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

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

The covering radius problem for sets of perfect matchings

Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lov\'asz local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. This is an analogue of an extremal result vis-\'a-vis the covering radius of sets of permutations, which was studied by Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}). We also conclude with a conjecture of a more general problem in hypergraph matchings.Comment: 10 page

Stress dependent thermal pressurization of a fluid-saturated rock

Temperature increase in saturated porous materials under undrained conditions leads to thermal pressurization of the pore fluid due to the discrepancy between the thermal expansion coefficients of the pore fluid and of the solid matrix. This increase in the pore fluid pressure induces a reduction of the effective mean stress and can lead to shear failure or hydraulic fracturing. The equations governing the phenomenon of thermal pressurization are presented and this phenomenon is studied experimentally for a saturated granular rock in an undrained heating test under constant isotropic stress. Careful analysis of the effect of mechanical and thermal deformation of the drainage and pressure measurement system is performed and a correction of the measured pore pressure is introduced. The test results are modelled using a non-linear thermo-poro-elastic constitutive model of the granular rock with emphasis on the stress-dependent character of the rock compressibility. The effects of stress and temperature on thermal pressurization observed in the tests are correctly reproduced by the model

One-dimensional modelling of pulse wave propagation in human airway bifurcations in space-time variables

Airflow in the respiratory system is complicated as it goes through various regions with different geometries and mechanical properties. Three-dimensional (3-D) simulations are typically limited to local areas of the system because of their high computational cost. On the other hand, the one-dimensional (1-D) equations of flow in compliant tubes offer a good compromise between accuracy and computational cost when a global assessment of airflow in the system is required. The aim of the current study is to apply the 1-D formulation in space and time variables to study the propagation of a pulse wave in human airways; first in a simple system composed of just one bifurcation, trachea-main bronchi, according to the symmetrical Weibel model. Then extending the system to include a further generation, the bronchi branches. Pulse waveforms carry information about the functionality and morphology of the respiratory system and the 1-D modelling, in terms of space and time variables, represents an innovative approach for respiratory response interpretation. 1-D modelling in space-time variables has been extensively applied to simulate blood pressure and flow in the cardiovascular system. This work represents the first attempt to apply this formulation to study pulse waveforms in the human bronchial tree

The Complementary Role of Exports and R&D Investments as Sources of Productivity Growth

This paper examines two potential channels of knowledge acquisition that underlie firm productivity growth in the Taiwanese electronics industry: participation in the export market and investments in R&D and/or worker training. We focus on the argument that a firm's own investments in R&D are necessary for the firm to assimilate knowledge or expertise gained from foreign contacts and thus are an important component of the process of learning-by-exporting. Firm-level panel data from 1986, 1991, and 1996 is used to investigate a firm's decision to invest in these two activities and to assess the effects of these investments on the firm's future total factor productivity. The empirical model consists of four equations. The firm's decisions to export and invest in R&D and/or worker training are modeled with a bivariate probit model that recognizes the interdependence of the decisions. We then estimate how participation in these investment activities alters the firm's future productivity trajectory while controlling for the potential selection bias introduced by endogenous firm exit. The primary empirical findings are that, on average, firms that export but do not invest in R&D and/or worker training have significantly higher future productivity than firms that do not participate in either activity. In addition, firms that export and invest in R&D and/or worker training have significantly higher future productivity than firms that only export. These findings are consistent with the hypothesis that export experience is an important source of productivity growth for Taiwanese firms and that firm investments in R&D and worker training facilitate their ability to benefit from their exposure to the export market.

Productivity and the Decision to Export: Micro Evidence from Taiwan and South Korea

While there is widespread empirical evidence indicating exporting producers have higher productivity than nonexporters, the mechanisms that generate this pattern are less clear. One view is that exporters acquire knowledge of new production methods, inputs, and product designs from their international contacts, and this learning results in higher productivity for exporters relative to their more insulated domestic counterparts. Alternatively, the higher productivity of exporters may simply reflect the self-selection of more efficient producers into a highly competitive export market. In this paper we use micro data collected in the manufacturing censuses in South Korea and Taiwan to study the linkages between a producer's total factor productivity and choice to participate in the export market. We find differences between the countries in the importance of selection and learning forces. In Taiwan, transitions of firms in and out of the export market reflect systematic variations in productivity as predicted by self-selection models. Firms with higher productivity, ex ante, tend to enter the export market and exporters with low productivity tend to exit. Moreover, in several industries, entry into the export market is followed by relative productivity improvements, a result consistent with learning-by-exporting forces. In South Korea, the evidence of self-selection on the basis of productivity is much weaker. In addition, unlike Taiwan, we find no significant productivity changes following entry or exit from the export market that are consistent with learning from exporting. Comparison of the two countries suggests that in Korea factors other than production efficiency play a more prominent role as determinants determinants of the export decision.