840 results on '"Iterated Integrals"'
Search Results
152. THE METHOD OF ITERATED SPLINES FOR INTEGRAL EQUATIONS WITH VANISHING DELAY.
- Author
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BICA, A. M., CURILǍ, M., and CURILǍ, S.
- Subjects
SPLINES ,NUMERICAL solutions to integral equations ,ITERATED integrals ,VOLTERRA equations ,NUMERICAL solutions to Fredholm equations ,NUMERICAL solutions to delay differential equations ,NUMERICAL solutions to initial value problems - Abstract
The method of iterated splines is presented for constructing the numerical solution of Volterra and Fredholm integral equations with vanishing delay. We study as particular cases two-point boundary value problems of even order and initial value problems with vanishing delay, including the well-known pantograph equation. The convergence of the method is proved by providing the error estimate. The numerical stability regarding the choice of the first iteration is investigated. The accuracy, the convergence and the numerical stability are illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]
- Published
- 2018
153. An Iterated Projection Approach to Variational Problems Under Generalized Convexity Constraints.
- Author
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Carlier, Guillaume and Dupuis, Xavier
- Subjects
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CONVEX domains , *GRAPHICAL projection , *ITERATED integrals , *CONSTRAINTS (Physics) , *NUMERICAL analysis - Abstract
The principal-agent problem in economics leads to variational problems subject to global constraints of b-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Figalli et al. (J Econ Theory 146(2):454-478, 2011) identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra's iterated projection algorithm to handle the b-convexity constraint in the framework of (Figalli et al. in J Econ Theory 146(2):454-478, 2011). Our method also turns out to be simple for convex envelope computations. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
154. Slow reflection.
- Author
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Freund, Anton
- Subjects
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HIERARCHY (Linguistics) , *ITERATED integrals , *ARITHMETIC , *TRANSFINITE numbers - Abstract
We describe a “slow” version of the hierarchy of uniform reflection principles over Peano Arithmetic ( PA ). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower complexity) and introduce a new provably total function. At the same time the consistency of PA plus slow reflection is provable in PA + Con ( PA ) . We deduce a conjecture of S.-D. Friedman, Rathjen and Weiermann: Transfinite iterations of slow consistency generate a hierarchy of precisely ε 0 stages between PA and PA + Con ( PA ) (where Con ( PA ) refers to the usual consistency statement). [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
155. An Erdös-Révész type law of the iterated logarithm for reflected fractional Brownian motion.
- Author
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Dębicki, K. and Kosiński, K.
- Subjects
ITERATED integrals ,GAUSSIAN function ,WIENER processes ,MATHEMATICAL functions ,DIFFERENTIAL equations - Abstract
Let $B_{H}=\{B_{H}(t):t\in \mathbb R\}$ be a fractional Brownian motion with Hurst parameter H ∈ (0,1). For the stationary storage process $Q_{B_{H}}(t)=\sup _{-\infty
f(t)\, \text { i.o.})}$ equals 0 or 1. Using this criterion we find that, for a family of functions f ( t), such that $z_{p}(t)=\mathbb P(\sup _{s\in [0,f_{p}(t)]}Q_{B_{H}}(s)>f_{p}(t))/f_{p}(t)=\mathcal C(t\log ^{1-p} t)^{-1}$ , for some $\mathcal C>0$ , ${\mathbb P(Q_{B_{H}}(t) > f_{p}(t)\, \text { i.o.})= 1_{\{p\ge 0\}}}$ . Consequently, with $\xi _{p} (t) = \sup \{s:0\le s\le t, Q_{B_{H}}(s)\ge f_{p}(s)\}$ , for p ≥ 0, $\lim _{t\to \infty }\xi _{p}(t)=\infty $ and $\limsup _{t\to \infty }(\xi _{p}(t)-t)=0$ a.s. Complementary, we prove an Erdös-Révész type law of the iterated logarithm lower bound on ξ ( t), i.e., $\liminf _{t\to \infty }(\xi _{p}(t)-t)/h_{p}(t) = -1$ a.s., p > 1; $\liminf _{t\to \infty }\log (\xi _{p}(t)/t)/(h_{p}(t)/t) = -1$ a.s., p ∈ (0,1], where h ( t) = (1/ z ( t)) p loglog t. [ABSTRACT FROM AUTHOR]- Published
- 2017
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- View/download PDF
156. Empirical Bayesian Estimation in the Model of Competing Risks.
- Author
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Abdushukurov, A. and Muminov, A.
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BAYESIAN analysis , *COMPETING risks , *EXPONENTIAL functions , *ITERATED integrals , *DIRICHLET principle - Abstract
We study empirical semi-parametric Bayesian estimates of exponential functionals in the model of competing risks. For these estimates we establish the properties of the uniform strong consistency and iterated logarithm type laws. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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157. Calculation of adjoint-weighted kinetic parameters with the reactor Monte Carlo code RMC.
