Understand: Academics

With Robert Gilmore

Chaotic data generated by a three-dimensional dynamical system can be embedded into R^3 in a number of inequivalent ways. However, when lifted into R^5 they all become equivalent, indicating that they all belong to a single universality class sharing a common chaos-generating mechanism. We present a complete invariant determining this universality class and distinguishing attractors generated by distinct mechanisms. This invariant is easily computable from an appropriately ``dressed'' return map of any particular three-dimensional embedding.

With Robert Gilmore

Embeddings are diffeomorphisms between some dynamical phase space and a reconstructed image. Different embeddings may or may not be equivalent under isotopy. We regard embeddings as representations of the dynamical phase space. We determine the topological labels required to distinguish inequivalent representations of three-dimensional dissipative dynamical systems when the embeddings are into R^k, k=3,4,5,.... Three representation labels are required for embeddings into R^3, and only one is required in R^4. In R^5 there is a single ``universal'' representation.

With Robert Gilmore

Baker-Campbell-Hausdorff formulas are exceedingly useful for disentangling operators so that they may be more easily evaluated on particular states. We present such a disentangling theorem for general bilinear and linear combinations of multiple boson creation and annihilation operators. This work generalizes a classical result of Schwinger.

With Robert Gilmore

Takens has shown that a dynamical system may be reconstructed from scalar data taken along some trajectory of the system. A reconstruction is considered successful if it produces a system diffeomorphic to the original. However, if the original dynamical system is symmetric, it is natural to search for reconstructions that preserve this symmetry. These generally do not exist. We demonstrate that a differential reconstruction of nonlinear dynamical system preserves at most a two-fold symmetry.

With Robert Gilmore

Ideally an embedding of an N-dimensional dynamical system is N-dimensional. Ideally, an embedding of a dynamical system with symmetry is symmetric. Ideally, the symmetry of the embedding is the same as the symmetry of the original system. This ideal often cannot be achieved. Differential embeddings of the Lorenz system, which possesses a twofold rotation symmetry, are not ideal. While the differential embedding technique happens to yield an embedding of the Lorenz attractor in three dimensions, it does not yield an embedding of the entire flow. An embedding of the flow requires at least four dimensions. The four dimensional embedding produces a flow restricted to a twisted three dimensional manifold in R^4. This inversion symmetric three-manifold cannot be projected into any three dimensional Euclidean subspace without singularities.

With Ryan Michaluk and Robert Gilmore

An algorithm inspired by Genome sequencing is proposed which “reconstructs” a single long trajectory of a dynamical system from many short trajectories. This procedure is useful in situations when many data sets are available but each is insufficiently long to apply a meaningful analysis directly. The algorithm is applied to the Rössler and Lorenz dynamical systems as well as to experimental data taken from the Belousov-Zhabotinskii chemical reaction. Topological information was reliably extracted from each system and geometrical and dynamical measures were computed.

With Robert Gilmore

Embeddings are diffeomorphisms between some unseen physical attractor and a reconstructed image. Different embeddings may or may not be equivalent under isotopy. We regard embeddings as representations of the attractor, review the labels required to distinguish inequivalent representations for an important class of dynamical systems, and discuss the systematic ways inequivalent embeddings become equivalent as the embedding dimension increases until there is finally only one “universal” embedding in a suitable dimension.

The interaction of a magnetic dipole with a point charge leads to an apparent paradox when analyzed using the 3-vector formulation of the Lorentz force. Specifically, the dipole is subject to a torque in some frames and not in others. We show that when analyzed according to the covariant 4-vector formulation the paradox disappears. The torque that arises in certain frames is connected to the time-space components of the torque in the rest frame, giving rise to "hidden" momentum.

