Conventional Partial and Complete Solutions of the Fundamental Equations of Fluid Mechanics in the Problem of Periodic Internal Waves with Accompanying Ligaments Generation

The problem of generating beams of periodic internal waves in a viscous, exponentially stratified fluid by a band oscillating along an inclined plane is considered by the methods of the theory of singular perturbations in the linear and weakly nonlinear approximations. The complete solution to the linear problem, which satisfies the boundary conditions on the emitting surface, is constructed taking into account the previously proposed classification of flow structural components described by complete solutions of the linearized system of fundamental equations without involving additional force or mass sources. Analyses includes all components satisfying the dispersion relation that are periodic waves and thin accompanying ligaments, the transverse scale of which is determined by the kinematic viscosity and the buoyancy frequency. Ligaments are located both near the emitting surface and in the bulk of the liquid in the form of wave beam envelopes. Calculations show that in a nonlinear description of all components, both waves and ligaments interact directly with each other in all combinations: waves-waves, waves-ligaments, and ligaments-ligaments. Direct interactions of the components that generate new harmonics of internal waves occur despite the differences in their scales. Additionally, the problem of generating internal waves by a rapidly bi-harmonically oscillating vertical band is considered. If the difference in the frequencies of the spectral components of the band movement is less than the buoyancy frequency, the nonlinear interacting ligaments generate periodic waves as well. The estimates made show that the amplitudes of such waves are large enough to be observed under laboratory conditions.

Nonlinear Approximations to Critical and Relaxation Processes

We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration.

Nonlinear Approximations to Critical and Relaxation Processes

We discuss methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D and 3D channels are brought up. Power series for the permeability derived for small values of amplitude are employed to calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. The technique developed for critical phenomena is applied then for relaxation phenomena. The concept of time-translation invariance is discussed, its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the timetranslation symmetry complete (or partial) restoration. We estimate typical time extent, amplitude and direction for such restorative process. The new technique is based on explicit introduction of origin in time. After some transformations we come to the exponential and generalized, exponential-type solution with explicit finite time scale, which was only implicit in initial parametrization with polynomial approximation. The concept of crash as a relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is put forward. COVID-19 related mini-crash in the time series for Shanghai Composite is discussed as an illustration.

Nonlinear Approximations to Critical and Relaxation Processes

We discuss methods for calculation of critical indices and amplitudes from the perturbative expansions. They are demonstrated for the Stokes flow through 2D and 3D channels enclosed by two wavy walls. Efficient formulas for the permeability are derived in the form of series for small values of amplitude. Various power-laws are found in the regime of large amplitudes, based only on expansions at small amplitudes. Lubrication approximation is shown to break down, but accurate formulas for the effective permeability for arbitrary values of the wave amplitude are derived from the expansions. The technique developed for critical phenomena is applied then for relaxation phenomena. The concept of time-translation invariance is discussed, its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation symmetry complete (or partial) restoration. We estimate typical time extent, amplitude and direction for such restorative process. The new technique is based on explicit introduction of origin in time. After some transformations we come to the exponential and generalized, exponential-type solution with explicit finite time scale, which was only implicit in initial parametrization with polynomial approximation. The concept of crash as a phenomenon, consisting of time-translation invariance breaking and restoration, is put forward. %Concrete form of symmetry breaking/restoration is suggested, using polynomial regression transformed into exponential and Gompertz approximants. COVID-19 related mini-crash in the time series for Shanghai Composite is discussed as an illustration.

Nonlinear approximations of functions having mixed smoothness

For multivariate Besov-type classes $U^a_{p,\theta}$ of functions having nonuniform mixed smoothness $a\in\rr^d_+$, we obtain the asumptotic order of entropy numbers $\epsilon_n(U^a_{p,\theta},L_q)$ and non-linear widths $\rho_n(U^a_{p,\theta},L_q)$ defined via pseudo-dimension. We obtain also the asymptotic order of optimal methods of adaptive sampling recovery in $L_q$-norm of functions in $U^a_{p,\theta}$ by sets of a finite capacity which is measured by their cardinality or pseudo-dimension.

Augmented Single Loop Single Vector Algorithm Using Nonlinear Approximations of Constraints in Reliability-Based Design Optimization

Reliability-based design optimization (RBDO) aims at minimizing a function of probabilistic design variables, given a maximum allowed probability of failure. The most efficient methods available for solving moderately nonlinear problems are single loop single vector (SLSV) algorithms that use a first-order approximation of the probability of failure in order to rewrite the inherently nested structure of the loop into a more efficient single loop algorithm. The research presented in this paper takes off from the fundamental idea of this algorithm. An augmented SLSV algorithm is proposed that increases the rate of convergence by making nonlinear approximations of the constraints. The nonlinear approximations are constructed in the following way: first, the SLSV experiments are performed. The gradient of the performance function is known, as well as an estimate of the most probable failure point (MPP). Then, one extra experiment, a probe point, per performance function is conducted at the first estimate of the MPP. The gradient of each performance function is not updated but the probe point facilitates the use of a natural cubic spline as an approximation of an augmented MPP estimate. The SLSV algorithm using probing (SLSVP) also incorporates a simple and effective move limit (ML) strategy that also minimizes the heuristics needed for initiating the optimization algorithm. The size of the forward finite difference design of experiment (DOE) is scaled proportionally with the change of the ML and so is the relative position of the MPP estimate at the current iteration. Benchmark comparisons against results taken from the literature show that the SLSVP algorithm is more efficient than other established RBDO algorithms and converge in situations where the SLSV algorithm fails.

Nonlinear Approximations in Cryptanalysis Revisited

This work studies deterministic and non-deterministic nonlinear approximations for cryptanalysis of block ciphers and cryptographic permutations and embeds it into the well-understood framework of linear cryptanalysis. For a deterministic (i.e., with correlation ±1) nonlinear approximation we show that in many cases, such a nonlinear approximation implies the existence of a highly-biased linear approximation. For non-deterministic nonlinear approximations, by transforming the cipher under consideration by conjugating each keyed instance with a fixed permutation, we are able to transfer many methods from linear cryptanalysis to the nonlinear case. Using this framework we in particular show that there exist ciphers for which some transformed versions are significantly weaker with regard to linear cryptanalysis than their original counterparts.

Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically.