We wish to transmit messages to and from a hypersonic vehicle around which a plasma sheath has formed. For long distance transmission, the signal carrying these messages must be necessarily low frequency, typically 2 GHz, to which the plasma sheath is opaque. The idea is to use the plasma properties to make the plasma sheath appear transparent.

Using weakly nonlinear approach we have to solve two boundary problem for linear second order ODE. We used sweep-method, involving solution of a linear system of equations with a tridiagonal matrix. Also we give a brief comparison with some analytical results.

This is joint work with Alan Newell and Vladimir Zakharov.

Cellular DNA knotting is driven by DNA compaction, topoisomerization, supercoiling-promoted strand collision, and DNA self-interactions resulting from transposition, site-specific recombination, and transcription. Type II topoisomerases are the ubiquitous, essential enzymes that interconvert DNA topoisomers to resolve knots. These enzymes pass one DNA helix through another by creating an enzyme-bridged transient break in the DNA. Explicitly how type II topoisomerases recognize their substrate and decide where to unknot DNA is unknown. Uniquely combining biology, chemistry, physics, and mathematics, we investigate the physiological effects of DNA knotting, the biophysics of knotting/unknotting, and the unknotting mechanism of human topoisomerase IIα.

In the presence of decreased tissue oxygen levels, red blood cells release an increased amount of ATP which triggers a signal to travel upstream and cause blood vessels to dilate so that more blood is supplied to the region of demand. A theoretical model analyzing this mechanism is presented here. In the model, arterioles regulate blood flow by dilating or constricting in response to changes in metabolism as well as to changes in pressure and wall shear stress. The model predicts that responses to these three stimuli can account for the increase in blood flow that occurs in response to an increase in oxygen demand. In addition, the model predicts that vasomotion (spontaneous rhythmic variations in vessel diameter) occurs under some conditions.

In the context of the classical fluctuation-dissipation theorem, the average linear response to external fluctuations is represented as a simple time autocorrelation function of an unperturbed dynamical system, which is easy to compute numerically via time averaging along a single long-time trajectory. However, the majority of real-world climate models are chaotic forced-dissipative nonlinear systems with complex dynamics, for which the fluctuation-dissipation theorem in its classical setting is not valid. Here we test a universal linear response approach which is valid for a wide variety of dynamical systems, but works only for a short time.

We examine a selective list of combinatorial optimization problems in NP with respect to inapproximability (Arora and Lund, 1997) given that the ground set of elements N has additional characteristics. For each of the problems, the set N is expressed explicitly by subsets of N either as a partition or in the form of a cover. The problems examined are generalizations of well known classical graph problems and include the minimal spanning tree, the assignment problem, a number of elementary machine scheduling problems, bin-packing, and the TSP. We conclude that for all these generalized problems the existence of PTAS (polynomial time approximation scheme) is impossible unless P = NP. This suggests a partial characterization for a family of inapproximable problems. For the generalized Euclidean TSP we prove inapproximability even if the subsets are of cardinality two.

This is joint work with James Orlin.

The evolution of meandering channels is a complex morpho-dynamic process that has been the focus of research among geomorphologists and river engineers for decades. The evolution of a meandering channel is the result of the interaction between flow and sediment material. A numerical model including a depth-averaged two-dimensional hydrodynamic flow algorithm, a sediment transport equation, and a bank erosion routine was developed to simulate the evolution of channel meandering. The sediment transport equation calculates both bed load and suspended load assuming equilibrium sediment transport. The bank erosion routine simulates two interactive processes: basal erosion and bank failure.

An important aspect of this model is that bank erosion does not guarantee the retreat of a bank line if eroded bank material remains at the toe of the bank. Whether or not a bank retreats or advances depends on the balance of sediment load at near-bank regions where sediment may come from upstream, bank erosion, and secondary flow. Modeling results clearly demonstrated the evolution of meandering from low to high sinousities through downstream translation, lateral extension, upstream and downstream rotation. The essential processes leading to meandering formation are well replicated with this model. The growth of sand bars determines the hydrodynamic flow field that pushes toward the concaving banks. Bank material from the caving banks will supplement sediment deposits on point bars when bed and bank material are the same, such as in the laboratory experiments. At this point, this model properly simulates key laboratory experiments of channel meandering. It is also very similar to some features observed in the field, such as observed on the Lower Yellow River in China.

Phenotypes result from complex interactions between many molecules. These interactions can be modeled using networks. Analyzing these network models is one of the pillars of a new discipline: Systems Biology. Many aspects of the analysis of these networks are reasonably mature for the case where the network is known. Our research takes the next logical step: searching over the network topology space. To enable this step, new technology that enables approximate evaluation of very large sets of related systems of differential equations is needed. Our approach to this evaluation combines trust region approaches with Markov Chain Monte Carlo techniques to quickly screen network designs for promising explanations to the expression of a given phenotype.

This is joint work with Matej Boguzsak, Andrew Paek, Jay Konieczka, and Parker Antin.

Simulation of massive data transmission in a high speed fiber communication systems requires parallel algorithm for affordable computation time. In this talk the first implementation of such an algorithm is presented. The algorithm is based on short term predictability of the solutions of considered matematical model. Major part of calculations could be made in advance in a parallel fashion and subsequently used for generation of the final solution.

Rupture abdominal aortic aneurysms (AAA) represent the 13th leading cause of death within the U.S. This catastrophic event can come unexpectedly as this disease is most often asymptomatic. The current research presentation will focus on the use of finite element modeling in AAA rupture assessment. Detailed information on anisotropic constitutive modeling and its implementation into a commercially available finite element code will be presented. Ongoing efforts in computational modeling of AAA will also be discussed.

Using the variation found in extant human populations, mathematics is employed to elucidate our understanding of human evolution and the processes that led to contemporary DNA sequences. The gene genealogy is modeled as a stochastic process that proceeds backwards in time over which we can define a probability space. In this talk we will construct the coalescent process for the standard neutral model and generalize to more complicated demographies and deviations from neutrality. We will also discuss how modern-day analysis of DNA sequences is performed, and show recent results being discovered at the University of Arizona.

When solving the flow and transport through heterogeneous porous media, some type of upscaling or coarsening is needed due to scale disparity. I will describe multiscale/upscaling techniques used for solving stochastic flow equations. These techniques allow us to simulate the flow and transport processes on the coarse grid and thus reduce the computational cost. I will show how the proposed coarse-scale models are used in uncertainty quantification, which involves sampling the media properties conditioned to coarse-scale measurements, as well as present numerical results.

We shall examine some linear and nonlinear equations with random coefficients and examine their behavior under homogenization, i.e. suitable rescaling.

We obtain an exact vector solitary solution for the amplification of an optical pulse with a time width short compared with both population and polarization decay time. This dissipative soliton results from the balance between the gain from inverted resonant two-level atoms and the linear loss of the host material. We suppose that the excited state of the active centers is degenerate on the projection of the angular moment. Numerical simulations demonstrate the stability of these vector dissipative solitons in the presence of both linear birefringence and group velocity dispersion of the host material.