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We study weight one specializations of the Euler system of Beilinson–Flach elements introduced by Kings, Loeffler and Zerbes, with a view towards a conjecture of Darmon, Lauder and Rotger relating logarithms of units in suitable number fields to special values of the Hida–Rankin p-adic L-function. We show that the latter conjecture follows from expected properties of Beilinson–Flach elements and prove the analogue of the main theorem of Castella and Hsieh about generalized Kato classes. [ABSTRACT FROM AUTHOR]

Vafa–Witten (VW) theory is a topologically twisted version of N = 4 supersymmetric Yang–Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU(N) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU(2) and is considered on the complex projective plane P 2 . This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU(3) involves an iterated integral of modular forms. Moreover, the modular anomaly for SU(3) can be traded for a holomorphic anomaly, which is shown to factor into a product of the partition functions for lower rank gauge groups. The SU(3) partition function is mathematically an example of a mock modular form of depth two. [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]

The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull-back differential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati’s method. The main concepts are the Picard-Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial, and the iterated integral. [ABSTRACT FROM AUTHOR]

We give a new proof of the duality of multiple zeta values, which makes no use of the iterated integrals. The same method is also applicable to Ohno's relation for (q -)multiple zeta values. [ABSTRACT FROM AUTHOR]

The signature of a path γ is a sequence whose n-th term is the order-n iterated integrals of γ. It arises from solving multidimensional linear differential equations driven by γ. We are interested in relating the path properties of γ with its signature. If γ is C¹, then an elegant formula of Hambly and Lyons relates the length of γ to the tail asymptotics of the signature. We show an analogous formula for the multidimensional Brownian motion, with the quadratic variation playing a similar role to the length. In the proof, we study the hyperbolic development of Brownian motion and also obtain a new subadditive estimate for the asymptotic of signature, which may be of independent interest. As a corollary, we strengthen the existing uniqueness results for the signatures of Brownian motion. [ABSTRACT FROM AUTHOR]

The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense. [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]

In this paper we give two idelic representations of the multiple zeta values—one using iterated integrals over the finite ideles and the other using iterated integrals over the idele class group. The two representations lead to stuffle and shuffle relations, respectively. Using iterated integrals over the ideles, we recover in a unified way the double shuffle relations for multiple zeta values. [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]

In this paper, we present a feasible predictor-corrector interior-point method for symmetric cone optimization problem in the large neighbourhood of the central path. The method is generalization of Ai-Zhang's predictor-corrector algorithm to the symmetric cone optimization problem. Starting with a feasible point in given large neighbourhood of the central path, the algorithm still terminates in at most iterations. This matches the best known iteration bound that is usually achieved by short-step methods, thereby, closing the complexity gap between long- and short-step interior-point methods for symmetric cone optimization. The preliminary numerical results on a selected set of NETLIB problems show advantage of the method in comparison with the version of the algorithm that is not based on the predictor-corrector scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a non-monotone line search method for solving optimization problems on Stiefel manifold. The main novelty of our approach is that our method uses a search direction based on a linear combination of descent directions and a Barzilai-Borwein line search. The feasibility is guaranteed by projecting each iterate on the Stiefel manifold through SVD (singular value decomposition) factorizations. Some theoretical results for analysing the algorithm are presented. Finally, we provide numerical experiments for comparing our algorithm with other state-of-the-art procedures. The code is available online. The experimental results show that the proposed algorithm is competitive with other approaches and for particular problems, the computational performance is better than the state-of-the-art algorithms. [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]

The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of ZF between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of ZF. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna–Wagner principles and obtain some results related to their failure. [ABSTRACT FROM AUTHOR]

We examine recursive monotonic functions on the Lindenbaum algebra of $EA$. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and (φ ∧ Con(φ)). We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function ƒ, if there is an iterate of Con that bounds ƒ everywhere, then ƒ must be somewhere equal to an iterate of Con. [ABSTRACT FROM AUTHOR]

We determine the proof-theoretic ordinals (i) of C-TI[α], the transfinite induction along α , for any hyperarithmetical level C, in the first order setting and (ii) of any combination of iterated arithmetical comprehension and C-TI[α] for C ≡ Πik, Σik (i = 0,1) in the second order setting. [ABSTRACT FROM AUTHOR]

