Application of a new discreet form of Gauss’ theorem for measuring volume

Volume measurements are useful in many branches of science and medicine. They are usually accomplished by acquiring a sequence of cross sectional images through the object using an appropriate scanning modality, for example x-ray computed tomography (CT), magnetic resonance (MR) or ultrasound (US). In the cases of CT and MR, a dividing cubes algorithm can be used to describe the surface as a triangle mesh. However, such algorithms are not suitable for US data, especially when the image sequence is multiplanar (as it usually is). This problem may be overcome by manually tracing regions of interest (ROIs) on the registered multiplanar images and connecting the points into a triangular mesh. In this paper we describe and evaluate a new discreet form of Gauss’ theorem which enables the calculation of the volume of any enclosed surface described by a triangular mesh. The volume is calculated by summing the vector product of the centroid, area and normal of each surface triangle. The algorithm was tested on computer-generated objects, US-scanned balloons, livers and kidneys and CT-scanned clay rocks. The results, expressed as the mean percentage difference ± one standard deviation were 1.2 ± 2.3, 5.5 ± 4.7, 3.0 ± 3.2 and −1.2 ± 3.2% for balloons, livers, kidneys and rocks respectively. The results compare favourably with other volume estimation methods such as planimetry and tetrahedral decomposition.

Exploring the fundamental theorem of arithmetic in excel 2007

This paper discusses how fundamentals of number theory, such as unique prime factorization and greatest common divisor can be made accessible to secondary school students through spreadsheets. In addition, the three basic multiplicative functions of number theory are defined and illustrated through a spreadsheet environment. Primes are defined simply as those natural numbers with just two divisors. One focus of the paper is to show the ease with which spreadsheets can be used to introduce students to some basics of elementary number theory. Complete instructions are given to build a spreadsheet to enable the user to input a positive integer, either with a slider or manually, and see the prime decomposition. The spreadsheet environment allows students to observe patterns, gain structural insight, form and test conjectures, and solve problems in elementary number theory.

The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

Population Monte Carlo Algorithm in High Dimensions

The population Monte Carlo algorithm is an iterative importance sampling scheme for solving static problems. We examine the population Monte Carlo algorithm in a simplified setting, a single step of the general algorithm, and study a fundamental problem that occurs in applying importance sampling to high-dimensional problem. The precision of the computed estimate from the simplified setting is measured by the asymptotic variance of estimate under conditions on the importance function. We demonstrate the exponential growth of the asymptotic variance with the dimension and show that the optimal covariance matrix for the importance function can be estimated in special cases.

Elliptic curves, group law, and efficient computation

This thesis is about the derivation of the addition law on an arbitrary elliptic curve and efficiently adding points on this elliptic curve using the derived addition law. The outcomes of this research guarantee practical speedups in higher level operations which depend on point additions. In particular, the contributions immediately find applications in cryptology. Mastered by the 19th century mathematicians, the study of the theory of elliptic curves has been active for decades. Elliptic curves over finite fields made their way into public key cryptography in late 1980’s with independent proposals by Miller [Mil86] and Koblitz [Kob87]. Elliptic Curve Cryptography (ECC), following Miller’s and Koblitz’s proposals, employs the group of rational points on an elliptic curve in building discrete logarithm based public key cryptosystems. Starting from late 1990’s, the emergence of the ECC market has boosted the research in computational aspects of elliptic curves. This thesis falls into this same area of research where the main aim is to speed up the additions of rational points on an arbitrary elliptic curve (over a field of large characteristic). The outcomes of this work can be used to speed up applications which are based on elliptic curves, including cryptographic applications in ECC. The aforementioned goals of this thesis are achieved in five main steps. As the first step, this thesis brings together several algebraic tools in order to derive the unique group law of an elliptic curve. This step also includes an investigation of recent computer algebra packages relating to their capabilities. Although the group law is unique, its evaluation can be performed using abundant (in fact infinitely many) formulae. As the second step, this thesis progresses the finding of the best formulae for efficient addition of points. In the third step, the group law is stated explicitly by handling all possible summands. The fourth step presents the algorithms to be used for efficient point additions. In the fifth and final step, optimized software implementations of the proposed algorithms are presented in order to show that theoretical speedups of step four can be practically obtained. In each of the five steps, this thesis focuses on five forms of elliptic curves over finite fields of large characteristic. A list of these forms and their defining equations are given as follows: (a) Short Weierstrass form, y2 = x3 + ax + b, (b) Extended Jacobi quartic form, y2 = dx4 + 2ax2 + 1, (c) Twisted Hessian form, ax3 + y3 + 1 = dxy, (d) Twisted Edwards form, ax2 + y2 = 1 + dx2y2, (e) Twisted Jacobi intersection form, bs2 + c2 = 1, as2 + d2 = 1, These forms are the most promising candidates for efficient computations and thus considered in this work. Nevertheless, the methods employed in this thesis are capable of handling arbitrary elliptic curves. From a high level point of view, the following outcomes are achieved in this thesis. - Related literature results are brought together and further revisited. For most of the cases several missed formulae, algorithms, and efficient point representations are discovered. - Analogies are made among all studied forms. For instance, it is shown that two sets of affine addition formulae are sufficient to cover all possible affine inputs as long as the output is also an affine point in any of these forms. In the literature, many special cases, especially interactions with points at infinity were omitted from discussion. This thesis handles all of the possibilities. - Several new point doubling/addition formulae and algorithms are introduced, which are more efficient than the existing alternatives in the literature. Most notably, the speed of extended Jacobi quartic, twisted Edwards, and Jacobi intersection forms are improved. New unified addition formulae are proposed for short Weierstrass form. New coordinate systems are studied for the first time. - An optimized implementation is developed using a combination of generic x86-64 assembly instructions and the plain C language. The practical advantages of the proposed algorithms are supported by computer experiments. - All formulae, presented in the body of this thesis, are checked for correctness using computer algebra scripts together with details on register allocations.

