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Concept# Closed-form expression

Summary

In mathematics, an expression is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (+, −, ×, ÷, and integer powers) and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions . However, the set of basic functions depends on the context.
The closed-form problem arises when new ways are introduced for specifying mathematical objects, such as limits, series and integrals: given an object specified with such tools, a natural problem is to find, if possible, a closed-form expression of this object, that is, an expression of this object in terms of previous ways of specifying it.
Example: roots of polynomials
The quadratic formula
:x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.
is a closed form of the solutions
of the general quadratic equation ax^2+bx+c=0.
More generally, in the context of polynomial equations,

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In this thesis we describe a path integral formalism to evaluate approximations to the probability density function for the location and orientation of one end of a continuum polymer chain at thermodynamic equilibrium with a heat bath. We concentrate on those systems for which the associated energy density is at most quadratic in its variables. Our main motivation is to exploit continuum elastic rod models for the approximate computation of DNA looping probabilities. We first re-derive, for a polymer chain system, an expression for the second order correction term due to quadratic fluctuations about a unique minimal energy configuration. The result, originally stated for a quantum mechanical system by G. Papadopoulos (1975), relies on an elegant algebraic argument that carries over to the real-valued path integrals of interest here. The conclusion is that the appropriate expression can be evaluated in terms of the energy of the minimizer and the inverse square root of the determinant of a matrix satisfying a certain non-linear system of differential equations. We then construct a change of variables, which establishes a mapping between the solutions of the aforementioned non-linear Papadopoulos equations and a matrix satisfying an initial value problem for the classic linear system of Jacobi equations associated with the second variation of the energy functional. This conclusion is trivial if no cross-term is present in the second variation, but ceases to be so otherwise. Cross-terms are always present in the application of rod models to DNA. We therefore can conclude that the second order fluctuation correction term to the probability density function for a chain is always given by the inverse square root of the determinant of a matrix of solutions to the Jacobi equations. We believe this conclusion to be original for the real-valued case when the second-variation involves cross-terms. Similar results are known for quantum mechanical systems, and, in this context, a connection between the so called Van-Hove-Morette determinant, which involves partial derivatives of the classical action with respect to the boundary values of the configuration variable, and the Jacobi determinant have also been established. We next apply the formula described above to the specific context of rods, for which the configuration space is that of framed curves, or curves in R3 × SO(3). An immediate application of our theory is possible if the rod model encompasses bend, twist, stretch and shear. However the constrained case, where the rod is considered to be inextensible and unshearable, is more standard in polymer physics. In this last case, our results are more delicate as the Lagrangian description breaks down, and the Hamiltonian formulation must be invoked. It is known that the unconstrained local minimizers approach constrained minimizers as the coefficients in the shear and extension terms of the energy are sent to infinity. Here we observe that the Hamiltonian form of the unconstrained Jacobi system similarly has a limit, so that the fluctuation correction in the path integral can still be expressed as the square root of the determinant of a matrix solution of a set of Jacobi equations appropriate to the constrained problem. As in reality DNA or biological macromolecules are certainly at least slightly shearable and extensible, the limit of the fluctuation correction is undoubtedly physically appropriate. The above theory provides a computationally highly tractable approach to the estimation of the appropriate probability density functions. For application to sequence-dependent models of DNA the associated systems of equations has non-constant coefficients, which is of little consequence for a numerical treatment, but precludes the possibility of finding closed form expressions. On the other hand the theory also applies to simplified homogeneous models. Accordingly, we conclude by applying our approach in a completely analytic and closed-form way to the computation of the approximate probability density function for a uniform, non-isotropic, intrinsically straight and untwisted rod to form a circular loop.

This thesis was carried out within the framework of a scientific cooperation project entitled “Application of High Power Electromagnetics to Human Safety” developed by the EPFL, the National University of Colombia and Los Andes University, Colombia. The project was funded by the Swiss Agency for Development and Cooperation (SDC) through the EPFL Centre Coopéation & Développement (CODEV). The Scientific Cooperation aimed at the study and development of techniques for the generation of high power electromagnetic signals for the disruption or preemptive activation of Improvised Explosive Devices (IEDs) during humanitarian clearance activities. The results and conclusions of the thesis will be applied to the construction of a resonant radiator, which can be used for securing humanitarian demining operations in Colombia. The thesis is devoted to the analysis of a specific type of resonant radiators known as Switched Oscillators (SWO). An SWO is a radiator constituted by a high voltage charging circuit that drives a quarter-wave transmission line resonator connected to an antenna. An SWO can produce high-amplitude, short duration, electromagnetic fields, with a moderate bandwidth, when compared to the main resonance frequency. The outcome of the thesis can be also be used in electromagnetic compatibility applications, for the production of resonant, high power electromagnetic fields, with the aim of testing the immunity of electronic systems against Intentional Electromagnetic Interference (IEMI) attacks. The thesis is divided in three parts. The first part deals with the electrostatic design of an SWO. A method for producing an optimized design of the electrodes forming the spark gap of the SWO is presented. The method is based on the generation of a curvilinear coordinate space on which the electrodes are conformal to one of the coordinate axis of the space. Laplace equation is solved in the interelectrodic space, obtaining an analytical solution for the electrostatic distribution. Furthermore, using appropriate mathematical manipulations, we derive an analytical expression for the impedance of the transmission line formed by the proposed electrodes. The second part of the thesis is devoted to the analysis of SWOs in the frequency domain. An original analysis approach, based on the chain-parameter technique, is proposed in which the SWO and the connected antenna are described using a two-port network using which a transfer function between the input voltage and the radiated field is established. A closed form expression of the resonance frequency of the SWO is also obtained. The developed technique makes it possible to study the response of an SWO when connected to an arbitrary antenna with a frequency-dependent input impedance. The final part of the thesis presents the construction and test of an SWO prototype. The prototype design is based on the theoretical developments presented in the first two parts of the thesis. The realized SWO is experimentally characterized using different antennas. It is characterized by an input voltage of 30 kV and a resonance frequency of 433 MHz. Radiated electric fields using monopole antennas were in the order of 10 kV/m at a distance of 1.5 m. The prototype is used for testing the validity of the electrodynamic model for the analysis of SWOs connected to frequency dependent antennas. Different monopole antennas connected to the SWO are considered and the radiated fields are measured and compared with theoretical calculations. It is shown that the developed theoretical model is able to reproduce with a good accuracy the behavior of the SWO connected to a frequency dependent antenna.

We study the optimal strategy for a sailboat to reach an upwind island under the hypothesis that the wind direction fluctuates according to a Brownian motion and the wind speed is constant. The work is motivated by a concrete problem which typically arises during sailing regattas, namely finding the best tacking strategy to reach the upwind buoy as quickly as possible. We assume that there is no loss of time when tacking. We first guess an optimal strategy and then we establish its optimality by using the dynamic programming principle. The Hamilton Jacobi Bellmann equation obtained is a parabolic PDE with Neumann boundary conditions. Since it does not admit a closed form solution, the proof of optimality involves an intricate estimate of derivatives of the value function. We explicitly provide the asymptotic shape of the value function. In order to do so, we prove a result on large time behavior for solutions to time dependent parabolic PDE using a coupling argument. In particular, a boat far from the island approaches the island at $\frac{1}{2} + \frac{\sqrt{2}}{\pi} = 95.02\%$ of the boat's speed.