Geometric characterization for the least Lagrangian action ofn-body problems

For n-body problems with quasihomogeneous potentials in ?^k (2[ n/2] ? k) we prove that the minimum of the Lagrangian action integral defined on the zero mean loop space is exactly the circles with center at the origin and the configuration of the n-bodies is always a regular n - 1 simplex with fixed side length.     

Geometric characterization for the least Lagrangian action ofn-body problems

For n-body problems with quasihomogeneous potentials in ℝ^k (2[ n/2] ⩽ k) we prove that the minimum of the Lagrangian action integral defined on the zero mean loop space is exactly the circles with center at the origin and the configuration of the n-bodies is always a regular n - 1 simplex with fixed side length.     

A potential well for an integral functional with singular Lagrangian

This paper concerns a functional of the form

$$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$

on the Sobolev space \(H_0^1(\Omega )\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N\) with \(N\ge 3\) and \(0\in \Omega \). The hypotheses on L ensure that \(u\equiv 0\) is a critical point of \(\Phi \), but allow the Lagrangian to be singular at \(x=0\). It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of \(\Phi \) at \(u\equiv 0\)), Legendre (ellipticity at \(u\equiv 0\)) and Weierstrass [strict convexity of \(L(x,s,\xi )\) with respect to \(\xi \)] from the calculus of variations are not sufficient ensure that \(u\equiv 0\) is a local minimum of \(\Phi \). Using recent criteria for the existence of a potential well of a \(C^1\)-functional on a real Hilbert space, conditions implying that \(u\equiv 0\) lies in a potential well of \(\Phi \) are established. They are shown to be sharp in some cases.

     

Iterative proximal regularization of the modified Lagrangian functional for solving the quasi-variational Signorini inequality

The iterative Uzawa method with a modified Lagrangian functional is used to numerically solve the semicoercive Signorini problem with friction (quasi-variational inequality).     

An inverse‐free ADI algorithm for computing Lagrangian invariant subspaces

The numerical computation of Lagrangian invariant subspaces of large‐scale Hamiltonian matrices is discussed in the context of the solution of Lyapunov equations. A new version of the low‐rank alternating direction implicit method is introduced, which, in order to avoid numerical difficulties with solutions that are of very large norm, uses an inverse‐free representation of the subspace and avoids inverses of ill‐conditioned matrices. It is shown that this prevents large growth of the elements of the solution that may destroy a low‐rank approximation of the solution. A partial error analysis is presented, and the behavior of the method is demonstrated via several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Droplet minimizers for the Cahn-Hilliard free energy functional

We prove theorems characterizing the minimizers for the Cahn-Hilliard free energy functional, which is used to describe the liquid vapor phase transition (or the 2 state magnetization transition). In particular, we exactly determine the critical density for droplet formation, and the geometry of the droplets.     

Duffing equation and action functional

In this paper, we investigate the periodic nonlinear second order ordinary differential equation with friction

     

Lectures on the Inverse Scattering Transform

In 1967 a new method of mathematical physics was discovered by Gardner, Greene, Kruskal, and Miura. Their significant results have received much attention, and numerous developments have occurred within the decade. In this paper a survey of some of these developments is made. By and large the emphasis is on relating the key ideas via typical examples.     

A retarded action functional for the Dirac polaron

The ground-state characteristics of the one-dimensional (Dirac) polaron are considered as an example of a system with a retarded interaction functional. Using a path-integral approach, we estimate both the ground-state energy and the effective mass of the polaron within and beyond the most general Gaussian approximation. The leading-order Gaussian contribution to the self-energy slightly improves the Feynman estimate and belongs to the lowest upper bounds available. The next-to-leading non-Gaussian corrections do not significantly perturb the results obtained. For comparison, a lower bound to the ground-state energy is calculated using the Lieb-Yamazaki method. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 231–238, February, 2000.     

Cyclic by prime fixed point free action

Let the finite group be acting on a (solvable) group and suppose that no non-trivial element of is fixed under the action of all the elements of . Assume furthermore that . A long standing conjecture is that then the Fitting height of is bounded by the length of the longest chain of subgroups of . Even though this conjecture is known to hold for large classes of groups , it is still unknown for some relatively uncomplicated groups. In the present paper we prove the conjecture for all finite groups that have a normal cyclic subgroup of square free order and prime index. Since many of these groups have natural modules where they act faithfully and coprimely but without regular orbits, the result is new for many of the groups we consider.

     

Lectures on Algebraic topology

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Lectures on Algebraic topology de Marvin Greenberg, W. A. Benjamin, N. Y. 1967. US$5.95

     

Lectures on Wonderful Varieties

These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.     

Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.     

Lectures on the shift operator. III

This is the continuation of the papers I and II published in Vols. 39, 47, of the same Zapiski. Contents: Lecture 8 (the Malmquist-Walsh lemma; the triangulation theorem for the truncated shift operator; the spectral mapping theorem and the Fredholm spectrum; the unicellularity of the Volterra integration operator); Lecture 9 (invariant subspaces of H^R for the half plane; continuous contraction semigroups and their cogenerators; Cayley transforms of H²; Lax's theorems; individual theorems for semigroups of shifts).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 104–127, 1976.