- Author
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Qiu, Yishu, Wang, Zijie, Li, Kaiwen, Yuan, Yuan, Wang, Kan, and Fratoni, Massimiliano
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NUCLEAR reactors , *MONTE Carlo method , *DELAYED neutrons , *ITERATED integrals , *ESTIMATION theory , *NUCLEAR fission - Abstract
In this work, the capability of computing adjoint-weighted kinetic parameters, including effective delayed neutron fraction and neutron generation time, was implemented in the Reactor Monte Carlo (RMC) code based on the iterated fission probability (IFP) method. Three algorithms, namely, the Non-Overlapping Blocks (NOB) algorithm, the Multiple Overlapping Blocks (MOB) algorithm and the superhistory algorithm, were implemented in RMC to investigate their accuracy, computational efficiency and estimation of variance. The algorithms and capability of computing kinetic parameters in RMC were verified and validated by comparison with MCNP6 as well as experimental results through a set of multi-group problems and continuous-energy problems. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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158. Embeddings of Lorentz-type spaces involving weighted integral means.
- Author
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Gogatishvili, Amiran, Křepela, Martin, Pick, Luboš, and Soudský, Filip
- Subjects
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EMBEDDINGS (Mathematics) , *LORENTZ spaces , *PROBLEM solving , *ITERATED integrals , *MATHEMATICAL inequalities , *DUALITY theory (Mathematics) - Abstract
We solve the problem of characterizing weights on ( 0 , ∞ ) for which the inequality involving two possibly different general inner weighted means ( ∫ 0 ∞ ( ∫ 0 t f ⁎ ( s ) m 2 u 2 ( s ) d s ) p 2 m 2 w 2 ( t ) d t ) 1 p 2 ≤ C ( ∫ 0 ∞ ( ∫ 0 t f ⁎ ( s ) m 1 u 1 ( s ) d s ) p 1 m 1 w 1 ( t ) d t ) 1 p 1 holds, where p 1 , p 2 , m 1 , m 2 ∈ ( 0 , ∞ ) and p 2 > m 2 . The proof is based on a new approach combining duality techniques with sharp weighted estimates for iterated integral and supremum operators. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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159. Pseudo-Fubini Real-Entire Functions on the Plane
- Author
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Universidad de Sevilla. Departamento de Análisis Matemático, Bernal González, Luis, Calderón Moreno, María del Carmen, Jung, Andreas, Universidad de Sevilla. Departamento de Análisis Matemático, Bernal González, Luis, Calderón Moreno, María del Carmen, and Jung, Andreas
- Abstract
In this note, it is proved the existence of a c -dimensional vector space of real-entire functions all of whose nonzero members are non-integrable in the sense of Lebesgue but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space can be chosen to be dense in the space of all real C∞ -functions on the plane endowed with the topology of uniform convergence on compacta for all derivatives of all orders. If the condition of being entire is dropped, then a closed infinite dimensional subspace satisfying the same properties can be obtained.
- Published
- 2022
160. Duality Formulas for Arakawa–Kaneko Zeta Values and Related Variants
- Author
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Ce Xu
- Subjects
010101 applied mathematics ,Pure mathematics ,Polylogarithm ,Logarithm ,Iterated integrals ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Duality (optimization) ,Function (mathematics) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we present some new identities for multiple polylogarithm functions by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish several duality formulas for Arakawa–Kaneko zeta values and Kaneko–Tsumura $$\eta $$ -values. At the end of the paper, we study a variant of Kaneko–Tsumura $$\eta $$ -function with r-complex variables and establish two formulas about the values of this variant; these two formulas were proved previously by Yamamoto.
- Published
- 2021
161. The Alternating Block Decomposition of Iterated Integrals and Cyclic Insertion on Multiple Zeta Values
- Author
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Steven Charlton
- Subjects
Combinatorics ,Identity (mathematics) ,Conjecture ,Mathematics - Number Theory ,Iterated integrals ,General Mathematics ,FOS: Mathematics ,Pi ,Block (permutation group theory) ,Structure (category theory) ,Number Theory (math.NT) ,11M32 ,Mathematics - Abstract
The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and Lison\v{e}k states that by inserting all cyclic permutations of some initial blocks of 2's into the multiple zeta value $ \zeta(1,3,\ldots,1,3) $ and summing, one obtains an explicit rational multiple of a power of $ \pi $. Hoffman gives a conjectural identity of a similar flavour concerning $ 2 \zeta(3,3,\{2\}^m) - \zeta(3,\{2\}^m,(1,2)) $. In this paper we introduce the 'generalised cyclic insertion conjecture', which we describe using a new combinatorial structure on iterated integrals -- the so-called alternating block decomposition. We see that both the original BBBL cyclic insertion conjecture, and Hoffman's conjectural identity, are special cases of this 'generalised' cyclic insertion conjecture. By using Brown's motivic MZV framework, we establish that some symmetrised version of the generalised cyclic insertion conjecture always holds, up to a rational; this provides some evidence for the generalised conjecture., Comment: 40 pages, 1 figure created with Inkscape. Added an observation due to Panzer about the structure of D_odd cycle I(\ell_1, \ldots, \ell_n), in the odd weight case. Added a reference to the recent paper from Hirose and Sato, which proves Hoffman's conjectural identity exactly
- Published
- 2021
162. Uniformly hyperbolic viable sets in affine IFS.
- Author
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Glavan, Vasile and Guţu, Valeriu
- Subjects
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SHADOWING theorem (Mathematics) , *ITERATED integrals , *FRACTALS , *CONTROL theory (Engineering) , *SET-valued maps , *MATHEMATICAL symmetry - Published
- 2016
163. On the set of orthogonally additive functions with orthogonally additive second iterate.
- Author
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Baron, Karol
- Subjects
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ORTHOGONAL functions , *ORTHOGONAL series , *ADDITIVES , *ITERATED integrals , *TOPOLOGY - Abstract
Let E be a real inner product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all orthogonally additive functions from E into E with the Tychonoff topology. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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164. Minkowski’s question mark measure.
