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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]

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]

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 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]

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]

Summary: Any homogeneous controller asymptotically stabilizing disturbed integrator chains can be coupled with the proposed robust homogeneous filtering observer. The type of convergence (in finite‐time, in fixed‐time to any ball or exponential) is determined by the sign of the system homogeneity degree (HD). Filtering high‐gain observers and sliding‐mode‐based observers/differentiators are important particular cases corresponding to zero and negative HDs, respectively. Stabilization accuracy is estimated in the presence of possibly unbounded sampling noises featuring locally bounded iterated integrals. Accurate stabilization and differentiation are demonstrated for extremely large noises. [ABSTRACT FROM AUTHOR]

We consider a standard one‐dimensional Brownian motion on the time interval [0,1] conditioned to have vanishing iterated time integrals up to order N. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as N→∞ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by N and show that they converge in finite‐dimensional distributions as N→∞ to a collection of independent zero‐mean Gaussian random variables whose variances follow a scaled semicircle. The fluctuation result is a consequence of a limit theorem for Legendre polynomials which quantifies their completeness and orthogonality property. In the proof of the latter, we encounter a Catalan triangle. [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]

A milestone in probability theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables {ti}i=1∞ with mean 0 and variance 1 In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph H. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random k‐uniform hypergraphs, we obtain the Central Limit Theorem and LIL for the number of Hamilton cycles. [ABSTRACT FROM AUTHOR]

Abstract: An affine iterated function system (IFS) is a finite collection of affine invertible contractions and the invariant set associated to the mappings is called self‐affine. In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self‐affine set is the affinity dimension for Lebesgue almost every translation vectors. Similar statement was proven by Jordan, Pollicott, and Simon in 2007 for the dimension of self‐affine measures. In this article, we have an orthogonal approach. We introduce a class of self‐affine systems in which, given translation vectors, we get the same results for Lebesgue almost all matrices. The proofs rely on Ledrappier–Young theory that was recently verified for affine IFSs by Bárány and Käenmäki, and a new transversality condition, and in particular they do not depend on properties of the Furstenberg measure. This allows our results to hold for self‐affine sets and measures in any Euclidean space. [ABSTRACT FROM AUTHOR]

For a martingale ( X n) converging almost surely to a random variable X, the sequence ( X n- X) is called martingale tail sum. Recently, Neininger ( Random Structures Algorithms 46 (2015), 346-361) proved a central limit theorem for the martingale tail sum of Régnier's martingale for the path length in random binary search trees. Grübel and Kabluchko (in press) gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane-oriented recursive trees. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 493-508, 2017 [ABSTRACT FROM AUTHOR]

The self-affine measure [ABSTRACT FROM AUTHOR]

A remarkable result by Denjoy and Wolff states that every analytic self-map ϕ of the open unit disc D of the complex plane, except an elliptic automorphism, has an attractive fixed point to which the sequence of iterates {ϕn}n≥1 converges uniformly on compact sets: if there is no fixed point in D, then there is a unique boundary fixed point that does the job, called the Denjoy-Wolff point. This point provides a classification of the analytic self-maps of D into four types: maps with interior fixed point, hyperbolic maps, parabolic automorphism maps and parabolic non-automorphism maps. We determine the convergence of the Aleksandrov-Clark measures associated to maps falling in each group of such classification. [ABSTRACT FROM AUTHOR]

In this paper, we consider intrinsic Diophantine approximation in the sense of Mahler [‘Some suggestions for further research’, Bull. Aust. Math. Soc. 29 (1984) 101–108.] on the Cantor set and similar fractals. We begin by obtaining a Dirichlet-type theorem for the limit set of a rational iterated function system. Next, we investigate the rigidity of this result by applying a random affine transformation to such a fractal and determining the intrinsic Diophantine theory of the image fractal. The final two sections concern the optimality of the Dirichlet-type theorem established at the beginning. The first of these seeks to show optimality in the sense that any proof using the same method as ours cannot prove a better approximation exponent, in a precise sense. This is done by introducing a new height function on the rationals intrinsic to the fractal and studying the Diophantine properties of points on the fractal with respect to this new height function. In the final section, we use a result of S. Ramanujan to give a lower bound on the periods of rationals which could cause exceptions to the optimality of the approximation exponent (this time with the usual height function). We give a heuristic argument suggesting that there are only finitely many rationals with periods so large; if this is true, then the approximation exponent is optimal for the Cantor set. [ABSTRACT FROM AUTHOR]

For a transcendental meromorphic function f, various properties of the set[formula]were obtained in [8] and [9]. Here we establish analogous properties for the smaller sets[formula]introduced in [5], and[formula]We deduce a symmetry result for Julia sets J(f), and also indicate some techniques for showing that certain invariant curves lie in I′(f), Z(f) and J(f). 2000 Mathematics Subject Classification 30D05, 37F10, 37F50. [ABSTRACT FROM AUTHOR]

We consider the Abel equationdydx=p(x)y2+q(x)y3, a≤x≤b,where p and q are polynomials, and a is a fixed constant. We denote the solution of (1) by y(x, c), where y(a, c) = c. Standard existence theorems ensure that y(x, c) is well-defined and analytic in both its arguments, for c sufficiently small. If y(b, c) = c, then we call y(x, c) a periodic solution. Likewise, if y(b, c) ≡ c for all c close to 0, then we say that the system has a centre between a and b. The numbers a and b are not important; by a simple transformation, we can always choose a = 0 and b = 1, and we shall usually do so from now on.Abel equations arise in several circumstances, but perhaps the main reason for their recent study is connected to the family of systemsx˙=−y+M(x,y), y˙=x+N(x,y),where M and N are homogeneous polynomials of the same degree n. A transformation due to Cerkas allows us to bring these systems to the form (1), where p and q are now trigonometric polynomials. It is not hard to show that, setting a = 0 and b = 2π, the definitions of periodic solution and centre for (1) coincide with their usual definitions in the planar system (2). There are also transformations to Abel-type equations for more general systems; see [8, 10].This trigonometric Abel equation has been used in a large number of works in order to estimate the number of limit cycles or obtain centre conditions, as well as in more general investigations relating the derivatives of the return map with iterated integrals. However, studying system (2) in whatever form is by no means easy, and a natural question is to ask whether we can still capture the essence of this problem if we take p and q to be polynomials, in the hope that the calculations will become easier. The recent series of investigations by Briskin, Françoise and Yomdin [3, 4, 5] seems to indicate that this could be the case.Our interest here is to see what conditions the existence of a centre in (1) imposes on the defining equations. For ease of reference, we shall always denote the antiderivative of the polynomials p and q as P and Q; that is,P(x)=∫axp(ξ)dξ, Q(x)=∫axq(ξ)dξ. 2000 Mathematics Subject Classification 34C25, 34C99. [ABSTRACT FROM AUTHOR]