The observed accelerated cosmic expansion is problematic in that it seems to require an otherwise unobserved dark energy for its origin. A possible alternative explanation has been recently given, which attempts to account for this expansion in terms of a hypothesized matter-anti-matter repulsion. This repulsion or anti-gravity is derived by applying the CPT theorem to general relativity. We show that this proposal cannot work for two reasons: 1) it incorrectly predicts the behavior of photons and 2) the CPT transformation itself is not consistently applied.

The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integral between arbitrary manifolds may be similarly reduced to a lower dimensional integral.

The recently proposed Cooperstock-Tieu galaxy model claims to explain the flat rotation curves without dark matter. The purpose of this note is to show that this model is internally inconsistent and thus cannot be considered a valid solution. Moreover, by making the solution consistent the ability to explain the flat rotation curves is lost.

Mach's principle states that the local inertial properies of matter are determined by the global matter distribution in the universe. In 1958 Cocconi and Salpeter suggested that due to the quadrupolar assymetry of matter in the local galaxy about the earth, inertia on earth would be slightly anisotropic, leading to unequal level splittings of nuclei in a magnetic field [1,2]. Hughes, et al., Drever, and more recently Prestage, et al. found no such quadrupole splitting [3-5]. However, recent cosmological overservations show an anisotropy in the Cosmic Microwave Background, indicating anisotropy of the matter at much greater distances. Since the in- ertial interaction acts as a power law of order unity, the effect of this matter would far outweigh the relatively local contribution from the galaxy [1,6]. Thus, the present article extends the work of Cocconi and Salpeter to higher multipoles leading to unequal level splittings that should be measurable by magnetic resonance experiments on nuclei of appropriate spin.

Note: PhD Oral qualifying report.

Of all the mysteries of quantum mechanics, the existence of half-integer spin is perhaps the hardest to accept. In this talk I want to take a look at why spin exists. It turns out that spin owes its existence to some rather deep and counter intuitive properties of three dimensional space. However, these properties have implications in classical physics, not just quantum mechanics. I will describe these properties, some of their classical manifestations, and how they give rise to spin. As a bonus, we will see why in two dimensions there exist particles with arbitrary spin, so-called anyons.

Supplementary videos:

Note: Beamer presentation. Report version is in the works.

Demonstratration of the group theoretical origin of Maxwell's equations. The equations are constraints on a classical field to suppress non-physical degrees of freedom which are not present in the fundamental quantum description of the field. This is mostly complete, but there are some details which are glossed over that I hope to fix in the future.

The twin paradox is analyzed in situations where no acceleration is necessary for the twins to reunite. Sorry if the format makes it difficult to understand - I should add a complete article version at some point.

Note: Beamer presentaiton.

The Master Analytic Representation for the root space A1 is constructed. This gives all of the unitary irreducible representations of the two real forms of this root space, su(2) and su(1, 1). This procedure is carried out in a generalization of Schwinger’s presentation for angular momentum.

When considering maps in several complex variables one may want to consider whether the maps are immersive, submersive, or locally diffeomorphic. These same questions are easily formulated in terms of functions of real variables using the Jacobian determinant. This article uses the natural correspondence between complex and real maps to extend the real result to the complex case, expressing this result entirely in terms of the complex functions (the complex Jacobian). To do this we employ a result of Sylvester on the determinant of block matrices.

The Flux Rule for calculating the EMF due to a changing magnetic flux is critically examined. First, the rule is derived from Maxwell’s equations in a way that unifies the two contributions to the flux change. Then it is shown that so-called “failures of the flux rule” are not problems with the actual rule, but rather in trying to improperly deduce a stronger local result from the weaker global result that the rule actually provides.

This method requires a little bit of complex analysis, but otherwise is at the level of elementary calculus and requires no special machinery.

The Doppler shift equations are derived exploiting invariance principles in the more general cases, keeping the hard work to a minimum.

Problems and solutions.

Note: All problems in chapter 1-7. A few problems from chapters 8-10.

Problems and solutions.

Note: Only chapter 2.

Problems and solutions.

Note: Only chatpers 2 and 3 are finished.