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We consider the Sobolev norms of the pointwise product of two functions, and estimate from above and below the constants appearing in two related inequalities.
We consider some integral-geometric quantities that have recently arisen in harmonic analysis and elsewhere, derive some sharp geometric inequalities relating them, and place them in a wider context.
We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…
We establish weak-type $(1,1)$ bounds for the maximal function associated with ergodic averaging operators modeled on a wide class of thin deterministic sets $B$. As a corollary we obtain the corresponding pointwise convergence result on…
We develop a new tool, the time inhomogeneous Poisson equation in the whole space and with a terminal condition at infinity, to study the asymptotic behavior of the non-autonomous multi-scale stochastic system with irregular coefficients,…
In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some…
Balancing square and rectangular tables by rotation has been a interesting way to illustrate the intermediate value theorem. The aim of this note is to show that the balancing act but with non-rectangular tables can be a nice application of…
In this note we briefly survey and propose some open problems related to isoparametric theory.
In this paper, we investigate the ergodicity in total variation of the process $X_t$ related to some integro-differential operator with unbounded coefficients and describe the speed of convergence to the respective invariant measure. Some…
For a Dunford-Schwartz operator in the $L^p-$space, $1\leq p< \infty$ , of an arbitrary measure space, we prove pointwise convergence of the conventional and Besicovitch weighted ergodic averages. Pointwise convergence of various types of…
We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \}…
The paper introduces a new estimation method for the standard linear regression model. The procedure is not driven by the optimisation of any objective function rather, it is a simple weighted average of slopes from observation pairs. The…
The ergodic hypothesis outgrew from the ancient conception of motion as periodic or quasi periodic. It did cause a revision of our views of motion, particularly through Boltzmann and Poincar\'e: we discuss how Boltmann's conception of…
We derive consistency and asymptotic normality results for quasi-maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous-time setting. The special feature of our…
Single particle tracking has become a standard tool to investigate diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual…
Synchronization is an important dynamical phenomenon in coupled nonlinear systems, which has been studied extensively in recent years. However, analysis focused on individual orbits seems hard to extend to complex systems while a global…
We study uniquely ergodic dynamical systems over locally compact, sigma-compact Abelian groups. We characterize uniform convergence in Wiener/Wintner type ergodic theorems in terms of continuity of the limit. Our results generalize and…
Approximate inference algorithm is one of the fundamental research fields in machine learning. The two dominant theoretical inference frameworks in machine learning are variational inference (VI) and Markov chain Monte Carlo (MCMC).…
We establish variation and oscillation inequalities for convolution products of probability measures on Z.
The last decade has witnessed a number of important and exciting developments that had been achieved for improving recurrence plot based data analysis and to widen its application potential. We will give a brief overview about important and…