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An experimental study was conducted in the CICLoPE long-pipe facility to investigate the correlation between wall-pressure and turbulent velocity fluctuations in the logarithmic region, at high friction Reynolds numbers (
$4794 \lesssim Re_\tau \lesssim 47\,015$). Hereby, we explore the scalability of employing wall-pressure to effectively estimate off-the-wall velocity states (e.g. to be of use in real-time control of wall-turbulence). Coherence spectra for wall-pressure and streamwise (or wall-normal) velocity fluctuations collapse when plotted against 
$\lambda _x/y$ and thus reveals a Reynolds-number-independent scaling with distance-from-the-wall. When the squared wall-pressure fluctuations are considered instead of the linear wall-pressure term, the coherence spectra for the wall-pressure-squared and velocity are higher in amplitude at wavelengths corresponding to large-scale streamwise velocity fluctuations (e.g. at 
$\lambda _x/y = 60$, the coherence value increases from roughly 0.1 up to 0.3). This higher coherence typifies a modulation effect, because low-frequency content is introduced when squaring the wall-pressure time series. Finally, quadratic stochastic estimation is employed to estimate turbulent velocity fluctuations from the wall-pressure time series only. For each 
$Re_\tau$ investigated, the estimated time series and a true temporal measurement of velocity inside the turbulent pipe flow yield a normalised correlation coefficient of 
$\rho \approx 0.6$ for all cases. This suggests that wall-pressure sensing can be employed for meaningful estimation of off-the-wall velocity fluctuations and thus for real-time control of energetic turbulent velocity fluctuations at high-
$Re_\tau$ applications.
In the dynamical systems approach to turbulence, unstable periodic orbits (UPOs) provide valuable insights into system dynamics. Such UPOs are usually found by shooting-based Newton searches, where constructing sufficiently accurate initial guesses is difficult. A common technique for constructing initial guesses involves detecting recurrence events by comparing past and future flow states using their 
$L_2$-distance. An alternative method uses dynamic mode decomposition (DMD) to generate initial guesses based on dominant frequencies identified from a short time series, which are signatures of a nearby UPO. However, DMD struggles with continuous symmetries. To address this drawback, we combine symmetry-reduced DMD (SRDMD) introduced by Marensi et al. (2023, J. Fluid Mech., vol. 954, A10), with sparsity promotion. This combination provides optimal low-dimensional representations of the given time series as a time-periodic function, allowing any time instant along this function to serve as an initial guess for a Newton solver. We also discuss how multi-shooting methods operate on the reconstructed trajectories, and we extend the method to generate initial guesses for travelling waves. We demonstrate SRDMD as a method complementary to recurrent flow analysis by applying it to data obtained by direct numerical simulations of three-dimensional plane Poiseuille flow at the friction Reynolds number 
$\textit{Re}_\tau \approx51$ (
$\textit{Re}=802$), explicitly taking a continuous shift symmetry in the streamwise direction into account. The resulting unstable relative periodic orbits cover relevant regions of the state space, highlighting their potential for describing the flow.
By methods of harmonic analysis, we identify large classes of Banach spaces invariant of periodic Fourier multipliers with symbols satisfying the classical Marcinkiewicz type conditions. Such classes include general (vector-valued) Banach function spaces Φ and/or the scales of Besov and Triebel–Lizorkin spaces defined on the basis of Φ.
We apply these results to the study of the well-posedness and maximal regularity property of an abstract second-order integro-differential equation, which models various types of elliptic and parabolic problems arising in different areas of applied mathematics. In particular, under suitable conditions imposed on a convolutor c and the geometry of an underlying Banach space X, we characterize the conditions on the operators A, B, and P on X such that the following periodic problem
\begin{equation*}\partial P \partial u + B \partial u + {A} u + c \ast u = f \qquad \textrm{in } {\mathcal D}'({\mathbb{T}}; X)\end{equation*}
is well-posed with respect to large classes of function spaces. The obtained results extend the known theory on the maximal regularity of such problem.
The present paper develops a unified approach when dealing with short- or long-range dependent processes with finite or infinite variance. We are concerned with the convergence rate in the strong law of large numbers (SLLN). Our main result is a Marcinkiewicz–Zygmund law of large numbers for 
$S_{n}(f)= \sum_{i=1}^{n}f(X_{i})$, where 
$\{X_i\}_{i\geq 1}$ is a real stationary Gaussian sequence and 
$f(\!\cdot\!)$ is a measurable function. Key technical tools in the proofs are new maximal inequalities for partial sums, which may be useful in other problems. Our results are obtained by employing truncation alongside new maximal inequalities. The result can help to differentiate the effects of long memory and heavy tails on the convergence rate for limit theorems.
