Cycles in a random graph near the critical point

Let G(n, M) be a graph chosen at random from the family of all labelled graphs with n vertices and M(n) = 0.5n + s(n) edges, where s³(n)n^?2→∞ but s(n) = o(n). We find the limit distribution of the length of shortest cycle contained in the largest component of G(n, M), as well as of the longest cycle outside it. We also describe the block structure of G(n, M) and derive from this result the limit probability that G(n, M) contains a cycle with a diagonal. Finally, we show that the probability tending to 1 as n-→∞ the length of the longest cycle in G(n, M) is of the order s²(n)/n.     

Sequence of bifurcations in a three-dimensional system near a critical point

The purpose of this paper is to present a new approach to a bifurcation problem proposed by W. F. Langford and to provide a rigorous proof for the existence of a family of invariant tori.     

On the location of critical point for the Poisson equation in plane

The location of the unique critical point of Δu=−1 is investigated by conformal mapping method in complex variables. It is found that if the domain is given by r=1+?p(θ), the critical point coincides with the center of mass up to the order of ?. However, the two do not exactly match in general as shown by simple examples.     

Component behavior near the critical point of the random graph process

We study the behavior of a random graph process (G(n, M))₀²ⁿ for M(n) = n/2 + s and ∣s∣³n^?;2 → ∞. Among others we find the number of components in G(n, M) and estimate the number of vertices and edges in the kth largest component of G(n, M), for any natural number k, Moreover, it is shown that, with probability 1 –o(1), when M(n) = n/2 + s, s³n^?2 →?∞, then during a random graph process in some step M¹ > M a “new” largest component will emerge, whereas when s³n^?2→∞, the largest component of G(n, M) remains largest until the very end of the process.     

Structure of the correlation function of a fluctuating system at the critical point

Donets Physicotechnical Institute, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 2, pp. 223–238, August, 1990.     

The third neighbourhood of a point in a plane

Summary An unexceptionally birational model is constructed, in the form of a non singular algebraic threefold variety W, of the three dimensional aggregate of plane curve elements P₀P₁P₂P₃ of the third rank with origin at a fixed point P₀ of the plane. The geometric properties of W are then examined, and a base for surfaces and curves on W found. In particular the mapping on W of the singular (for instance cuspidal) curve elements is clarified. To Enrico Bompiani on his scientific Jubilee     

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

Convergence near a point and the Arzela-Ascoli-type theorems

Theorems are proved giving necessary and sufficient conditions for the convergence of a sequence of continuous (differentiable) functions to a continuous (differentiable) function. The concepts of convergence near a point and equipotential convergence near a point are introduced. These concepts are introduced locally; on a segment, they are equivalent to the quasiuniform convergence and to the uniform convergence of a sequence of functions, respectively.Translated from Ukrainskii Matemahcheskii Zhurnal, Vol. 45, No. 8, pp. 1090–1095, August, 1993.     

Dynamics of flow near a separation point

Dynamical behavior of a boundary layer separation is studied using the experimental approach. The dynamical nature of the phenomenon is demonstrated on a simple case of a flat–plate boundary layer in adverse pressure gradient. The Time–Resolved PIV technique was utilized for monitoring instantaneous structure of the separation region and its time development. Distinctive coherent structures and their dynamical behavior were identified using the Proper Orthogonal Decomposition method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling of the controlled motion ofa point on a plane

A solution of the problem of feedback control of the motion of a point on a plane is presented. The equations of the controlprogramme (the objective) are set up as a system of differential equations with a given set of singular trajectories in the domain of admissible positions of the controlled point, as well as a given topological structure of the partition into trajectories. These equations define the vector field of velocities of the programmed motions of the point and are used to find the corresponding control forces.     

Hopf bifurcation near a double singular point

The nonlinear equation f(x,λ,) = 0, f:X × R²→X, where X is a Banach space and f satisfies a Z₂-symmetry relation is considered. Interest centres on a certain type of double singular point, where the solution x is symmetric and f_x has a double zero eigenvalue, with one eigenvector symmetric and one antisymmetric.

We show that under certain nondegeneracy conditions, which are stated both algebraically and geometrically, there exists a path of Hopf bifurcations or imaginary Hopf bifurcations passing through the double singular point, and for which x is not symmetric except at the double singular point. An easy geometrical test is found to decide which type of phenomenon occurs. A biproduct of the analysis is that explicit expressions are obtained for quantities which help to provide a reliable numerical method to compute these paths. A pseudo-spectral method was used to obtain numerical results for the Brusselator equations to illustrate the theory.