We analyze the models L[T2n] , where T2n is a tree on ω × κ12n + 1 projecting to a universal Π1n set of reals, for n > 1. Following Hjorth's work on L[T2n] , we show that under Det(Π12n , the models L[T2n] are unique, that is they do not depend of the choice of the tree T2n. This requires a generalization of the Kechris–Martin theorem to all pointclasses Π12n + 1. We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of Μ#2n + 1. We then show that the GCH holds in L[T2n], for every n < ω, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy. [ABSTRACT FROM AUTHOR]

We analyze the models L[T2n] , where T2n is a tree on ω × κ12n + 1 projecting to a universal Π1n set of reals, for n > 1. Following Hjorth's work on L[T2n] , we show that under Det(Π12n , the models L[T2n] are unique, that is they do not depend of the choice of the tree T2n. This requires a generalization of the Kechris–Martin theorem to all pointclasses Π12n + 1. We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of Μ#2n + 1. We then show that the GCH holds in L[T2n], for every n < ω, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy. [ABSTRACT FROM AUTHOR]

We determine continuous bijections f, acting on a real interval into itself, whose k-fold iterate is the quasi-arithmetic mean of all its subsequent iterates from f0 up to fn (where 0⩽k⩽n). Namely, we prove that if at most one of the numbers k, n is odd, then such functions consist of at most three affine pieces. [ABSTRACT FROM AUTHOR]

In this paper, we prove a family of identities among multiple zeta values, which contains as a special case a conjectural identity of Hoffman. We use the iterated integrals on ℙ 1 ∖ { 0 , 1 , ∞ , z } for our proof. [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]

In 1985, Odoni showed that in characteristic 0 the Galois group of the n-th iterate of the generic polynomial with degree d is as large as possible. That is, he showed that this Galois group is the n-th wreath power of the symmetric group Sd. We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper. [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]

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 In presence of very noisy data, standard regularization methods generally fail in reconstructing satisfying solutions. To this end, iterated regularization techniques can be implemented. This class of regularization approaches can be seen as an iterative refinement of an initial solution obtained from classical regularization methods. In the present paper, an iterated multiplicative regularization for dealing with force reconstruction problems is discussed. More specifically, the associated non-stationary formulation is compared to its stationary counterpart through numerical and experimental validations. It is shown that the proposed stationary version outperforms the non-stationary formulation regarding the quality of reconstructed solutions. Highlights • A stationary and non-stationary iterated multiplicative regularization are introduced. • The formulation is highly flexible to describe prior knowledge of the sources. • The method is validated numerically and experimentally. • The proposed approach is more robust than standard approaches. [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]

We treat synchronization for iterated function systems generated by diffeomorphisms on compact manifolds. Synchronization here means the convergence of orbits starting at different initial conditions when iterated by the same sequence of diffeomorphisms. The iterated function systems admit a description as skew product systems of diffeomorphisms on compact manifolds driven by shift operators. Under open conditions including transitivity and negative fiber Lyapunov exponents, we prove the existence of a unique attracting invariant graph for the skew product system. This explains the occurrence of synchronization. The result extends previous results for iterated function systems by diffeomorphisms on the circle, to arbitrary compact manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we present a new algorithm for solving generalized variational inequality problems(GVIP for short) in finite-dimensional Euclidean space. In this method, our next iterate point is obtained by projecting the current iterate point onto a half-space. This half-space can separate strictly the current iterate point from the solution set of GVIP. Moreover, this method works without needing the current point belongs to the feasible set. Comparing with methods in Konnov [A combined relaxation method for variational inequalities with nonlinear constraints. Math Program. 1998;80(2):239-252] and Fang and Chen [Subgradient extragradient algorithm for solving multi-valued variational inequality. Appl Math Comput. 2014;229(3-4):123-130], our method can get rid of an auxiliary procedure in each iteration which is used to ensure the current iterate point belongs to feasible set. Consequently, our method is more simpler than those algorithms. The global convergence is proved under mild assumptions. Numerical results show that this method is much more efficient than the method in Li and He [An algorithm for generalized variational inequality with pseudomonotone mapping. J Comput Appl Math. 2009;228:212-218]. [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]

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]

This paper is devoted to study some chaotic properties of iterated function systems (IFSs). Specially, a new notion named thick chaotic IFSs is introduced. The relationship between thick chaos and another properties of some notions in dynamical systems are studied. [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]