- Author
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Mantica, Giorgio
- Subjects
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MINKOWSKI space , *ORTHOGONAL polynomials , *LOGARITHMIC functions , *CHRISTOFFEL-Darboux formula , *ITERATED integrals - Abstract
Minkowski’s question mark function is the distribution function of a singular continuous measure: we study this measure from the point of view of logarithmic potential theory and orthogonal polynomials. We conjecture that it is regular, in the sense of Ullman–Saff–Stahl–Totik and moreover that it belongs to a Nevai class; we provide numerical evidence of the validity of these conjectures. In addition, we study the zeros of its orthogonal polynomials and the associated Christoffel functions, for which asymptotic formulae are derived. As a by-product, we compute upper and lower bounds to the Hausdorff dimension of Minkowski’s measure. Rigorous results and numerical techniques are based upon Iterated Function Systems composed of Möbius maps. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
165. One-Step Generalized Estimating Equations With Large Cluster Sizes.
- Author
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Lipsitz, Stuart, Fitzmaurice, Garrett, Sinha, Debajyoti, Hevelone, Nathanael, Hu, Jim, and Nguyen, Louis L.
- Subjects
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GENERALIZED estimating equations , *EXCHANGEABLE bonds , *ITERATED integrals , *CLUSTER analysis (Statistics) , *MATRIX multiplications - Abstract
Medical studies increasingly involve a large sample of independent clusters, where the cluster sizes are also large. Our motivating example from the 2010 Nationwide Inpatient Sample (NIS) has 8,001,068 patients and 1049 clusters, with average cluster size of 7627. Consistent parameter estimates can be obtained naively assuming independence, which are inefficient when the intra-cluster correlation (ICC) is high. Efficient generalized estimating equations (GEE) incorporate the ICC and sum all pairs of observations within a cluster when estimating the ICC. For the 2010 NIS, there are 92.6 billion pairs of observations, making summation of pairs computationally prohibitive. We propose a one-step GEE estimator that (1) matches the asymptotic efficiency of the fully iterated GEE; (2) uses a simpler formula to estimate the ICC that avoids summing over all pairs; and (3) completely avoids matrix multiplications and inversions. These three features make the proposed estimator much less computationally intensive, especially with large cluster sizes. A unique contribution of this article is that it expresses the GEE estimating equations incorporating the ICC as a simple sum of vectors and scalars. [ABSTRACT FROM PUBLISHER]
- Published
- 2017
- Full Text
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166. Iterated limits for aggregation of randomized INAR(1) processes with Poisson innovations.
- Author
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Barczy, Mátyás, Nedényi, Fanni, and Pap, Gyula
- Subjects
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ITERATED integrals , *AGGREGATION (Statistics) , *POISSON distribution , *AUTOREGRESSIVE models , *BROWNIAN motion - Abstract
We discuss joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α ∈ ( 0 , 1 ) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ ( x ) ( 1 − x ) β , x ∈ ( 0 , 1 ) , with lim x ↑ 1 ψ ( x ) = ψ 1 ∈ ( 0 , ∞ ) , different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ ( − 1 , 0 ) , β = 0 , β ∈ ( 0 , 1 ) or β ∈ ( 1 , ∞ ) , when taking first the limit as N → ∞ and then the time scale n → ∞ , or vice versa. In fact, we give a partial solution to an open problem of Pilipauskaitė and Surgailis [13] by replacing the random-coefficient AR(1) process with a certain randomized INAR(1) process. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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167. CAUCHY TRANSFORMS OF SELF-SIMILAR MEASURES: STARLIKENESS AND UNIVALENCE.
- Author
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XIN-HAN DONG, KA-SING LAU, and HAI-HUA WU
- Subjects
- *
CAUCHY transform , *SELF-similar processes , *UNIVALENT functions , *ITERATED integrals , *BOUNDARY value problems - Abstract
For the contractive iterated function system Skz = e2πik/m + ρ(z - e2πik/m) with 0 < ρ < 1, k = 0, · · ·, m - 1, we let K ⊂ C be the attractor, and let μ be a self-similar measure defined by μ = 1/m Σk=0m-1 μoSk-1. We consider the Cauchy transform F of μ. It is known that the image of F at a small neighborhood of the boundary of K has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of F away from K; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
168. Evaluating multiple polylogarithm values at sixth roots of unity up to weight six.
- Author
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Henn, J.M., Smirnov, A.V., and Smirnov, V.A.
