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Trust region method is a robust method for optimization problems. In this paper, we propose a novel adaptive nonmonotone technique based on trust region methods for solving unconstrained optimization. In order to accelerate the convergence of trust region methods, an adaptive trust region is generated according to the Hessian of the iterate point. Both the nonmonotone techniques and this adaptive strategies can improve the trust region methods in the sense of convergence. We prove that the proposed method is locally superlinear convergence under some standard assumptions. Numerical results show that the new method is effective and has a high speed of convergence in practice. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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.

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]

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]

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]

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]

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]

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]

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]