Graph-based prior and forward models for inverse problems on manifolds with boundaries

This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Matérn-type Gaussian field priors that enable flexible modeling near the boundaries, representing boundary values by superposition of harmonic functions with appropriate Dirichlet boundary conditions. We also investigate the graph-based approximation of forward models from PDE parameters to observed quantities. In the construction of graph-based prior and forward models, we leverage the ghost point diffusion map algorithm to approximate second-order elliptic operators with classical boundary conditions. Numerical results validate our graph-based approach and demonstrate the need to design prior covariance models that account for boundary conditions.

Influence of Impulse Moments On Single-Frequency Torsional Oscillations of Nonlinearly Elastic Bodies

Analytical study of the impulse moment influences on the nonlinear torsional oscillations in the homogeneous constant cross-section of a body under classical boundary conditions of the first, second, and third types has been developed. For the case when the elastic material properties meet the body close to the power law of elasticity, mathematical models of the process are obtained. They are the boundary value problems for an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of pulse momentum on the oscillatory process. The peculiarities of resonant oscillations are established. Relative torsional oscillations of a nonlinear elastic body that rotates around the axis with a constant portable angular velocity are considered, taking into account the periodic action of pulse momentum acting in a fixed cross-section. The reliability of the obtained calculation formulas is confirmed.

Classical boundary field theory of Jacobi sigma models by Poissonization

In this paper, we are going to construct the classical field theory on the boundary of the embedding of \mathbb{R} \times S^{1}ℝ×S1 into the manifold MM by the Jacobi sigma model. By applying the poissonization procedure and by generalizing the known method for Poisson sigma models, we express the fields of the model as perturbative expansions in terms of the reduced phase space of the boundary. We calculate these fields up to the second order and illustrate the procedure for contact manifolds.

The generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow

This paper studies the generation of Tollmien–Schlichting waves by free-stream turbulence in transonic flow over a half-infinite flat plate with a roughness element using an asymptotic approach. It is assumed that the Reynolds number (denoted Re) is large, and that the free-stream turbulence is uniform so it can be modelled as vorticity waves. Close to the plate, a Blasius boundary layer forms at a thickness of $$O(\mathrm{{Re}}^{-{1}/{2}})$$ O ( Re - 1 / 2 ) , and a vorticity deformation layer is also present with thickness $$O(\mathrm{{Re}}^{-{1}/{4}})$$ O ( Re - 1 / 4 ) . The report shows that there is no mechanism by which the vorticity waves can penetrate from the vorticity deformation layer into the classical boundary layer; therefore, a transitional layer is introduced between them in order to prevent a discontinuity in vorticity. The flow in the interaction region in the vicinity of the roughness element is then analysed using the triple-deck model for transonic flow. A novel asymptotic expansion is used to analyse the upper deck, which enables a viscous–inviscid interaction problem to be derived. In order to make analytical progress, the height of the roughness element is assumed to be small, and from this, we find an explicit formula for the receptivity coefficient of the Tollmien–Schlichting wave far downstream of the roughness.

Existence of solution in the bending of thin plates with Gurtin–Murdoch surface elasticity

We consider the well-posedness of classical boundary value problems in a theory of bending of thin plates which incorporates the effects of surface elasticity via the Gurtin–Murdoch surface model. We employ the fundamental solution of the governing system of equations to develop integral-type solutions of the corresponding Dirichlet, Neumann, and Robin boundary value problems. Using the boundary integral equation method, we subsequently establish results for the existence of a solution in the appropriate function spaces.

ПРО КРАЙОВІ ЗАДАЧІ ДЛЯ РІВНЯННЯ ПУАССОНА У БАГАТОЛИСТІЙ ОБЛАСТІ, СКЛАДЕНІЙ ІЗ РІЗНИХ КРУГОВИХ СЕГМЕНТІВ

The non-classical boundary problem of the mathematical physics for the two-dimensional Poisson equation is considered. As the area, in which the solution is sought, the area, made up of different circular segments, folded into a multi-sheet plate of a book structure, is taken. All sheets are different from each other, both in their physical properties and in geometric dimensions, and are interconnected by a chord common to all sheets. The problem statement is given and its exact solution is obtained.The solution to the problem is considered in bipolar coordinate systems, each of which is associated with one of the segments. In this case, all coordinate systems have a common parameter - the length of the rectilinear segment boundary. As a method for solving the problem, the classical method of separation of variables is used – the Fourier method. Although the Dirichlet problem is considered as a basic one, however, the proposed method can be applied in the case when conditions of other types are given on the arcs of separate circles: Neumann or the third main problem.The statement of the considered problem differs from the classical one in that the conjugation conditions of fields on the line of connection of segments are added to the traditional boundary conditions. These conditions represent the equality of the values of the functions and the equality to zero of the sum of linear combinations of their normal derivatives. The solution is constructed (selected) in such a way that the first of the field conjugation conditions is fulfilled automatically for any choice of unknown functions. The boundary conditions on the segments and the second conjugation condition make it possible to determine all the unknown functions of the problem. To apply the Fourier method, it is necessary that all boundary functions are equal to zero at the corner points of the segments. If this condition is violated, a modification of the method that allows one to obtain an exact solution in this case is proposed. As an application, such problems are considered: a) on the torsion of a composite rod, the cross-section of which is two different segments; b) the stationary heat conductivity problem for two glued half-segment with sources of heat inside the area. Exact analytical solutions to these new problems have been obtained.

Space‐time structure as a property derived from the operation of consciousness at the quantum-classical boundary

The space‐time is empirically perceived as a pre-existing property of the universe. However, a special kind of perception that takes place in near-death-experiences (NDEs) is challenging this idea. Here, I will illustrate how understanding of this particular state of consciousness (named the bodiless consciousness) helps us re-think the space‐time structure of the physical world. I first speculate that the bodiless consciousness perceives the physical world as nonlocal 4D. I then propose that the space‐time is a “derived” feature subsequent to the emergence of perception of the bodiless consciousness, rather than a pre-existing and unchangeable property. Next, I explain that the space structure only takes place in the classical (or macroscopic) world rather than in the quantum (or microscopic) world, due to its intrinsic imperceptibility to the bodiless consciousness. Without a presupposed structure of the space, the strangeness of the quantum world is expected. Then, I bring up the old measurement problem. I will argue that it is the bodiless consciousness that may entangle with the superposed state of an observed system and trigger the collapse. Finally, I will briefly discuss the potential relationship between electromagnetic wave and consciousness.