- Subjects
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VECTOR spaces , *FEYNMAN integrals , *ITERATED integrals , *MATHEMATICAL regularization , *LINEAR equations - Abstract
We evaluate multiple polylogarithm values at sixth roots of unity up to weight six, i.e. of the form G ( a 1 , … , a w ; 1 ) where the indices a i are equal to zero or a sixth root of unity, with a 1 ≠ 1 . For w ≤ 6 , we construct bases of the linear spaces generated by the real and imaginary parts of G ( a 1 , … , a w ; 1 ) and obtain a table for expressing them as linear combinations of the elements of the bases. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
169. Duality results for iterated function systems with a general family of branches.
- Author
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Mengue, Jairo K. and Oliveira, Elismar R.
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DUALITY theory (Mathematics) , *ITERATED integrals , *MATHEMATICAL functions , *METRIC spaces , *COST functions , *KANTOROVICH method - Abstract
Given , , and compact metric spaces, we consider two iterated function systems and , where and are contractions. Let be the set of probabilities with -marginal being holonomic with respect to and -marginal being holonomic with respect to . Given and , let be the set of probabilities in having -marginal and -marginal . Let be the relative entropy of with respect to and be the relative entropy of with respect to . Given a cost function , let . We will prove the duality equation: In particular, if and are single points and we drop the entropy, the equation above can be rewritten as the Kantorovich duality for the compact spaces and a continuous cost function . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
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170. On martingale tail sums for the path length in random trees.
- Author
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Sulzbach, Henning
- Subjects
COMPUTER algorithms ,CENTRAL limit theorem ,ITERATED integrals ,LOGARITHMIC functions ,RANDOM variables - Abstract
For a martingale ( X
n ) converging almost surely to a random variable X, the sequence ( Xn - 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]- Published
- 2017
- Full Text
- View/download PDF
171. Embedding properties of hereditarily just infinite profinite wreath products.
- Author
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Klopsch, Benjamin and Vannacci, Matteo
- Subjects
- *
PROFINITE groups , *EMBEDDINGS (Mathematics) , *PERMUTATION groups , *FINITE simple groups , *ITERATED integrals - Abstract
We study infinitely iterated wreath products of finite permutation groups w.r.t. product actions. In particular, we prove that, for every non-empty class of finite simple groups X , there exists a finitely generated hereditarily just infinite profinite group W with composition factors in X such that any countably based profinite group with composition factors in X can be embedded into W . Additionally we investigate when infinitely iterated wreath products of finite simple groups w.r.t. product actions are co-Hopfian or non-co-Hopfian. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
172. ON PERMUTATIONAL INVARIANCE OF THE METRIC DISCREPANCY RESULTS.
- Author
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KATUSI FUKUYAMA and YUTARO NODA
- Subjects
- *
LOGARITHMS , *ITERATED integrals , *PERMUTATION groups , *MATHEMATICAL symmetry , *IRREGULARITIES of distribution (Number theory) - Abstract
Let {Nk} be a sequence of non-zero real numbers. We prove that the law of the iterated logarithm for discrepancies of the sequence {nkx} is permutational invariant if ∣nk+1/nk∣→∞ is satisfied. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
173. Convergence rates in precise asymptotics for a kind of complete moment convergence.
- Author
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Kong, Lingtao and Dai, Hongshuai
- Subjects
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STOCHASTIC convergence , *DIFFERENTIAL equations , *ASYMPTOTIC theory of algebraic ideals , *MOMENTS method (Statistics) , *ITERATED integrals , *RANDOM fields - Abstract
Liu and Lin ( Statist. Probab. Lett. 2006) introduced a kind of complete moment convergence which includes complete convergence as a special case. In this paper, we study the convergence rates of the precise asymptotics for complete moment convergence introduced by Liu and Lin (2006) and get the corresponding convergence rates. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
174. On integrality of p-adic iterated integrals.
- Author
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Chatzistamatiou, Andre
- Subjects
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P-adic analysis , *ITERATED integrals , *DIVISOR theory , *ZETA functions , *WITT group - Abstract
The purpose of this paper is to prove integrality for certain p -adic iterated Coleman integrals. As underlying geometry we will take the complement of a divisor D ⊂ X with good reduction, where X is the projective line or an elliptic curve over the Witt vectors of a perfect characteristic p field. As a corollary we prove a lower bound for the valuations of p -adic multiple zeta values. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
175. On Weakly Hyperbolic Iterated Function Systems.
- Author
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Arbieto, Alexander, Junqueira, André, and Santiago, Bruno
- Subjects
- *
HYPERBOLIC functions , *ITERATED integrals , *FUNCTION spaces , *EXISTENCE theorems , *PARAMETER estimation - Abstract
We study weakly hyperbolic iterated function systems on compact metric spaces, as defined by Edalat (Inform Comput 124(2):182-197, 1996), but in the more general setting of compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint and the ergodicity of invariant measure. We also define weakly hyperbolic iterated function systems for complete metric spaces and compact parameter space, extending the above mentioned definition. Furthermore, we study the question of existence of attractors in this setting. Finally, we prove a version of the results by Barnsley and Vince (Ergodic Theory Dyn Syst 31(4):1073-1079, 2011), about drawing the attractor (the so-called the chaos game), for compact parameter space. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
176. Kernel Estimation of a Characteristic Function.
- Author
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Abdushukurov, A. and Norboev, F.
