Improved estimation of Fourier coefficients for ill-posed inverse problems

In this dissertation we present and solve an ill-posed inverse problem which involves reproducing a function f(x) or its Fourier coefficients from the observed values of the function. The observations of the f(x) are made at n equidistant points on the unit interval with p observations being made at each point. The observations are effected by a random error with a known distribution. First of all we present a very simple estimator for the Fourier coefficients of f(x). Then we present an iteration algorithm for improving the estimator for the Fourier coefficients. We show that the improved estimator we use is a simplified and improved version of the Maximum Likelihood Estimator. Second, we introduce the mean squared error (MSE) for the estimators, which is the main measure of estimator performance. We show that a singly iterated estimator has a smaller MSE then a non-iterated estimator and a multiply iterated estimator has a smaller MSE then a singly iterated estimator. We also prove that the errors in estimating the Fourier Coefficients by the singly and multiply improved methods are normally distributed. Third, we prove a theorem showing that as the sample size goes to infinity, the MSE of our estimator asymptotically approaches the theoretical minimum. That shows that our results are theoretically the best possible results. Fourth, we perform simulations which numerically approximate MSE for a given set of f, error distributions, as well as the number of observation points. We approximate the MSE for the non-iterated error coefficient approximation as well as the singly iterated and multiply iterated ones. We show that indeed the MSE decreases with each iteration. We also plot an error histogram in each case showing that the errors are normally distributed. Finally, we look at some ways in which our problem can be expanded. Possible expansions include working on the problem in multiple dimensions, taking measurements of f at random points, or both of the above

Ramjet Acceleration of Microscopic Black Holes within Stellar Material

In this work, we present a case that Microscopic Black Holes (MBH) of mass 1016kg–3×1019kg experience acceleration as they move within stellar material at low velocities. The accelerating forces are caused by the fact that an MBH moving through stellar material leaves a trail of hot rarefied gas. The rarefied gas behind an MBH exerts a lower gravitational force on the MBH than the dense gas in front of it. The accelerating forces exceed the gravitational drag forces when MBH moves at Mach number M<M0<1. The equilibrium Mach number M0 depends on MBH mass and stellar material characteristics. Our calculations open the possibility of MBH orbiting within stars including the Sun at Mach number M0. At the end of this work, we list some unresolved problems which result from our calculations

Drone Launched Short Range Rockets

A concept of drone launched short range rockets (DLSRR) is presented. A drone or an aircraft rises DLSRR to a release altitude of up to 20 km. At the release altitude, the drone or an aircraft is moving at a velocity of up to 700 m/s and a steep angle of up to 68&deg; to the horizontal. After DLSRRs are released, their motors start firing. DLSRRs use slow burning motors to gain altitude and velocity. At the apogee of their flight, DLSRRs release projectiles which fly to the target and strike it at high impact velocity. The projectiles reach a target at ranges of up to 442 km and impact velocities up to 1.88 km/s. We show that a rocket launched at high altitude and high initial velocity does not need expensive thermal protection to survive ascent. Delivery of munitions to target by DLSRRs should be much less expensive than delivery by a conventional rocket. Even though delivery of munitions by bomber aircraft is even less expensive, a bomber needs to fly close to the target, while a DLSRR carrier releases the rockets from a distance of at least 200 km from the target. All parameters of DLSRRs, and their trajectories are calculated based on theoretical (mechanical and thermodynamical) analysis and on several MatLab programs