Doklady Mathematics最新文献

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.

New cases of integrable ninth-order dynamical systems that are homogeneous in terms of some of their variables are presented, in which a system on the tangent bundle of a four-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has dissipation of different signs. The external field is introduced using some unimodular transformation, and it generalizes previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

This paper is devoted to the localisation of random 3-CNF formulas that are polynomially solvable by the resolution algorithm. It is shown that random formulas with the number of clauses proportional to the square of the number of variables, are polynomially solvable with probability close to unity when the proportionality coefficient exceeds the found threshold.