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In this article we use techniques of proof mining to analyse a result, due to Yonghong Yao and Muhammad Aslam Noor, concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and…
In this paper we study quantitative recurrence and the shrinking target problem for dynamical systems coming from overlapping iterated function systems. Such iterated function systems have the important property that a point often has…
We study a conical extension of averaged nonexpansive operators and the role it plays in convergence analysis of fixed point algorithms. Various properties of conically averaged operators are systematically investigated, in particular, the…
A fundamental aspect of the quantum-to-classical limit is the transition from a non-commutative algebra of observables to commutative one. However, this transition is not possible if we only consider unitary evolutions. One way to describe…
Given $d \ge 1$, let $(A_i)_{i\ge 1}$ be a sequence of random $d\times d$ real matrices and $Q$ be a random vector in $\mathbb{R}^d$. We consider fixed points of multivariate smoothing transforms, i.e. random variables $X\in \mathbb{R}^d$…
We study the average shifted convolution sum $$ B(H,N):= \frac{1}{H} \sum_{h \sim H} \sum_{n \sim N} A_{\pi_1}(n)\, A_{\pi_2}(n+h), $$ where $A_{\pi_i}(n)$ denotes the Fourier coefficients of a Hecke--Maass cusp form $\pi_i$ for…
In two recent papers we introduced some new techniques for constructing an extension of a probability-preserving system $T:\mathbb{Z}^d\curvearrowright (X,\mu)$ that enjoys certain desirable properties in connexion with the asymptotic…
It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p$-space, $1\leq p<\infty$, or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge…
It is shown that the homogeneous ergodic bilinear averages with M\"{o}bius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},\mu)$, and $a,b \in \mathbb{Z}$, then for…
We show that the averaged equation for a one-frequency fast-oscillating Hamiltonian system is the result of symplectic reduction of a certain natural system on the corresponding $S^1$-bundle with respect to the circle action. Furthermore,…
In this paper, we consider the mean value of the product of two real valued multiplicative functions with shifted arguments. The functions $F$ and $G$ under consideration are close to two nicely behaved functions $A$ and $B$, such that the…
We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain…
Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. Familiar…
We study the structure of multiple correlation sequences defined by measure preserving actions of commuting transformations. When the iterates of the transformations are integer polynomials we prove that any such correlation sequence is the…
It is a folklore conjecture that the M\"obius function exhibits cancellation on shifted primes; that is, $\sum_{p\le X}\mu(p+h) \ = \ o(\pi(X))$ as $X\to\infty$ for any fixed shift $h>0$. This appears in print at least since Hildebrand in…
We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.
Exponential averages that appear in integral fluctuation theorems can be recast as a sum over moments of thermodynamic observables. We use two examples to show that such moment series can exhibit non-uniform convergence in certain singular…
For non uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds…
We identify a recently proposed shifting operation on classical phase space as a gauge transformation for statistical mechanical microstates. The infinitesimal generators of the continuous gauge group form a non-commutative Lie algebra,…
This paper deals with a modifed iterative projection method for approximating a solution of hierarchical fixed point problems for nearly nonexpansive mappings. Some strong convergence theorems for the proposed method are presented under…