- Subjects
- *
KERNEL (Mathematics) , *RANDOM variables , *DISTRIBUTION (Probability theory) , *ITERATED integrals , *LOGARITHMS - Abstract
This article deals with the study of asymptotic properties of a nonparametric kernel estimator of a characteristic function. Uniformly strong consistency of kernel estimator with fixed and expanding interval is established. The law of iterated logarithm type is also proved. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
177. Elliptic double zeta values.
- Author
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Matthes, Nils
- Subjects
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ELLIPTIC curves , *ITERATIVE methods (Mathematics) , *EISENSTEIN series , *MATHEMATICAL combinations , *THETA functions - Abstract
We study an elliptic analogue of multiple zeta values, the elliptic multiple zeta values of Enriquez, which are the coefficients of the elliptic KZB associator . Originally defined by iterated integrals on a once-punctured complex elliptic curve, it turns out that they can also be expressed as certain linear combinations of indefinite iterated integrals of Eisenstein series and multiple zeta values. In this paper, we prove that the Q -span of these elliptic multiple zeta values forms a Q -algebra, which is naturally filtered by the length and is conjecturally graded by the weight. Our main result is a proof of a formula for the number of Q -linearly independent elliptic multiple zeta values of lengths one and two for arbitrary weight. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
178. An infinite family of cubics with emergent reducibility at depth 1.
- Author
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Preszler, Jason I.
- Subjects
MATHEMATICS research ,ITERATED integrals ,EXPONENTIATION ,ALGEBRA software ,COEFFICIENTS (Statistics) - Abstract
A polynomialf(x) has emergent reducibility at depthniff◦k(x) is irreducible for 0≤ k ≤ n −1 butf◦n(x) is reducible. In this paper we prove that there are infinitely many irreducible cubicsf ∈[x] withf ◦freducible by exhibiting a one parameter family with this property. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
179. Multiple Dedekind zeta functions.
- Author
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Horozov, Ivan Emilov
- Subjects
- *
DEDEKIND sums , *ITERATED integrals , *ZETA functions , *EULER theorem , *EISENSTEIN series - Abstract
In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series [6]. We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein--Kronecker series [9]. Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1, 1; 1, 1). Based on such computations, we state two conjectures about MDZV. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
180. A law of the iterated logarithm for Grenander’s estimator.
- Author
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Dümbgen, Lutz, Wellner, Jon A., and Wolff, Malcolm
- Subjects
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ESTIMATION theory , *LOGARITHMS , *ITERATED integrals , *EMPIRICAL research , *BROWNIAN motion - Abstract
In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f ( t 0 ) > 0 , f ′ ( t 0 ) < 0 , and f ′ is continuous in a neighborhood of t 0 , then b l a lim sup n → ∞ ( n 2 log log n ) 1 / 3 ( f ̂ n ( t 0 ) − f ( t 0 ) ) = | f ( t 0 ) f ′ ( t 0 ) / 2 | 1 / 3 2 M almost surely where M ≡ sup g ∈ G T g = ( 3 / 4 ) 1 / 3 and T g ≡ argmax u { g ( u ) − u 2 } ; here G is the two-sided Strassen limit set on R . The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom’s switching relation, and properties of Strassen’s limit set analogous to distributional properties of Brownian motion; see Strassen [26] . [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
181. COMPUTING INTEGRALS OVER POLYNOMIALLY DEFINED REGIONS AND THEIR BOUNDARIES IN 2 AND 3 DIMENSIONS
- Author
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Michael Wester, Yuzita Yaacob, and Stanley Steinberg
- Subjects
d: area integral ,line integral ,volume integral ,iterated integrals ,cylindrical algebraic decomposition (cad) ,Information technology ,T58.5-58.64 - Abstract
This study uses the cylindrical algebraic decomposition algorithms implemented in Mathematica to produce procedures to analytically compute integrals over polynomially defined regions and their boundaries in two and three dimensions. Using these results, we can implement the divergence theorem in three dimensions or the Green’s theorems in two dimensions. These theorems are of central importance in the applications of multidimensional integration. They also provide a strong correctness test for the implementation of our results in a computer algebra system. The resulting software can solve many of the two and some of the three dimension a lintegration problems in vector calculu stextbooks. The three dimensional results are being extended. The results in this paper are being included inanautomated student assistant for vector calculus.