The effect of red blood cell orientation on the electrical impedance of pulsatile blood with implications for impedance cardiography

Impedance cardiography is an application of bioimpedance analysis primarily used in a research setting to determine cardiac output. It is a non invasive technique that measures the change in the impedance of the thorax which is attributed to the ejection of a volume of blood from the heart. The cardiac output is calculated from the measured impedance using the parallel conductor theory and a constant value for the resistivity of blood. However, the resistivity of blood has been shown to be velocity dependent due to changes in the orientation of red blood cells induced by changing shear forces during flow. The overall goal of this thesis was to study the effect that flow deviations have on the electrical impedance of blood, both experimentally and theoretically, and to apply the results to a clinical setting. The resistivity of stationary blood is isotropic as the red blood cells are randomly orientated due to Brownian motion. In the case of blood flowing through rigid tubes, the resistivity is anisotropic due to the biconcave discoidal shape and orientation of the cells. The generation of shear forces across the width of the tube during flow causes the cells to align with the minimal cross sectional area facing the direction of flow. This is in order to minimise the shear stress experienced by the cells. This in turn results in a larger cross sectional area of plasma and a reduction in the resistivity of the blood as the flow increases. Understanding the contribution of this effect on the thoracic impedance change is a vital step in achieving clinical acceptance of impedance cardiography. Published literature investigates the resistivity variations for constant blood flow. In this case, the shear forces are constant and the impedance remains constant during flow at a magnitude which is less than that for stationary blood. The research presented in this thesis, however, investigates the variations in resistivity of blood during pulsataile flow through rigid tubes and the relationship between impedance, velocity and acceleration. Using rigid tubes isolates the impedance change to variations associated with changes in cell orientation only. The implications of red blood cell orientation changes for clinical impedance cardiography were also explored. This was achieved through measurement and analysis of the experimental impedance of pulsatile blood flowing through rigid tubes in a mock circulatory system. A novel theoretical model including cell orientation dynamics was developed for the impedance of pulsatile blood through rigid tubes. The impedance of flowing blood was theoretically calculated using analytical methods for flow through straight tubes and the numerical Lattice Boltzmann method for flow through complex geometries such as aortic valve stenosis. The result of the analytical theoretical model was compared to the experimental impedance measurements through rigid tubes. The impedance calculated for flow through a stenosis using the Lattice Boltzmann method provides results for comparison with impedance cardiography measurements collected as part of a pilot clinical trial to assess the suitability of using bioimpedance techniques to assess the presence of aortic stenosis. The experimental and theoretical impedance of blood was shown to inversely follow the blood velocity during pulsatile flow with a correlation of -0.72 and -0.74 respectively. The results for both the experimental and theoretical investigations demonstrate that the acceleration of the blood is an important factor in determining the impedance, in addition to the velocity. During acceleration, the relationship between impedance and velocity is linear (r2 = 0.98, experimental and r2 = 0.94, theoretical). The relationship between the impedance and velocity during the deceleration phase is characterised by a time decay constant, ô , ranging from 10 to 50 s. The high level of agreement between the experimental and theoretically modelled impedance demonstrates the accuracy of the model developed here. An increase in the haematocrit of the blood resulted in an increase in the magnitude of the impedance change due to changes in the orientation of red blood cells. The time decay constant was shown to decrease linearly with the haematocrit for both experimental and theoretical results, although the slope of this decrease was larger in the experimental case. The radius of the tube influences the experimental and theoretical impedance given the same velocity of flow. However, when the velocity was divided by the radius of the tube (labelled the reduced average velocity) the impedance response was the same for two experimental tubes with equivalent reduced average velocity but with different radii. The temperature of the blood was also shown to affect the impedance with the impedance decreasing as the temperature increased. These results are the first published for the impedance of pulsatile blood. The experimental impedance change measured orthogonal to the direction of flow is in the opposite direction to that measured in the direction of flow. These results indicate that the impedance of blood flowing through rigid cylindrical tubes is axisymmetric along the radius. This has not previously been verified experimentally. Time frequency analysis of the experimental results demonstrated that the measured impedance contains the same frequency components occuring at the same time point in the cycle as the velocity signal contains. This suggests that the impedance contains many of the fluctuations of the velocity signal. Application of a theoretical steady flow model to pulsatile flow presented here has verified that the steady flow model is not adequate in calculating the impedance of pulsatile blood flow. The success of the new theoretical model over the steady flow model demonstrates that the velocity profile is important in determining the impedance of pulsatile blood. The clinical application of the impedance of blood flow through a stenosis was theoretically modelled using the Lattice Boltzman method (LBM) for fluid flow through complex geometeries. The impedance of blood exiting a narrow orifice was calculated for varying degrees of stenosis. Clincial impedance cardiography measurements were also recorded for both aortic valvular stenosis patients (n = 4) and control subjects (n = 4) with structurally normal hearts. This pilot trial was used to corroborate the results of the LBM. Results from both investigations showed that the decay time constant for impedance has potential in the assessment of aortic valve stenosis. In the theoretically modelled case (LBM results), the decay time constant increased with an increase in the degree of stenosis. The clinical results also showed a statistically significant difference in time decay constant between control and test subjects (P = 0.03). The time decay constant calculated for test subjects (ô = 180 - 250 s) is consistently larger than that determined for control subjects (ô = 50 - 130 s). This difference is thought to be due to difference in the orientation response of the cells as blood flows through the stenosis. Such a non-invasive technique using the time decay constant for screening of aortic stenosis provides additional information to that currently given by impedance cardiography techniques and improves the value of the device to practitioners. However, the results still need to be verified in a larger study. While impedance cardiography has not been widely adopted clinically, it is research such as this that will enable future acceptance of the method.

