The Tur\'{a}n number and probabilistic combinatorics

In this short expository article, we describe a mathematical tool called the probabilistic method, and illustrate its elegance and beauty through proving a few well-known results. Particularly, we give an unconventional probabilistic proof of a classical theorem concerning the Tur\'{a}n number $T(n,k,l)$. Surprisingly, this proof cannot be found in existing literature.Comment: 5 pages; to appear in Amer. Math. Monthly 201

Compositional controls on melting and dissolving a salt into a ternary melt

We explore theoretically the controls on dissolution of salt A, in an undersaturated brine of salts A and B. We show that, as the concentration of B increases, the dissolution rate of A decreases, for brine of given temperature. We also show that there is a sharper decrease in dissolution rate with increasing concentration, for concentrations of B above a critical value, where B limits the equilibrium concentration. We explore the implications of the predictions for dissolution of KCl or NaCl, by a mixed brine of NaCl and KCl, a common reaction that may arise in dissolution of evaporites. We predict that, with mixed-composition brine, KCl crystals dissolve more rapidly than NaCl crystals, unless the (far-field) brine is nearly saturated in KCl. We also predict that the dissolution rate of these salts is largely independent of fluid temperature and is controlled by compositional diffusion

Automating the determination of wave speed using the pu-loop method

The PU-loop (pressure-velocity loop) is a method for determining wave speed and relies on the linear relationship between the pressure and velocity in the absence of reflected waves. This linearity of the PU-loop during early systole, which is directly related to wave speed, has always been established by eye. This paper presents a new technique that establishes this linearity and thus determining wave speed online. Pressure and flow were measured in the ascending aorta of 11 anesthetised dogs. The slope of the PU-loop, indicating wave speed was determined by eye and by using the new technique. The difference between the slopes of the two methods is in the order of 3%. The new technique is convenient and allows for the online assessment of wave speed, which could be used as a bedside tool for the assessment of arterial compliance

Diffusion-controlled dissolution of a binary solid into a ternary liquid with partially-molten zone formation

We build a theoretical model of equilibrium dissolution of a homogeneous, solid mixture of two salts A and B, KCl and NaCl being used as the type example, into an aqueous solution of the two salts, with diffusive transport. We find that there are two sharp dissolution fronts, separating fluid, a partially molten zone containing a single solid and mixed solid. The phase change happens almost entirely at the two sharp fronts. In equilibrium, the leading front exhibits a small amount of precipitation of NaCl, simultaneous with complete dissolution of KCl. There is a unique surface in the space of far-field fluid KCl concentration, far-field fluid NaCl concentration and solid composition, dividing conditions where NaCl is the solid in the partially molten zone, from conditions where KCl is the solid in the partially molten zone. The movement rates of the dissolution fronts decrease as the concentration of either salt in the far-field fluid is increased. The movement rates of the dissolution fronts increase as either far-field temperature is increased, but this effect is smaller than that of concentration. In most circumstances, the dissolution front for a given salt moves more slowly, the more of that salt is present in the original solid, although the mass dissolution rate is not greatly affected by the solid composition

Determination of wave speed and distensibility of flexible tubes using diameter and velocity

It is well accepted that wave speed is one of the key factors describing wave propagation in arteries. Local wave speed is directly related to the mechanical properties of the arterial wall and is widely used to determine the arterial distensibility . Several methods have been proposed for determining wave speed in arteries, such as foot-to-foot and PU-loop methods. In this paper, we suggest a new method for the determination of wave speed and wall distensibility, using noninvasive measurements. The theoretical foundation of this method is based on the 1-D conservation of mass and momentum equations of flow in flexible tubes. We simultaneously measured pressure, diameter and velocity at the same site, sequentially along silicon and latex tubes which are 1 m in length and of different diameters. We compared the results of the new method, ln(D)U-loop, with those determined by the PU-loop method. Wave speeds determined by both methods are comparable, although wave speeds determined by the new technique are slightly smaller than those determined by PU-loop method. We also compared distensibility calculated by the new method with those calculated using the traditional method (Dt), Dt = 3DdP/AdA, where A and dA are the cross sectional area and its change respectively, and dP is the change in pressure. The results of both methods are in agreement. We conclude that the new technique has the advantage of using only noninvasive parameters which is of clinical relevance

In Diamond Health

In contrast to the sparkling gemstone, diamond-like carbon (DLC) is a thin film coating that is dense, inert, low friction and hard wearing. Interdisciplinary research, involving materials scientists, physicists, mechanical engineers, biomedical specialists and clinicians, is continuously expanding the potential and applications of DLC and enhanced carbon-based materials in the medical sector

A new approach to investigate wave dissipation in viscoelastic tubes: Application of wave intensity analysis

Wave dissipation in elastic and viscoelastic medium has been investigated extensively in the frequency domain. The aim of this study is to examine the pattern of wave dissipation in the time-domain using wave intensity analysis. A single semi-sinusoidal pulse was generated in 8 mm and 16 mm diameter tubes; each is of 200 cm in length. Pressure and flow measurements were taken at intervals of 5 cm along the tube. In order to examine the effect of the wall mechanical properties on wave dissipation, we also modified the wall of the 16 mm tube; a thread of strong cotton was wound with a pitch of approximately 30deg around the circumference of the tube in the longitudinal direction. The separated forward pressure, wave intensity and wave energy were calculated using wave intensity analysis. The amplitudes of the forward pressure wave, wave intensity and wave energy dissipated exponentially with distance. In the 8 mm diameter tube, the dissipation of forward pressure, wave intensity and wave energy were greater than those in 16 mm tube. For the same sized of tube, there was no significant difference in the dissipation of forward pressure, wave intensity and wave energy between the modified and normal wall tubes. It is concluded that the size of tube has a significant effect on the wave dissipation but the mechanical properties of the wall do not have a discernable effect on wave dissipatio

Determination of wave intensity in flexible tubes using measured diameter and velocity

Wave intensity (WI) is a hemodynamics index, which is the product of changes in pressure and velocity across the wave-front. Wave Intensity Analysis, which is a time domain technique allows for the separation of running waves into their forward and backward directions and traditionally uses the measured pressure and velocity waveforms. However, due to the possible difficulty in obtaining reliable pressure waveforms non-invasively, investigating the use of wall displacement instead of pressure signals in calculating WI may have clinical merits. In this paper, we developed an algorithm in which we use the measured diameter of flexible tube's wall and flow velocity to separate the velocity waveform into its forward and backward directions. The new algorithm is also used to separate wave intensity into its forward and backward directions. In vitro experiments were carried out in two sized flexible tubes, 12 mm and 16 mm in diameters, each is of 2 m in length. Pressure, velocity and diameter were taken at three measuring sites. A semi-sinusoidal wave was generated using a piston pump, which ejected 40 cc water into each tube. The results show that separated wave intensity into the forward and backward directions of the new algorithm using the measured diameter and velocity are almost identical in shape to those traditionally using the measured pressure and velocity. We conclude that the new algorithm presented in this work, could have clinical advantages since the required information can be obtained non-invasively