Doklady Mathematics最新文献

An addition to the article “A new spectral measure of complexity and its capabilities for detecting signals in noise” is presented.

We perform a comparative analysis of the accuracy of second-order TVD (Total Variation Diminishing), third-order RBM (Rusanov–Burstein–Mirin), and fifth-order in space and third-order in time A-WENO (Alternative Weighted Essentially Non-Oscillatory) difference schemes for solving a special Cauchy problem for shallow water equations with discontinuous initial data. The exact solution of this problem contains a centered rarefaction wave, but does not contain a shock wave. It is shown that in the centered rarefaction wave and its influence area, the solutions of these three schemes converge with different orders to different invariants of the exact solution. This leads to a decrease in the accuracy of these schemes when they used to calculate the vector of base variables of the considered Cauchy problem. The P-form of the first differential approximation of the difference schemes is used for theoretical justification of these numerical results.

An upper estimate for the maximum width of a forbidden foothold zone that a multi-legged walking robot can overcome in static stability mode is presented. By using mathematical models of six- and four-legged robots, it is shown that the estimate cannot be improved. For this purpose, foot placement sequences are formed for which the estimate is attained. The dependence of the maximum width of the zone on the body length is found for the six-legged robot model.

A parametrization of Brent equations is proposed which leads to a several times reduction of the number of unknowns and equations. The arising equations are solved numerically using a nonlinear least squares method. Matrix multiplication algorithms that are faster than previously known ones are obtained. In particular, ((4,4,4;48))- and ((2,4,5;32))-algorithms are found.

We obtain a Dini type blow-up condition for solutions of the differential inequality (sumlimits_{|alpha | = m} {{partial }^{alpha }}{{a}_{alpha }}(x,u) geqslant g({text{|}}u{text{|)}};{text{in}};{kern 1pt} {{mathbb{R}}^{n}},) where (m,n geqslant 1) are integers and ({{a}_{alpha }}) and g are some functions.

This article is devoted to the improvement of signal recognition methods based on information characteristics of the spectrum. A discrete function of the normalized ordered spectrum is established for a single window function included in the discrete Fourier transform. Lemmas on estimates of entropy, imbalance, and statistical complexity in processing a time series of independent Gaussian variables are proved. New concepts of one- and two-dimensional spectral complexities are proposed. The theoretical results were verified by numerical experiments, which confirmed the effectiveness of the new information characteristic for detecting a signal mixed with white noise at low signal-to-noise ratios.

We consider sets removable for bounded harmonic functions on a stratified set with flat interior strata. It is proved that relatively closed sets of finite Hausdorff ((n - 2))-measure are removable for bounded harmonic functions on an n-dimensional stratified set satisfying the strong sturdiness condition.

Linear motion of a point particle influenced by two forces varying according to power laws with arbitrary exponents is considered. Exponents are found for which the governing equation is nonlinear and the oscillation period is independent of the initial data (tautochronic motion). The equations are brought to Hamiltonian form, and the Hamiltonian normal form method is used to prove that there exist only two variants of tautochronic motion, namely, when the exponents are equal to 1 and –3 (variant 1) and when the exponents are equal to 0 and –1/2 (variant 2). For the other power laws, the motion of the point particle is not tautochronic. The Hamiltonian normal form of tautochronic motion is the Hamiltonian of a linear oscillator. The canonical transformation reducing the original Hamiltonian to normal form is expressed in terms of elementary functions. Hamiltonians of tautochronic motions can be used to test computer codes for calculating Hamiltonian normal forms.

We prove pseudocompactness of a Tychonoff space X and the space (mathcal{P}(X)) of Radon probability measures on it with the weak topology under the condition that the Stone–Čech compactification of the space (mathcal{P}(X)) is homeomorphic to the space (mathcal{P}(beta X)) of Radon probability measures on the Stone–Čech compactification of the space X.

Let a hyperelliptic curve (mathcal{C}) of genus g defined over an algebraically closed field K of characteristic 0 be given by the equation ({{y}^{2}} = f(x)), where (f(x) in K[x]) is a square-free polynomial of odd degree (2g + 1). The curve (mathcal{C}) contains a single “infinite” point (mathcal{O}), which is a Weierstrass point. There is a classical embedding of (mathcal{C}(K)) into the group (J(K)) of K-points of the Jacobian variety J of (mathcal{C}) that identifies the point (mathcal{O}) with the identity of the group (J(K)). For (2 leqslant g leqslant 5), we explicitly find representatives of birational equivalence classes of hyperelliptic curves (mathcal{C}) with a unique base point at infinity (mathcal{O}) such that the set (mathcal{C}(K) cap J(K)) contains at least six torsion points of order (2g + 1). It was previously known that for (g = 2) there are exactly five such equivalence classes, and, for (g geqslant 3), an upper bound depending only on the genus g was known. We improve the previously known upper bound by almost 36 times.