Shakedown stress analysis of insulated rail joint (IRJ)

A finite element numerical simulation is carried out to examine stress distributions on railhead in the cicinity of the endpost of an insulated rail joint. The contact patch and pressure distribution are considered using modified Hertzian simulation. A combined elasto-plastic material modelling available in Abaqus is employed in the simulation. A dynamic load factor of 1.21 is considered in modelling for the wheel load based on a previous study as part of this on going research. Shakedown theorem is employed in this study. A peak pressure load which is above the shakedown limit is determined as input load. As a result, a progressive damage in the railhead has been captured as depicted in the equivalent plastic strain plot.

Elasto-plastic stress analysis of an insulated rail joint (IRJ) with a loading below shakedown limit

A finite element numerical simulation is carried out to examine stress distributions on railhead in the vicinity of the endpost of a insulated rail joint. The contact patch and pressure distribution are considered using modified Hertzian formulation. A combined elasto-plastic material modelling available in Abaqus is employed in the simulation. A dynamic load factor of 1.21 is considered in modelling for the wheel load based on a previous study as part of this on going research. Shakedown theorem is employed in this study. A peak pressure load which is above the shakedown limit is determined as input load. As a result, a progressive damage in the railhead has been captured as depicted in the equivalent plastic strain plot.

A Stochastic View of Optimal Regret through Minimax Duality

We study the regret of optimal strategies for online convex optimization games. Using von Neumann's minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary's action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen's inequality for a concave functional--the minimizer over the player's actions of expected loss--defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary. Peter L. Bartlett, Alexander Rakhlin

Corrigendum to “Shifting : one-inclusion mistake bounds and sample compression” [J. Comput. System Sci. 75 (1) (2009) 37–59]

H. Simon and B. Szörényi have found an error in the proof of Theorem 52 of “Shifting: One-inclusion mistake bounds and sample compression”, Rubinstein et al. (2009). In this note we provide a corrected proof of a slightly weakened version of this theorem. Our new bound on the density of one-inclusion hypergraphs is again in terms of the capacity of the multilabel concept class. Simon and Szörényi have recently proved an alternate result in Simon and Szörényi (2009).