- Published
- 2012
- Full Text
- View/download PDF
182. On the smallest disks enclosing graph-directed fractals.
- Author
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Deniz, Ali and Çakmak, Gökçe
- Subjects
- *
FRACTALS , *ITERATED integrals , *ATTRACTORS (Mathematics) , *PROXIMITY spaces , *SIMPLEX algorithm , *MATHEMATICAL models - Abstract
Highlights • The existence of the global minimum is shown for radii of disks that encloses an attractor of a graph-directed iterated function system. • An upper bound of the diameter of the smallest disk that bounds the attractor is given. • An algorithm is given to obtain the diameters of the smallest disks enclosing the attractors with any proximity. Abstract We consider the graph-directed iterated function systems and give upper bounds for the diameters of the smallest disks enclosing their attractors. We also give an algorithm to obtain these smallest enclosing disks with any proximity. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
183. Confluence relations for the multiple zeta values (Algebraic Number Theory and Related Topics 2017)
- Author
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Sato, Nobuo
- Subjects
multiple logarithms ,iterated integrals ,extended double shuffle relation ,11M32 ,multiple zeta values ,double shuffle relation ,regularized double shuffle relation ,duality relation ,hyperlogarithms ,33E20 - Abstract
This article is a research announcement of my upcoming joint paper with Minoru Hirose on a certain class of Q-linear relations among the multiple zeta values (MZVs), which we call confluence relations. As is well known, MZVs are iterated integrals of meromorphic one-forms dt/t and dt/t-1 on a projective line. Here we consider more general iterated integrals of three different one-forms dt/t, dt/t-1 and dt/t-z and define the confluence relations as limits as z → 1 of Q-linear relations among these iterated integrals. At first, we define standard relations among the iterated integrals which naturally arise by regarding them as functions of z and thus using their differential structure with respect to z, and then we consider their limits as z → 1. The confluence relations seem to give a very rich family of Q-linear relations among MZVs and we even propose a conjecture that they exhaust all the Q-linear relations among MZVs. As a good reason for our conjecture, we prove that the confluence relations imply the extended double shuffle relations (also the duality realtion). A small table up to weight 4 of the confluence relations is given at the end., Algebraic Number Theory and Related Topics 2017. December 4-8, 2017. edited by Hiroshi Tsunogai, Takao Yamazaki and Yasushi Mizusawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.
- Published
- 2020
184. Some results on multiple polylogarithm functions and alternating multiple zeta values
- Author
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Ce Xu
- Subjects
Pure mathematics ,Algebra and Number Theory ,Polylogarithm ,Iterated integrals ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
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.
- Published
- 2020
185. Construction of Homotopic Invariants of Maps from Spheres to Compact Closed Manifolds
- Author
-
I. S. Zubov
- Subjects
Statistics and Probability ,Hopf invariant ,Pure mathematics ,Iterated integrals ,Applied Mathematics ,General Mathematics ,Bibliography ,SPHERES ,Mathematics::Geometric Topology ,Mathematics - Abstract
We study the homotopic classifications of maps from circles and spheres to manifolds and compare the classical approach to define the Hopf invariant with the approach based on Chen’s iterated integrals. Bibliography: 5 titles.
- Published
- 2020
186. The polylog quotient and the Goncharov quotient in computational Chabauty–Kim theory II
- Author
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David Corwin and Ishai Dan-Cohen
- Subjects
Discrete mathematics ,Conjecture ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Galois group ,Boundary (topology) ,01 natural sciences ,Iterated integrals ,Line (geometry) ,0101 mathematics ,Quotient ,Mathematics - Abstract
This is the second installment in a multi-part series starting with Corwin–Dan-Cohen [arXiv:1812.05707v3]. Building on previous work by Dan-Cohen–Wewers, Dan-Cohen, and F. Brown, we push the computational boundary of our explicit motivic version of Kim’s method in the case of the thrice punctured line over an open subscheme of Spec Z \operatorname {Spec}\mathbb {Z} . To do so, we develop a refined version of the algorithm of Dan-Cohen–Wewers tailored specifically to this case. We also commit ourselves fully to working with the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth- 1 1 part of the mixed Tate Galois group studied extensively by Goncharov. An application was given in Corwin–Dan-Cohen [arXiv:1812.05707v3], where we verified Kim’s conjecture in an interesting new case.
- Published
- 2020
187. TOWARDS ALGEBRAIC ITERATED INTEGRALS FOR ELLIPTIC CURVES VIA THE UNIVERSAL VECTORIAL EXTENSION (Various aspects of multiple zeta values)
- Author
-
Fonseca, Tiago J. and Matthes, Nils
- Subjects
periods ,universal vectorial extension ,iterated integrals ,Elliptic curves ,11F67 ,11M32 - Abstract
For an elliptic curve E defined over a field k⊂C, we study iterated path integrals of logarithmic differential forms on Et, the universal vectorial extension of E. These are generalizations of the classical periods and quasi-periods of E, and are closely related to multiple elliptic polylogarithms and elliptic multiple zeta values. Moreover, if k is a finite extension of Q, then these iterated integrals along paths between k-rational points are periods in the sense of Kontsevich-Zagier.
- Published
- 2020
188. Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras
- Author
-
Li Guo, Xing Gao, and Yi Zhang
- Subjects
Pure mathematics ,Matching (statistics) ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,010102 general mathematics ,01 natural sciences ,Renormalization ,Iterated integrals ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Connection (algebraic framework) ,Algebraic number ,Associative property ,Mathematics - Abstract
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.
