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

Experimental and numerical study of the fragmentation of expanding warhead casings by using different numerical codes and solution techniques

AbstractThere has been increasing interest in numerical simulations of fragmentation of expanding warheads in 3D. Accordingly there is a pressure on developers of leading commercial codes, such as LS-DYNA, AUTODYN and IMPETUS Afea, to implement the reliable fracture models and the efficient solution techniques. The applicability of the Johnson–Cook strength and fracture model is evaluated by comparing the fracture behaviour of an expanding steel casing of a warhead with experiments. The numerical codes and different numerical solution techniques, such as Eulerian, Lagrangian, Smooth particle hydrodynamics (SPH), and the corpuscular models recently implemented in IMPETUS Afea are compared. For the same solution techniques and material models we find that the codes give similar results. The SPH technique and the corpuscular technique are superior to the Eulerian technique and the Lagrangian technique (with erosion) when it is applied to materials that have fluid like behaviour such as the explosive and the tracer. The Eulerian technique gives much larger calculation time and both the Lagrangian and Eulerian techniques seem to give less agreement with our measurements. To more correctly simulate the fracture behaviours of the expanding steel casing, we applied that ductility decreases with strain rate. The phenomena may be explained by the realization of adiabatic shear bands. An implemented node splitting algorithm in IMPETUS Afea seems very promising

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

On the Mathematical Structure for Discrete and Continuous Metric Point Sets

We have studied fundamental properties of continuous and discrete metric point sets. Our focus is pure geometrical objects. We show how geodesic lines and angles can be constructed from an imposed metric even in discrete spaces. Lines in discrete and continuous metric point sets are constructed and compared with Euclid’s five axioms. The angle between two lines is defined. Euclid’s axioms E1 and E2 are sufficient to achieve local angles and to define an infinite space. Axiom E3 is sufficient to define a space with more than one dimension. Axiom E4 is666 J. F. Moxnes and K. Hausken sufficient to define a homogenous space. Axiom E5 is sufficient to define a flat space. We study how the concepts of vector spaces could appear from the metric point set. We have constructed arrows from each point in the metric point set. These arrows can be conceived as lines with a direction. The sum of arrows from each point is constructed algebraically without parallel transport. A method is presented for constructing coordinates. We have constructed coordinates in a metric point space by assuming that the arrow from a specific point o in the metric point space defines a vector space at each point p. We comment on the force concept. Different parallel transports are constructed geometrically. The concepts of tensors and tensor fields are briefly addressed

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