sum of | G | -terms from S or else have all but | G / H | − 2 of its terms lying in a common H -coset for some H ≤ G . We show that the much weaker hypothesis | S | ≥ | G | + exp ⁡ ( G ) suffices to obtain a nearly identical conclusion, where for the case H is trivial we must allow all but | G / H | − 1 terms of S to be from the same H -coset. The bound on | S | is improved for several classes of groups G , yielding optimal lower bounds for | S | . We also generalize Olson's result for | G | -term subsums to an analogous one for n -term subsums when n ≥ exp ⁡ ( G ) , with the bound likewise improved for several special classes of groups. [ABSTRACT FROM AUTHOR]

In this paper, we give a proof for Sehgal-Guseman theorem of fixed point in rectangular b-metric spaces. Our result is supported with a suitable example. As a corollary of our results, we obtain fixed point results of contraction mappings in b-metric spaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Motivated by the fact that fractional q-integrals play important roles in numerous areas of mathematical, physical and engineering sciences, it is natural to consider the corresponding iterated fractional q-integrals. The main object of this paper is to define these iterated fractional q-integrals, to build the relations between iterated fractional q-integrals and certain families of generating functions for q-polynomials and to generalize two fractional q-identities which are given in a recent work [Fract. Calc. Appl. Anal. 10 (2007), 359–373]. As applications of the main results presented here, we deduce several bilinear generating functions, Srivastava-Agarwal type generating functions, multilinear generating functions and U(n + 1) type generating functions for the Rajković-Marinković-Stanković polynomials. [ABSTRACT FROM AUTHOR]