- Published
- 2020
189. The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
- Author
-
Samuel Abreu, Ruth Britto, and Claude Duhr
- Subjects
Statistics and Probability ,High Energy Physics - Theory ,General Physics and Astronomy ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,High Energy Physics - Phenomenology ,High Energy Physics - Phenomenology (hep-ph) ,High Energy Physics - Theory (hep-th) ,perturbative quantum field theory ,Modeling and Simulation ,iterated integrals ,Feynman integrals ,Mathematical Physics ,Particle Physics - Theory ,Particle Physics - Phenomenology - Abstract
Dimensionally-regulated Feynman integrals are a cornerstone of all perturbative computations in quantum field theory. They are known to exhibit a rich mathematical structure, which has led to the development of powerful new techniques for their computation. We review some of the most recent advances in our understanding of the analytic structure of multiloop Feynman integrals in dimensional regularisation. In particular, we give an overview of modern approaches to computing Feynman integrals using differential equations, and we discuss some of the properties of the functions that appear in the solutions. We then review how dimensional regularisation has a natural mathematical interpretation in terms of the theory of twisted cohomology groups, and how many of the well-known ideas about Feynman integrals arise naturally in this context. This is Chapter 3 of a series of review articles on scattering amplitudes, of which Chapter 0 [arXiv:2203.13011] presents an overview and Chapter 4 [arXiv:2203.13015] contains closely related topics., Comment: 62 pages, see also the overview article arXiv:2203.13011. v3: journal version
- Published
- 2022
- Full Text
- View/download PDF
190. Modular iterated integrals associated with cusp forms
- Author
-
Nikolaos Diamantis
- Subjects
Cusp (singularity) ,Pure mathematics ,Mathematics - Number Theory ,business.industry ,Applied Mathematics ,General Mathematics ,Modular form ,Extension (predicate logic) ,Construct (python library) ,Modular design ,Iterated integrals ,11F37 ,FOS: Mathematics ,Number Theory (math.NT) ,Invariant (mathematics) ,business ,Mathematics - Abstract
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.
- Published
- 2022
191. A bound on the norm of overconvergent p-adic multiple polylogarithms
- Author
-
David Jarossay
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,010102 general mathematics ,16. Peace & justice ,01 natural sciences ,Iterated integrals ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,Norm (social) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
We generalize the definition of overconvergent $p$-adic multiple polylogarithms and of $p$-adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized $p$-adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent $p$-adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on $p$-adic cyclotomic multiple zeta values., 25 pages. This is Part I-1 of "$p$-adic cyclotomic multiple zeta values and $p$-adic pro-unipotent harmonic actions"
- Published
- 2019
192. Real-Analytic Non-Integrable Functions on the Plane with Equal Iterated Integrals
- Author
-
Luis Bernal-González, María del Carmen Calderón-Moreno, Andreas Jung, Universidad de Sevilla. Departamento de Análisis matemático, and Universidad de Sevilla. FQM127: Análisis Funcional no Lineal
- Subjects
Mathematics (miscellaneous) ,Applied Mathematics ,Fubini’s theorem ,Real analytic functions ,Iterated integrals - Abstract
In this note, a vector space of real-analytic functions on the plane is explicitly constructed such that all its nonzero functions are non-integrable but yet their two iterated integrals exist as real numbers and coincide. Moreover, it is shown that this vector space is dense in the space of all real continuous functions on the plane endowed with the compact-open topology.
- Published
- 2021
193. Nilpotence of orbits under monodromy and the length of Melnikov functions
- Author
-
Pavao Mardesić, Jessie Pontigo-Herrera, Dmitry Novikov, L. Ortiz-Bobadilla, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Department of Mathematics [Zagreb], Faculty of Science [Zagreb], University of Zagreb-University of Zagreb, Department of Mathematics (Weizmann Institute of Science), Weizmann Institute of Science [Rehovot, Israël], Instituto de Matematicas [México], and Universidad Nacional Autónoma de México (UNAM)
- Subjects
Physics ,Pure mathematics ,Sequence ,Polynomial ,Conjecture ,Melnikov function ,Abelian integrals ,010102 general mathematics ,Statistical and Nonlinear Physics ,Iterated integrals ,Condensed Matter Physics ,01 natural sciences ,Nilpotence class ,Foliation ,Displacement function ,Limit cycles ,Monodromy ,Simple (abstract algebra) ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Product (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Orbit (control theory) ,ComputingMilieux_MISCELLANEOUS - Abstract
Let F ∈ ℂ [ x , y ] be a polynomial, γ ( z ) ∈ π 1 ( F − 1 ( z ) ) a non-trivial cycle in a generic fiber of F and let ω be a polynomial 1-form, thus defining a polynomial deformation d F + e ω = 0 of the integrable foliation given by F . We study different invariants: the orbit depth k , the nilpotence class n , the derivative length d associated with the couple ( F , γ ) . These invariants bind the length l of the first nonzero Melnikov function of the deformation d F + e ω along γ . We analyze the variation of the aforementioned invariants in a simple but informative example, in which the polynomial F is defined by a product of four lines. We study as well the relation of this behavior with the length of the corresponding Godbillon–Vey sequence. We formulate a conjecture motivated by the study of this example.
- Published
- 2021
194. Functional linear regression with truncated signatures.
- Author
-
Fermanian, Adeline
- Subjects
- *
ITERATED integrals , *INFINITE series (Mathematics) , *EMPIRICAL research , *FUNCTIONAL analysis - Abstract
We place ourselves in a functional regression setting and propose a novel methodology for regressing a real output on vector-valued functional covariates. This methodology is based on the notion of signature, which is a representation of a function as an infinite series of its iterated integrals. The signature depends crucially on a truncation parameter for which an estimator is provided, together with theoretical guarantees. An empirical study on both simulated and real-world datasets shows that the resulting methodology is competitive with traditional functional linear models, in particular when the functional covariates take their values in a high dimensional space. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
195. Anti-periodic solutions of Liénard equations with state dependent impulses.
- Author
-
Belley, J-M. and Bondo, É.