We show that when k ≠ 2, 4, 8, the Euler class of any vector bundle over Σkℝℙm is zero if the rank of the bundle is not m + k, provided that m ≠ 3 when k = 6. If k = 2, 4, 8, we show that the Euler class of any vector bundle over Σkℝℙm is zero whenever the rank of the bundle is not kr + k, provided that m ≠ 6, 7 when k = 2, where r is the largest integer such that kr ≤ m. [ABSTRACT FROM AUTHOR]

We show the existence and uniqueness for self-similar measures for iterated function systems driven by weak contractions. Our main idea is using the duality theorem of Kantorovich-Rubinstein and equivalent conditions for weak contractions established by Jachymski. We also show collage theorems for such iterated function systems. [ABSTRACT FROM AUTHOR]

In this paper we consider shadowing and weak shadowing properties for iterated function systems IFS and give some results on these concepts. At first, a sufficient condition for shadowing property is given and by this result we present two IFS which have the shadowing property. It is proved that every uniformly expanding as well as every uniformly contracting IFS has the weak shadowing property. By an example we show that in IFS's shadowing property does not imply weak shadowing property. Finally we have the main result of the paper and prove that the weak shadowing is a generic property in the set of all IFS's. [ABSTRACT FROM AUTHOR]

Limit theorems for sub-linear expectations are challenging field which have raised a large number of issues of interest recently. The aim of this paper is to establish general strong law of large numbers and the Chover's law of the iterated logarithm for a sequence of random variables under a sub-linear expectation space. As applications, several results on strong laws of large numbers which contain Marcinkiewicz strong law of large numbers and the Chover's law of the iterated logarithm for extended independence and identically distributed random variables have been generalized to the sub-linear expectation space context. Our results of strong laws of large numbers are more general than some related results previously reported obtained by Zhang (2016) [23] , Cheng (2016) [5] , Liu et al. (2015) [10] and Chen (2016) [3] . There is no report on the Chover's law of the iterated logarithm under sub-linear expectation, and we provide a method to study this subject. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the iterated local transitivity (ILT) model for online social networks for signed networks. Signed networks focus on the type of relations (friendship or enmity) between the vertices (members of online social networks). The ILT model for signed networks provide an insight into how networks react to the addition of clone vertex. In this model, at each time step t and for already existing vertex x, a new vertex (clone) x′ is added which joins to x and neighbors of x. The sign of new edge yx′,y∈N[x] neighborhood of x is defined by calculating the number of positive and negative neighbors of x. We also discuss properties such as balance and clusterability, sign-compatibility and C-sign-compatibility. [ABSTRACT FROM AUTHOR]

In this article we introduce a one-parameter family of skew product (Gt)t ∈ [−ε, ε]maps exhibiting a heterodimensional cycle such that the number of isolated periodic orbits inside it has not super-exponential growth. The dynamics in the central direction of the mapsGtis described by a one-parameter family of system of iterated functions. [ABSTRACT FROM PUBLISHER]

The trend of globalization has recently seen the study of distributed scheduling problems. This study attempts to solve the distributed hybrid flowshop scheduling problem with multiprocessor tasks, and is the first attempt to address this problem. To solve this strongly NP-hard problem, a mixed integer linear programming formulation and self-tuning iterated greedy (SIG) algorithm that incorporates an adaptive cocktail decoding mechanism are presented to minimize the makespan. Comprehensive computational results demonstrate that the proposed SIG algorithm is extremely efficient and effective. This paper successfully expands the research area of distributed scheduling problems. [ABSTRACT FROM AUTHOR]

We consider a class of iterated function systems (IFSs) of contracting similarities of Rn, introduced by Hutchinson, for which the invariant set possesses a natural Hölder continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-Hölder) parameterization. We also give a related necessary condition for the invariant set of such an IFS to be homeomorphic to an interval. [ABSTRACT FROM AUTHOR]

In this paper we study the dimension theory of planar self-affine sets satisfying dominated splitting in the linear parts and the strong separation condition. The main result of this paper is the existence of dimension maximizing Gibbs measures (Käenmäki measures). To prove this phenomena, we show that the Ledrappier-Young formula holds for Gibbs measures and we introduce a transversality type condition for the strong-stable directions on the projective space. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the problem of prevalence in the class of quadratic stochastic operators acting on the $$L^{1}$$ space for the uniform topology. We obtain that the set of norm quasi-mixing quadratic stochastic operators is a dense and open set in the topology induced by a very natural metric. This shows the typical long-term behaviour of iterates of quadratic stochastic operators. [ABSTRACT FROM AUTHOR]