In classical works, the Hubble constant is defined in terms of a metric. In this paper, we argue that this is invalid for many reasons and define the Hubble constant in terms of matter, following the work by Milne and McCrea. It turns out that the Hubble law is a general law of Hamiltonian mechanics, and Milne and McCrea's theory of an expanding universe can be extended not only to the relativistic case, but also to a special class of isotropic Hamiltonians. This allows us to explain the accelerated expansion as a simple relativistic effect without using Einstein's lambda, dark energy, or new particles, more specifically, as an exact consequence of the classical Einstein action for particles moving along geodesics.
We consider the Cauchy problem for a (2+1)-dimensional nonlinear equation of high order. We prove that there exist two critical exponents q_1kr: = 2 and q_2kr: = 4 such that a local (in time) weak solution in a certain class does not exist for 1 < q ⩽ q_1kr , a unique local (in time) weak solution exists in a certain class, but there is no global (in time) weak solution (so that we have a finite-time blow-up) for q_1kr < q ⩽ q_2kr , whereas a global weak solution exists for q > q_2kr .
This work is concerned with the computational analysis of two strategies for the numerical solution of the Navier–Stokes equations in moving domains. Computational meshes in these domains preserve topological structure and admit movement of their nodes. Both strategies exploit Oseen linearization of the Navier–Stokes equations resulting in a linear system to be solved at every time step. The first strategy is based on a collocated finite volume method exploiting the Ostrogradsky–Gauss theorem for the time-space divergence operator. The second strategy is based on a finite element scheme for the Navier–Stokes equations written in a reference domain. We compare the accuracy of the methods on the same sequence of meshes for a problem with a known analytical solution. Also, we provide a comparative analysis of two approximate solutions for hemodynamics in the right ventricle of a patient given by a time series of computer tomography scans.
A method of guaranteed accuracy control of solutions to elliptic boundary value problems in complex domains obtained by the fictitious domain method (FDM) is justified. To this end, the apparatus of functional-type a posteriori estimates is used. First, a posteriori identities and estimates are established that characterize the error arising from a change in the mathematical model (modeling error). These identities are then used to derive and analyze guaranteed and completely computable estimates for the errors of conforming and non-conforming approximations that arise when using the FDM. Numerical experiments demonstrate the effectiveness of the proposed approach.
This study investigates the analytic and geometric properties of generic q -hypergeometric polynomials ϕ_h(z) and their monic counterparts Φ_h(z) . We establish that these polynomials satisfy an R_I -type recurrence relation and possess Sobolev-type orthogonality on the unit circle. Integral representations and differential relations are derived, while the geometric behavior of their zeros—including location and convergence under parameter variation—is characterized. The finite and infinite continued fractions 𝒢_h(z) = Φ_h - 2(z)/Φ_h - 1(z) and ℋ_h(z) = Φ_h(z)/Φ_h - 1(z) are shown to exhibit reciprocal duality. We extended our exploration to elliptic hypergeometric polynomials E_h(z) and their monic forms ℰ_h(z) , which similarly obey R_I -type recurrences. The theoretical framework is validated through the explicit example of basic hypergeometric para-orthogonal polynomials on the unit circle.
