A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

FROM the beginning of civilization one need must make itself felt, namely, for some form of calendar. In turn, this gives rise to the primitive study of astronomy, involving some knowledge of the sun and the moon. When the planets are added in the next stage, the astronomer needs an ephemeris or almanac ; the modern Nautical Almanac, with its perfection within narrow limits, is the final outcome. There have been many stages on the road, and some apparent interruptions. But the urge has always been the same, essentially a practical one. The Analytical Foundations of Celestial Mechanics By Aurel Wintner. (Princeton Mathematical Series, No. 5.) Pp. xii + 448. (Princeton, N.J.: Princeton University Press; London: Oxford University Press, 1941.) 36s. net.

The Analytical Foundations of Celestial Mechanics

The paper is concerned with the local and global bifurcation structure of positive solutions ${u,v\in H^1_0(\Omega)}$ of the system

$$\left\{\begin{array}{lll}-\Delta u + u = \mu_1u^3+\beta v^2u &\quad\text{in $\Omega$}\\-\Delta v + v = \mu_2v^3+\beta u^2v &\quad\text{in $\Omega$}\nonumber\end{array}\right.$$

of nonlinear Schrodinger (or Gross-Pitaevskii) type equations in ${\Omega\subset\mathbb {R}^N}$, N ≤ 3. The system arises in nonlinear optics and in the Hartree–Fock theory for a double condensate. Local and global bifurcations in terms of the nonlinear coupling parameter β of the system are investigated by using spectral analysis and by establishing a new Liouville type theorem for nonlinear elliptic systems which provides a-priori bounds of solution branches. If the domain is radial, possibly unbounded, then we also control the nodal structure of a certain weighted difference of the components of the solutions along the bifurcating branches.

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A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system

In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Transformation of Groups

Linear Equations of Mathematical Physics. Edited by S. G. Mikhlin. English translation by Harry Hochstadt. Pp. x, iii, 318. $10. 1967. (Holt, Rinehart & Winston, Inc.)

The aim of this survey is to give a profound introduction to equivariant degree theory, avoiding as far as possible technical details and highly theoretical background. We describe the equivariant degree and its relation to the Brouwer degree for several classes of symmetry groups, including also the equivariant gradient degree, and particularly emphasizing the algebraic, analytical, and topological tools for its effective calculation, the latter being illustrated by six concrete examples. The paper concludes with a brief sketch of the construction and interpretation of the equivariant degree.

A short treatise on the equivariant degree theory and its applications

A degree for S 1 -equivariant orthogonal maps and its applications to bifurcation theory

Existence of limit cycles for coupled van der Pol equations

Let H be a separable Hilbert space which is an orthogonal representation of a compact Lie group G and let Φ : H → R be a G -invariant strongly indefinite functional of the class  C 1 . To study critical orbits of Φ we have defined the degree for G -invariant strongly indefinite functionals, which is an element of the Euler ring U ( G ) . Using this degree we have formulated the Rabinowitz alternative for G -invariant strongly indefinite functionals. The abstract results are applied to the study of global bifurcations of weak solutions of non-cooperative systems of elliptic differential equations.

Sławomir Rybicki

Papers

A short treatise on the equivariant degree theory and its applications

A degree for S 1 -equivariant orthogonal maps and its applications to bifurcation theory

Existence of limit cycles for coupled van der Pol equations

Global bifurcations of critical orbits of G-invariant strongly indefinite functionals

Existence and continuation of periodic solutions of autonomous Newtonian systems