Coordinate shadows of semi-definite and Euclidean distance matrices

We consider the projected semi-definite and Euclidean distance cones onto a subset of the matrix entries. These two sets are precisely the input data defining feasible semi-definite and Euclidean distance completion problems. We classify when these sets are closed, and use the boundary structure of these two sets to elucidate the Krislock-Wolkowicz facial reduction algorithm. In particular, we show that under a chordality assumption, the "minimal cones" of these problems admit combinatorial characterizations. As a byproduct, we record a striking relationship between the complexity of the general facial reduction algorithm (singularity degree) and facial exposedness of conic images under a linear mapping.Comment: 21 page

Edible Marijuana: A New Frontier in the Culinary World

Cannabis, commonly known as marijuana, has a rich history as a source of fiber, food and medicine (Li 437). Since 1785, physicians and scientists alike have worked to discover the active chemical components and medical effectiveness of this plant (Touw 2; Aldrich). Despite its complicated legal history, marijuana has retained a place culturally and, in some countries, scientifically as an effective medical agent. As a medically edible ingredient, cannabis has also been more recently heralded as a new, even cutting edge flavor, opening a new frontier to the culinary community. After the isolation of the main active ingredient in cannabis, δ-9 tetrahydrocannabinol (THC), in 1964, various physicians and scientists conducted research demonstrating its therapeutic medical efficacy for a variety of illnesses including glaucoma, cancer, AIDS, anorexia, Crohn’s disease, multiple sclerosis, and pain management (Watson et. al. 548). Over a decade has passed since cannabis first became legalized for medical use in California in 1996 (Eddy 8). Subsequently, 18 other states, including the District of Columbia, legalized cannabis for medical use. Individuals who have received a doctor’s recommendation for its use are authorized to consume cannabis for medical purposes. Although it is commonly ingested by inhalation, many individuals prefer to consume marijuana through edible products. Due to an increased interest in careful cultivation practices, biochemical evaluation of the product, and quality control efforts, a market for medical edibles has developed. Many chefs have begun experimenting with cannabis, using it as a spice or flavor, to create a sensory food experience. Savvy chefs, such as Scott Van Rixel, Kristi Knoblich, Eric Underwood, Julie Dooley and Julianna Carella entered this new market with both entrepreneurial and altruistic interest in creating medical marijuana edibles, tapping into the new flavor frontier this product provides. It is interesting that in the culinary world’s constant search for new and exciting flavor discoveries, the medical edibles industry has not received much mainstream interest. Little data has been compiled supporting the potential this ancient plant may bring to cuisine (Watson et. al. 548). Now, due to the application of ancient and experimental techniques, these innovative chefs are blazing the trail for the culinary community. This paper investigates edible medical marijuana as a viable frontier and niche market from a legal, medical and culinary perspective. It reviews the etymology and the history of cannabis, as well as current legal, medicinal and cultivation guidelines. This is accomplished through an overview of four states’ dispensary policies, as well as the creative culinary accomplishments of representative edible establishments. This study also draws on the results of two surveys: one directed to the edible product developers, dispensers, and vendors, the other survey directed to authorized patients of the dispensaries. The data collected is intended to confirm the emergence of a unique culinary product, its value, increased distribution, and potential for success in this newly developed market

Derivation and Analysis of a Discrete Predator–Prey Model

We derive a discrete predator–prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark–Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark–Sacker bifurcation as a function of the model parameters

Optimality criteria without constraint qualications for linear semidenite problems

We consider two closely related optimization problems: a problem of convex Semi- Infinite Programming with multidimensional index set and a linear problem of Semidefinite Programming. In study of these problems we apply the approach suggested in our recent paper [14] and based on the notions of immobile indices and their immobility orders. For the linear semidefinite problem, we define the subspace of immobile indices and formulate the first order optimality conditions in terms of a basic matrix of this subspace. These conditions are explicit, do not use constraint qualifications, and have the form of criterion. An algorithm determining a basis of the subspace of immobile indices in a finite number of steps is suggested. The optimality conditions obtained are compared with other known optimality conditions

ON THE CONNECTIONS BETWEEN SEMIDEFINITE OPTIMIZATION AND VECTOR OPTIMIZATION

This paper works out connections between semidefinite optimization and vector optimization. It is shown that well-known semidefinite optimization problems are scalarized versions of a general vector optimization problem. This scalarization leads to the minimization of the trace or the maximal eigenvalue

Twisted and Nontwisted Bifurcations Induced by Diffusion

We discuss a diffusively perturbed predator-prey system. Freedman and Wolkowicz showed that the corresponding ODE can have a periodic solution that bifurcates from a homoclinic loop. When the diffusion coefficients are large, this solution represents a stable, spatially homogeneous time-periodic solution of the PDE. We show that when the diffusion coefficients become small, the spatially homogeneous periodic solution becomes unstable and bifurcates into spatially nonhomogeneous periodic solutions. The nature of the bifurcation is determined by the twistedness of an equilibrium/homoclinic bifurcation that occurs as the diffusion coefficients decrease. In the nontwisted case two spatially nonhomogeneous simple periodic solutions of equal period are generated, while in the twisted case a unique spatially nonhomogeneous double periodic solution is generated through period-doubling. Key Words: Reaction-diffusion equations; predator-prey systems; homoclinic bifurcations; periodic solutions.Comment: 42 pages in a tar.gz file. Use ``latex2e twisted.tex'' on the tex files. Hard copy of figures available on request from [email protected]

A sequential quadratic penalty method for nonlinear semidefinite programming

2003-2004 > Academic research: refereed > Publication in refereed journa