Introducing Randomness into First-Order and Second-Order Deterministic Differential Equations

We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially time correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities for correlated and uncorrelated noise are discussed. Analytical results are presented for a deterministic linear friction force and a stochastic force that is uncorrelated or exponentially correlated

The dynamics of cell proliferation

Summary The article provides a mathematical description based on the theory of differential equations, for the proliferation of malignant cells (cancer). A model is developed which enables us to describe and predict the dynamics of cell proliferation much better than by using ordinary curve fitting procedures. By using differential equations the ability to foresee the dynamics of cell proliferation is in general much better than by using polynomial extrapolations. Complex time relations can be revealed. The mass of each living cell and the number of living cells are described as functions of time, accounting for each living cell's age since cell-birth. The linkage between micro-dynamics and the population dynamics is furnished by coupling the mass increase of each living cell up against the mitosis rate. A comparison is made by in vitro experiments with cancer cells exposed to digitoxin, a new promising anti-cancer drug. Theoretical results for the total number of cells (living or dead) is found to be in good agreement with experiments for the cell line considered, assuming different concentrations of digitoxin. It is shown that for the chosen cell line, the proliferation is halted by an increased time from birth to mitosis of the cells. The delay is probably connected with changes in the Ca concentration inside the cell. The enhanced time between the birth and mitosis of a cell leads effectively to smaller mitosis rates and thereby smaller proliferation rates. This mechanism is different from the earlier results on digitoxin for different cell lines where an increased rate of apoptosis was reported. But we find it reasonable that cell lines can react differently to digitoxin. A development from enhanced time between birth and mitosis to apoptosis can be furnished, dependent of the sensitivity of the cell lines. This mechanism is in general very different from the mechanism appealed to by standard chemotherapy and radiotherapy where the death ratios of the cells are mainly affected. Thus the analysis supports the view that a quite different mechanism is invoked when using digitoxin. This is important, since by appealing to different types of mechanism in parallel during cancer treatment, more selectivity in the targeting of benign versus malignant cells can be invoked. This increases the probability of successful treatment. The critical digitoxin level concentration, i.e. the concentration level where the number of living cells is not increasing, is approximately 50 ng/ml for the cell line we investigated in this article. Therapeutic plasma concentration of digitoxin when treating cardiac congestion is about 15-33 ng/ml, but individual tolerances are * Corresponding author. Tel.: +47-51-831632/+47-51-831500 (Dept.); fax: +47-51-831550. E-mail addresses: john-f. large. The effect of digitoxin during cancer treatment is therefore very promising. The dynamic model constitutes a new powerful tool, supported by empirics, describing the mechanism or process by which the number of malignant cells during anti-cancer treatment can be studied and reduced

On reminder effects, drop-outs and dominance: evidence from an online experiment on charitable giving

We present the results of an experiment that (a) shows the usefulness of screening out drop-outs and (b) tests whether different methods of payment and reminder intervals affect charitable giving. Following a lab session, participants could make online donations to charity for a total duration of three months. Our procedure justifying the exclusion of drop-outs consists in requiring participants to collect payments in person flexibly and as known in advance and as highlighted to them later. Our interpretation is that participants who failed to collect their positive payments under these circumstances are likely not to satisfy dominance. If we restrict the sample to subjects who did not drop out, but not otherwise, reminders significantly increase the overall amount of charitable giving. We also find that weekly reminders are no more effective than monthly reminders in increasing charitable giving, and that, in our three months duration experiment, standing orders do not increase giving relative to one-off donations

The kinetics of lactate production and removal during whole-body exercise

<p>Abstract</p> <p>Background</p> <p>Based on a literature review, the current study aimed to construct mathematical models of lactate production and removal in both muscles and blood during steady state and at varying intensities during whole-body exercise. In order to experimentally test the models in dynamic situations, a cross-country skier performed laboratory tests while treadmill roller skiing, from where work rate, aerobic power and blood lactate concentration were measured. A two-compartment simulation model for blood lactate production and removal was constructed.</p> <p>Results</p> <p>The simulated and experimental data differed less than 0.5 mmol/L both during steady state and varying sub-maximal intensities. However, the simulation model for lactate removal after high exercise intensities seems to require further examination.</p> <p>Conclusions</p> <p>Overall, the simulation models of lactate production and removal provide useful insight into the parameters that affect blood lactate response, and specifically how blood lactate concentration during practical training and testing in dynamical situations should be interpreted.</p

Using the power balance model to simulate cross-country skiing on varying terrain

