Material interface reconstruction (MIR) is the problem of
constructing an interface (or interfaces) between regions
of materials over a given grid. Each grid cell has
associated volume fractions which are the material
percentages present in the given cell. Volume-of-Fluid
methods use these volume fractions to construct an
interface. Volume-of-Fluid methods naturally conserve
the mass of each fluid, automatically handle changes in
global topology of the front, and the work required to
update the front location is entirely local.
There are many iterative techniques to find a root or
zero of a given function. For any iterative technique, it
is often of interest to know which initial seeds lead to
which roots. When the iterative technique used is
Newton's Method, this question is known as Cayley's
Problem. In this project, I investigate two extensions of
Cayley's Problem. In particular, I study generalizations
of Cayley's Problem in the plane, and the associated
fractal structures that arise from using more
sophisticated numerical techniques.
Excess entropy is a time-symmetric, information-theoretic
invariant of stochastic processes representing the amount
of information shared between the past and the future. It
is frequently used to identify and distinguish
organization in complex systems but has been difficult to
calculate, since no closed-form expression was known. By
considering prediction \textit{and} retrodiction, we
present a closed-form expression for the excess entropy
in terms of the $\epsilon$-machine, the unique and
minimal deterministic presentation of a stochastic
process. We also clarify the relationship between excess
entropy and statistical complexity---the amount of stored
information necessary for optimal prediction.
Previous work has shown that spiking neural networks are
capable of a range of different behavior. Towards the
goal of quantifying the types of spiking behavior, we
systematically examine and compare entropies in the
behavior of small networks with a range of connectivity
patterns and note trends in the types of spike patterns
with which these entropies are correlated. Additionally,
we study mutual information between spike trains of
individual neurons in small networks and make conclusions
regarding how structural connectivity in a network may be
reflected in such measurements of its behavior.
In the study of quantum finite state machines (QFSM) , we are confronted
with the problem of how to sample the space of these machines for
numerical analysis. The construction of machines by hand is time consuming
and does not answer questions about the entire space. We wish to be able to
organize the space by dividing it into machine classes based on their
topology. Additionally, we wish to reduce these classes to a minimal set.
This reduces computation time of random sampling and also imbues each
model with added meaning. The analogous problem for stochastic finite
state machines is significantly simpler due to the use of stochastic, rather
than unitary, matrices. We introduce the complications introduced by
unitarity and then, for dimensions $N \le 4$, we describe an algorithm that
enumerates the QFSM classes. Finally we present a sampling technique that
will allow us to systematically explore the full range of behaviors allowed by
$N \le 4$ QFSM.
The resilience of corals is important for withstanding ecological changes in the surrounding environment as well as recovering from the impact of hurricanes and other significant ecological disturbances. However, the last several decades have demonstrated a loss of resilience resulting from human and environmental impacts; such declines have frequently resulted in phase shifts to a degraded macroalgal-dominated state. A major defining issue in current research is to identify when and how it is possible to reverse these phase shifts, allowing for the ecosystem to return to a coral-dominated state and subsequently regain resilience. We focus on the impacts of over-harvesting of herbivorous reef fish (which graze on algae, preventing coral from becoming algae-dominated) in the Caribbean. Through the extension and analysis of an analytic model developed by Mumby et al. (2007), we identify situations in which reductions in fishing effort allow hysteresis to switch the system into a coral-dominated state. In addition, we consider two distinct scenarios: habitat quality versus food limitation (algae) as the primary limiting factor of grazer recruitment.
We analyze the Basis Pursuit recovery method when
observing signals with general perturbations (i.e.,
\emph{additive}, as well as \emph{multiplicative} noise).
Previous studies have only considered partially perturbed
observations $Ax + e.$ Our model also incorporates
perturbations to the matrix $A$ in the form of $(A+E)x +
e.$ This completely perturbed model extends the previous
work of Cand\`es, Romberg and Tao on stable signal
recovery from incomplete and inaccurate measurements. Our
results show that, under suitable conditions, the
stability of the recovered signal is limited by the noise
level in the observation. Moreover, this accuracy is
within a constant multiple of the best-case
reconstruction using the technique of least squares.