- Subjects
- *
VAN der Pol equation , *MATHEMATICAL bounds , *DERIVATIVES (Mathematics) , *STOCHASTIC convergence , *ITERATED integrals , *THEORY of distributions (Functional analysis) - Abstract
Subject to a priori bounds, Liénard equations with state dependent impulsive forcing are shown to admit a unique absolutely continuous anti-periodic solution with first derivative of bounded variation on finite intervals. The point-wise convergence of a sequence of iterates to the solution is obtained, along with a bound for the rate of convergence. The results are applied to Josephson's and van der Pol's equations. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
196. Algorithms generating images of attractors of generalized iterated function systems.
- Author
-
Jaros, Patrycja, Maślanka, Łukasz, and Strobin, Filip
- Subjects
- *
ATTRACTORS (Mathematics) , *ITERATED integrals , *FRACTAL analysis , *DETERMINISTIC algorithms , *AFFINE transformations - Abstract
The paper is devoted to searching algorithms which will allow to generate images of attractors of generalized iterated function systems (GIFS in short), which are certain generalization of classical iterated function systems, defined by Mihail and Miculescu in 2008, and then intensively investigated in the last years (the idea is that instead of selfmaps of a metric space X, we consider mappings form the Cartesian product X×...× X to X). Two presented algorithms are counterparts of classical deterministic algorithm and so-called chaos game. The third and fourth one is fitted to special kind of GIFSs - to affine GIFS, which are, in turn, also investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
197. Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros' problem.
- Author
-
Draga, Szymon and Morawiec, Janusz
- Subjects
- *
POLYNOMIALS , *EQUATIONS , *ITERATED integrals , *RECURSIVE sequences (Mathematics) , *INTEGERS - Abstract
We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive iterates of the unknown function at a given point. As an application we solve a generalized problem of Zoltán Boros posed during the 50th ISFE. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
198. ON CUT SETS OF ATTRACTORS OF ITERATED FUNCTION SYSTEMS.
- Author
-
LORIDANT, BENOÎT, JUN LUO, SELLAMI, TAREK, and THUSWALDNER, JÖRG
- Subjects
- *
ATTRACTORS (Mathematics) , *ITERATED integrals , *INJECTIVE functions , *ISOMORPHISM (Mathematics) , *INTEGERS - Abstract
In this paper, we study cut sets of attractors of iterated function systems (IFS) in ℝd. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a criterion for the existence of cut points of IFS attractors. If the IFS attractors are self-affine tiles, our results become algorithmically checkable and can be used to exhibit cut points with the help of Hata graphs. This enables us to construct cut points of some self-affine tiles studied in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
199. Separating invertible key derivations from non-invertible ones: sequential indifferentiability of 3-round Even-Mansour.
- Author
-
Guo, Chun and Lin, Dongdai
- Subjects
ITERATED integrals ,PERMUTATIONS ,BLOCK ciphers ,CIPHERS ,CRYPTOGRAPHY - Abstract
Iterated Even-Mansour (IEM) scheme consists of a small number r of fixed n-bit permutations separated by $$r+1$$ round-key additions. When the permutations are public, independent and random, and a common round key derived from the master key by an idealized non-invertible key derivation (KD) function is used, 5 rounds was proved sufficient to obtain (full) indifferentiability from ideal ciphers by Andreeva et al. (CRYPTO 2013). The KD can be a random oracle, or a Davies-Meyer construction from a random permutation. This work considers such IEM with non-invertible KD in the sequential indifferentiability model of Mandal et al. (TCC 2012). As results, this work shows that in both cases mentioned before, 3 rounds yields sequential indifferentiability from ideal ciphers. As Andreeva et al. has proved 3-round IEM with idealized invertible key derivations not sequentially indifferentiable (by exhibiting an attack), a definitive separation between IEM with invertible key derivations and IEM with non-invertible key derivations is established. This is the most important implication of the results in this work. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
200. Another look at the integral of exponential Brownian motion and the pricing of Asian options.
- Author
-
Lyasoff, Andrew
- Subjects
OPTIONS (Finance) ,INTEGRAL representations ,PROBABILITY theory ,ITERATED integrals ,HYPERGEOMETRIC functions - Abstract
It is shown that Marc Yor's formula (Adv. Appl. Probab. 24:509-531, 1992) for the density of the integral of exponential Brownian motion taken over a finite time interval is an extremal member of a family of previously unknown integral formulae for the same density. The derivation is independent from the one by Yor and obtained from a simple time-reversibility feature, in conjunction with a Fokker-Planck type argument. Similar arguments lead to an independent derivation of Dufresne's result (Scand. Actuar. J. 90:39-79, 1990) for the law of the integral taken over an infinite time interval. The numerical aspects of the new formulae are developed, with concrete applications to Asian options. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
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