The simplest almost Riemannian problem with a Martinet point is studied. Extremal and optimal trajectories are studied by analytic, symbolic, and numeric techniques.
Analytically the following results were obtained: existence of optimal trajectories was proved, absence of abnormal trajectories was shawn, a Hamiltonian system for normal extremals was derived, its symmetry and Maxwell points was described. Symbolically all polynomial integrals of the Hamiltonian system of degree not greater than 54 were described. Numerically the optimal synthesis was constructed.

This study numerically investigates boundary layer separation criteria for flows over small surface irregularities on a flat plate at high Reynolds numbers using the double-deck model framework. By solving the Prandtl equations with self-induced pressure, critical amplitude values (i.e., the height of a hump or the depth of a pit) separating attached laminar flow from separated flow with a stationary vortex are determined for Gaussian-shaped irregularities. The results show that separation begins at points of zero curvature of the streamlined surface. Importantly, no geometric parameter (such as maximum curvature or tangent angle) remains invariant along the obtained critical amplitude values, refuting prior hypotheses of a universal critical curvature of the irregularity. Furthermore, the critical amplitude values differ for humps and pits of identical shape. Thus, a separation criterion based solely on the geometry of the irregularity is not attainable for arbitrary shapes.

Physics-informed neural networks (PINNs) have demonstrated great promise in solving partial differential equations without labeled data, yet their performance often deteriorates for highly nonlinear systems — particularly steady non-Newtonian flows governed by power-law or yield-stress rheologies. In this work, we present a systematic comparative study of PINN training strategies for Couette flow of pseudoplastic fluids between coaxial cylinders, governed by the Ostwald – de Waele and Herschel – Bulkley constitutive laws. We evaluate three established differential-form PINN variants — baseline (fixed loss weights), curriculum learning, and adaptive loss weighting — and introduce a novel variational PINN (vPINN) derived from the dissipation potential of Herschel – Bulkley fluids. Crucially, the proposed vPINN embeds the principle of minimum dissipation directly into the loss functional via the analytically integrated shear-dependent potential $\Phi(\dot{\gamma}) = q_0^{}\dot{\gamma}^2 + \frac{2q_1^{}\dot{\gamma}^{\,z+1}}{z+1}$, thereby enforcing physics through a variational principle rather than residual minimization. Using an exact analytical solution as ground truth, we benchmark all models on velocity and pressure reconstruction across varying gap geometries $\Bigl($dimensionless parameter $\gamma=\frac{R_1^{}}{R_2^{}-R_1^{}}\Bigr)$. While adaptive weighting improves pressure recovery by $6$–$8\,\%$ (MAE, $L_\infty^{}$, RSD), all differential PINNs exhibit nearly identical — and limited — accuracy for velocity prediction, with no benefit from curriculum scheduling. In contrast, the vPINN achieves a substantial and consistent gain in velocity accuracy: for $\gamma=4.0$, RSD drops from $1.09\,\%$ (all PINNs) to $0.85\,\%$ ($-22\,\%$); for the widest gap ($\gamma=0.67$), RSD falls from $6.0\,\%$ to $4.07\,\%$ ($-32\,\%$). MAE and $L_\infty^{}$ errors decrease by $27\,\%$ and $23\,\%$, respectively. These improvements arise because the variational formulation naturally mitigates spectral bias and avoids ill-conditioned gradients inherent to the highly nonlinear Navier – Stokes equation. The vPINN network also trains approximately twice as fast. Although vPINN currently predicts only velocity (pressure is recovered a posteriori), its consistent accuracy gains — especially where non-Newtonian effects dominate — establish variational PINNs as a compelling, physics-based alternative to residual-based approaches for complex rheological modeling.

This paper investigates the topological conjugacy of skew products defined on the total space of locally trivial fiber bundles. We prove that under certain restrictions on the chaotic dynamics of the homeomorphism on the base, namely, the inability to choose a closed connected subset of the supporting manifold which is also an invariant set of the base homeomorphism (indecomposable chaos), any homeomorphism conjugating the skew products must also be the skew product. Thus, the topological conjugacy of homeomorphisms on the base spaces is a necessary condition for the conjugacy of such skew products. As a consequence, we establish that the direct products of indecomposably chaotic homeomorphisms with homeomorphisms having finite $\omega$-limit sets are topologically conjugate if and only if they are conjugate componentwise.

The complementary subspaces (CS) method is aimed at numerically solving equations of the form $P_{\cal U}^{}(Av-f)=0$ emerging when evolutionary equations are integrated by spectral methods. Here $v$ must belong to the finite-dimensional space ${\cal V}$ comprised of functions satisfying the prescribed boundary conditions and $P_{\cal U}^{}$ is an orthogonal projection on a finite-dimensional space ${\cal U}$. The idea of the CS method is to find the solution to the original problem by solving a~modified problem, $P_{\cal G}^{}(Aw-\widetilde f)=0$, $w\in{\cal W}$, and either adding the correction $\widetilde f-f$ before solving the modified problem, or computing the correction $v-w$ afterwards. The spaces ${\cal W}$ and ${\cal G}$ are chosen in such a way that solving the modified problem requires less operations than the original one. The method was introduced in [23]; we propose now its modification allowing more freedom in choosing the spaces for the modified problem and in computing the correction. The algorithm is discussed in the general form and in the case of the spaces spanned by linear combinations of Chebyshev polynomials. The method is applied for numerical investigation of convective flows in a plane horizontal layer heated from below and rotating about an inclined axis with no-slip horizontal boundaries. For the employed values of control parameters the temporal behavior is comprised of repeating events and is possibly related to heteroclinic connections between unstable steady states.

In the mechanics of dry friction systems, there are well-known cases where the equilibria are not isolated and form sets that generally depend on the parameters of the problem. In this paper, it is shown by example that such nonisolated equilibria can be viewed as the limiting case of a finite set of equilibria of a frictionless system.