In the Crayfish swimmeret system, each module contains a
half-center oscillator responsible for controlling the
power stroke and return stroke motoneurons. The
half-center oscillator is made up of two strongly
connected neurons, each with their own phase response
properties. Each module is capable of oscillations
independent of other modules, however all modules are
connected to each other by weak intersegmental coupling.
Because of this, we may use the theory of weakly coupled
oscillators in an effort to deduce how phase maintenance
is achieved in the cord over varying frequencies. Before
applying this theory, we need to know the phase response
properties of the individual modules. To do this, we
analyze the half-center oscillator in one module, and
through an idealized phase model derive the phase
response curve (PRC) of the half-center oscillator from
the PRC of the individual cells. The work is then
compared to experimental data from the Crayfish swimmeret
system.
Neurons can have extensive spatial geometries. However, for ease of mathematical analysis, many theoretical studies model neurons as single-compartment objects ignoring the spatial anatomy of the cell. An issue with this approach is that many neurons are not electrotonically compact and single compartment models cannot be expected to fully capture their behavior. Furthermore, dendritic properties can have substantial effects on the dynamics of single neurons, as well as the activity in neuronal networks. Even when considering dendrites without active currents, the effect of coupling the dendrite to a neuronal oscillator are not always easy to understand. Intuitively, if the leakage reversal potential of the dendrite is lower (higher) than the average voltage of the oscillations, then the firing frequency of the neuronal oscillator will decrease (increase) as the radius of the dendrite increases. However, the dendritic ``load" can sometimes have surprising effects on a neuron's firing frequency. To gain insight into the mechanisms of these effects, we model a neuron as a passive dendrite attached to an isopotential somatic oscillator, i.e. a ``ball-and-stick" model, and use the theory of weak coupling to derive an equation for the change in frequency of the oscillator due to the presence of the thin dendrite.
The simplex algorithm of George Dantzig is widely considered one of the top ten algorithms of the century. Given a convex polytope $P$, an initial vertex of $P$, and a linear functional $c$, the simplex algorithm finds a vertex optimizing $c$ by following a path on the graph of $P$ that is monotone on $c$. The simplex path travels from vertex to vertex on the polytope, with ambiguities on the choice of the next vertex resolved by a pivot rule. Under essentially every known pivot rule, the theoretical number of simplex iterations needed is exponential due to some well-engineered yet peculiar examples. However, the algorithm is highly efficient in practice on most linear optimization problems.
Providing good upper bounds on the diameters of convex polytopes is particularly interesting, since the diameter is a lower bound on the number of iterations required for the simplex algorithm under any pivot rule. In this poster, we present recent advances on the upper bounds of diameters of transportation polytopes and network flow polytopes. Portions of this presentation are in collaboration with Jes\'us De Loera, Shmuel Onn and Francisco Santos.
Generalized Julia Sets: An Extension of Cayley's Problem
Owen Lewis
Time's Barbed Arrow: Irreversibility, Crypticity, and Stored Information
Christopher Ellison, J. Mahoney, J. Crutchfield
Information Theoretic Approaches to Spiking Models
Richard Watson
Quantum Finite State Machine Construction and Sampling
John Mahoney, J. Crutchfield, K. Wiesner
Hysteresis and Caribbean coral reefs: prevention of phase shifts to a macroalgal-dominated state
Julie Blackwood
General Perturbations in Compressed Sensing
Matthew Herman, T.Strohmer
Deriving Phase Response Properties of an
Idealized Half-Center Oscillator Model
Tamara Joy Schlichter, Timothy Lewis, Brian
Mulloney, and Carmen Smarandache
The effects of dendritic load on the firing frequency of oscillating neurons
Michael A. Schwemmer and Timothy J. Lewis
The simplex algorithm on transportation and network flow polytopes
Edward D. Kim, J. De Loera, S. Onn, F. Santos
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