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In the mid-seventeenth century Isaac Newton formalized the language necessary to describe the evolution of physical systems. Newton argued that the evolution of the state of a process can be described entirely in terms of the forces involved with the process. About a century and a half later, William Hamilton was able to establish the whole of Newtonian mechanics without ever using the concept of force. Rather, Hamilton argued that a physical system will evolve in such a way as to extremize the integral of the difference between the kinetic and potential energies. This paradigmatic reformulation allows for a type of reverse engineering of physical systems. This paper will use the Hamiltonian formulation of a nonlinear damped harmonic oscillator with third and fifth order nonlinearities to establish the existence of localized solutions of the governing model. These localized solutions are commonly known in mathematical physics as solitons. The data obtained from the variational method will be used to numerically integrate the equation of motion, and find the exact solution numerically.
Chaos theory examines the iterates of continuous functions to draw conclusions about long-term behavior. As this relatively new theory has evolved, one difficulty still present is the absence of universally agreed upon definitions. On the other hand, function spaces and equicontinuity are well established concepts with mathematical definitions that are universally accepted. We will present some theorems that display the natural connections between chaos and equicontinuity.
The minimum rank of a simple graph is defined to be the smallest possible rank over all symmetric real matrices whose -th entry (for ) is nonzero whenever is an edge in and is zero otherwise. Maximum nullity is taken over the same set of matrices, and the sum of maximum nullity and minimum rank is the order of the graph. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. This paper defines the graph families ciclos and estrellas and establishes the minimum rank and zero forcing number of several of these families. In particular, these families provide examples showing that the maximum nullity of a graph and its dual may differ, and similarly for the zero forcing number.
We describe a numerical approach to the solution of two-delay Volterra integral equations, and we carry out a nonlinear stability analysis on an interesting test equation by means of a parallel investigation both on the continuous and the discrete problem.
The Szegő kernel serves as one of the canonical functions associated to a region in the complex plane, and from it one can compute the Riemann (or Ahlfors) map, the essentially unique conformal transformation of the region to the unit disc. We provide an elementary description of the method that Kerzman and Stein used to compute the Szegő kernel, and subsequently, the Riemann and Ahlfors maps. A description, too, is provided for a new tool that generates visual representations of these functions and is included with the open-source computer algebra system Sage.
In 1939, Richard Rado showed that any complex matrix is partition regular over if and only if it satisfies the columns condition. Recently, Hogben and McLeod explored the linear algebraic properties of matrices satisfying partition regularity. We further the discourse by generalizing the notion of partition regularity beyond systems of linear equations to topological surfaces and graphs. We begin by defining, for an arbitrary matrix , the metric space (, ). Here, is the set of all matrices equivalent to that are (not) partition regular if is (not) partition regular; and for elementary matrices, and , we let . Subsequently, we illustrate that partition regularity is in fact a local property in the topological sense, and uncover some of the properties of partition regularity from this perspective. We then use these properties to establish that all compact topological surfaces are partition regular.
Given a surface of revolution with boundary, we study the extrinsic energy of smooth tangent unit-length vector fields. Fixing continuous tangent unit-length vector fields on the boundary of the surface of revolution, we ask if there is a unique smooth tangent unit-length vector field continuously achieving the boundary data and minimizing energy amongst all smooth tangent unit-length vector fields also continuously achieving the boundary data.
We consider the Stokes equation for a flow through a partially obstructed channel and determine the relationship between Dirichlet boundary values (velocities) and Neumann boundary values (forces) for the FEM discrete form. For the steady state case we find a linear relationship. For the transient case the relationship depends on the time stepping procedure and includes the relationship at prior states. We resolve the issue for trapezoid and Adams–Bashford-2 time stepping. Since Stokes flow may be considered as the startup phase of Navier–Stokes flow, we give particular attention to a flow with a startup function.