We obtain a complete set of explicit necessary and sufficient conditions for the isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented in terms of the Taylor coefficients of the Hamiltonian function and have the form of an infinite collection of polynomial equations.
We study self-similar solutions of the Riemann problem in the nonuniqueness region for weakly anisotropic elastic media with a negative nonlinearity parameter. We show that all discontinuities contained in the solutions in the nonuniqueness region have a stationary structure. We also show that in the nonuniqueness region one can construct two types of self-similar solutions.
We study the contact interaction of a periodic system of axisymmetric rigid indenters with two height levels with an elastic half-space in the absence of friction forces. To construct a solution of the problem, we use the localization method. We obtain analytical expressions for the characteristics of the contact interaction (the radius of contact spots and the distribution of contact pressure) as well as for the components of the internal stress tensor on the symmetry axes of indenters of both levels. We analyze the effect of the shape of the contact surface of indenters, which is described by a power function (with arbitrary integer exponent), and the spatial arrangement of indenters on the contact characteristics and the stressed state of the elastic half-space.
For an arbitrary partition (sigma) of the set (mathbb{P}) of all primes, a sufficient condition for the (sigma) -subnormality of a subgroup of a finite group is given. It is proved that the Kegel–Wielandt (sigma) -problem has a positive solution in the class of all finite groups all of whose nonabelian composition factors are alternating groups, sporadic groups, or Lie groups of rank 1.
We discuss various definitions of Sobolev and Besov classes on infinite-dimensional spaces, give a survey of the results on coincidence of some of these classes, and obtain a number of new results.
The paper is devoted to subgradient methods with switching between productive and nonproductive steps for problems of minimization of quasiconvex functions under functional inequality constraints. For the problem of minimizing a convex function with quasiconvex inequality constraints, a result is obtained on the convergence of the subgradient method with an adaptive stopping rule. Further, based on an analog of a sharp minimum for nonlinear problems with inequality constraints, results are obtained on the geometric convergence of restarted versions of subgradient methods. Such results are considered separately in the case of a convex objective function and quasiconvex inequality constraints, as well as in the case of a quasiconvex objective function and convex inequality constraints. The convexity may allow to additionally suggest adaptive stopping rules for auxiliary methods, which guarantee that an acceptable solution quality is achieved. The results of computational experiments are presented, showing the advantages of using such stopping rules.
We consider the energy operator of six-electron systems in the Hubbard model and study the structure of the essential spectrum and the discrete spectrum of the system for the second singlet state of the system. In the one- and two-dimensional cases, it is shown that the essential spectrum of the six-electron second singlet state operator is the union of seven closed intervals, and the discrete spectrum of the system consists of a single eigenvalue lying below (above) the domain of the lower (upper, respectively) edge of the essential spectrum of this operator. In the three-dimensional case, there are the following situations for the essential and discrete spectra of the six-electron second singlet state operator: (a) the essential spectrum is the union of seven closed intervals, and the discrete spectrum consists of a single eigenvalue; (b) the essential spectrum is the union of four closed intervals, and the discrete spectrum is empty; (c) the essential spectrum is the union of two closed intervals, and the discrete spectrum is empty; (d) the essential spectrum is a closed interval, and the discrete spectrum is empty. Conditions are found under which each of the situations takes place.
We prove analogs of Bernstein’s inequalities and inequalities of different metrics and different dimensions for entire functions of exponential type. Such inequalities are well known for Lebesgue spaces. In this paper we prove them for Morrey spaces.
We study the Riemann–Liouville space of fractional potentials on the half-line and establish its properties such as embeddings in Besov spaces, Liouville classes, and Lizorkin–Triebel spaces.
The triangle-free Krein graph Kre ((r)) is strongly regular with parameters (((r^{2}+3r)^{2},) (r^{3}+3r^{2}+r,0,r^{2}+r)) . The existence of such graphs is known only for (r=1) (the complement of the Clebsch graph) and (r=2) (the Higman–Sims graph). A.L. Gavrilyuk and A.A. Makhnev proved that the graph Kre ((3)) does not exist. Later Makhnev proved that the graph Kre ((4)) does not exist. The graph Kre ((r)) is the only strongly regular triangle-free graph in which the antineighborhood of a vertex Kre ((r)^{prime}) is strongly regular. The graph Kre ((r)^{prime}) has parameters (((r^{2}+2r-1)(r^{2}+3r+1),r^{3}+2r^{2},0,r^{2})) . This work clarifies Makhnev’s result on graphs in which the neighborhoods of vertices are strongly regular graphs without (3) -cocliques. As a consequence, it is proved that the graph Kre ((r)) exists if and only if the graph Kre ((r)^{prime}) exists and is the complement of the block graph of a quasi-symmetric (2) -design.
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