The current study adapts the power balance model to simulate cross-country skiing on varying terrain. We assumed that the skier’s locomotive power at a self-chosen pace is a function of speed, which is impacted by friction, incline, air drag, and mass. An elite male skier’s position along the track during ski skating was simulated and compared with his experimental data. As input values in the model, air drag and friction were estimated from the literature based on the skier’s mass, snow conditions, and speed. We regard the fit as good, since the difference in racing time between simulations and measurements was 2 seconds of the 815 seconds racing time, with acceptable fit both in uphill and downhill terrain. Using this model, we estimated the influence of changes in various factors such as air drag, friction, and body mass on performance. In conclusion, the power balance model with locomotive power as a function of speed was found to be a valid tool for analyzing performance in cross-country skiing

Simulation of natural fragmentation of rings cut from warheads

Natural fragmentation of warheads that detonates causes the casing of the warhead to split into various sized fragments through shear or radial fractures depending on the toughness, density, and grain size of the material. The best known formula for the prediction of the size distribution is the Mott formulae, which is further examined by Grady and Kipp by investigating more carefully the statistical most random way of portioning a given area into a number of entities. We examine the fragmentation behavior of radially expanding steel rings cut from a 25 mm warhead by using an in house smooth particle hydrodynamic (SPH) simulation code called REGULUS. Experimental results were compared with numerical results applying varying particle size and stochastic fracture strain. The numerically obtained number of fragments was consistent with experimental results. Increasing expansion velocity of the rings increases the number of fragments. Statistical variation of the material parameters influences the fragment characteristics, especially for low expansion velocities. A least square regression fit to the cumulative number of fragments by applying a generalized Mott distribution shows that the shape parameter is around 4 for the rings, which is in contrast to the Mott distribution with a shape parameter of ½. For initially polar distributed particles, we see signs of a bimodal cumulative fragment distribution. Adding statistical variation in material parameters of the fracture model causes the velocity numerical solutions to become less sensitive to changes in resolution for Cartesian distributed particles

Strain rate dependency and fragmentation pattern of expanding warheads

For the characterization of the behaviors of a metal material in events like expanding warheads, it is necessary to know its strength and ductility at high strain rates, around 104–105/s. The flyer plate impact testing produces the uniform stress and strain rates but the testing is expensive. The Taylor test is relatively inexpensive but produces non-uniform stress and strain fields, and the results are not so easily inferred for material modeling. In the split-Hopkinson bar (SHB), which may be used in compression, tension and torsion testing, the strain rates never exceeds 103/s. In the present work, we use the expanding ring test where the strain rate is 104–105/s. A streak camera is used to examine the expanding ring velocity, and a water tank is used to collect the fragments. The experimental results are compared with the numerical simulations using the hydrocodes AUTODYN, IMPETUS Afea and a regularized smooth particle (RSPH) software. The number of fragments increases with the increase in the expansion velocity of the rings. The number of fragments is similar to the experimental results. The RSPH software shows much the same results as the AUTODYN where the Lagrangian solver is used for the ring. The IMPETUS Afea solver shows a somewhat different fragmentation characteristic due to the node splitting algorithm that induces pronounced tensile splitting

A critical investigation of proposed electrostatic corrections to quantum mechanical volumes: the importance of variation and the irrelevance of imbalance

<p>The crystal density of neutral and ionic molecular crystals is remarkably well approximated by the enclosed volume of molecular surfaces, where these surfaces are defined as regions of constant and small electron density. Several workers have proposed that estimates may be improved if one includes quantities extracted from the electrostatic potential on the surface of the molecule. The variation of the potential and the imbalance of positive and negative values have been considered to be of importance. In this study we demonstrate that whereas variation is important for improving crystal density predictions, imbalance is not. We present a density functional theory study on a set of 44 neutral molecular crystals. Ten-fold cross-validations were performed on models that incorporate variation, imbalance and combinations of both. Geometries were optimised using B3LYP and basis sets of type 6-31G(d). Electron densities and electrostatic potentials were computed with B3LYP and M05. Regardless of functional, models that correct for variation yield a relative decrease of 15%–18% in root-mean-square error of prediction. This correction appears to sharpen the error distribution about zero. Models based on imbalance yield no improvement, and we argue that it plays an insignificant role.</p

Projected area and drag coefficient of high velocity irregular fragments that rotate or tumble

3 degrees of freedom (DOF) exterior ballistic computer models are used in fragment studies to calculate individual trajectories of each fragment based on drag coefficient and the projected (presented) area in the direction of velocity of center of mass. The expectation of a randomly distributed projected area is commonly used for fragments that tumble (random rotation) during flight. We forecast a model where the expected drag coefficient is dependent of shape and Mach number. Rotation or tumbling only affects the expected projected area. Models of projected areas during tumbling and rotation are presented. An examination of the data by McCleskey (1988) indicates that the volume of the fragment to the power of 2/3 is a better parameter to characterize the drag coefficient of the fragments than the maximum projected area. Hydrocode simulations are used to verify results and to study projected area and drag coefficient of fragments