Fermat Metrics

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered.

Optimal Algorithm of Full Focus Array Based on Sparse Matrix

Ultrasonic phased array all-focus imaging algorithm has the advantages of high accuracy and full range dynamic focusing, but the existing imaging time-consuming problem limits its practical industrial application. In order to meet the requirements of real-time imaging and clear and distinguishable image defects in all-focus array ultrasound scanning. In this paper, a sparse optimization model of TFM phased array is established. Based on the Fermat principle, the delay time of each element under two layers of media is calculated, and a two-layer media phased array all-focus imaging algorithm is established, using minimal redundancy. The sparse design of the MRLA is carried out, and the all-focus imaging and sparse all-focus imaging experiments of the two-layer medium are carried out, and the influence of the sparse emission array on the defect quantitative accuracy and the calculation efficiency of the all-focus algorithm is discussed. The final optimized sparse array is compared with the sound field beam pattern of the full array, the smallest redundant array, and the genetic algorithm optimized array, and an all-focus scanning environment is established based on the array. Scanning images can not only obtain the maximum amount of information under minimum redundancy, but also have good sidelobe characteristics. At the same time, the array is sparse, which greatly improves the efficiency of all-focus scanning, and provides a certain reference value for the research of all-focus imaging technology.

Image reconstruction by solving the inverse problem in ultrasonic transmission tomography system

Purpose The purpose of this study is to develop a reconstruction and measurement system for data analysis using ultrasonic transmission tomography. The problem of reconstruction from the projection is encountered in practical implementation, which consists in reconstructing an image that is an estimation of an unknown object from a finite set of projection data. Reconstructive algorithms used in transmission tomography are based on linear mathematical models, which makes it necessary to process non-linear data into estimates for a finite number of projections. The application of transformation methods requires building a mathematical model in which the projection data forming the known and unknown quantities are functions with arguments from a continuous set of real numbers, determining the function describing the unknown quantities sought in the form of inverse relation and adapting it to operate on discrete and noisy data. This was done by designing a tomographic device and proprietary algorithms capable of reconstructing two-dimensional images regardless of the size, shape, location or number of inclusions hidden in the examined object. Design/methodology/approach The application consists of a device and measuring sensors, as well as proprietary algorithms for image reconstruction. Ultrasonic transmission tomography makes it possible to analyse processes occurring in an object without interfering with the examined object. The proposed solution uses algorithms based on ray integration, the Fermat principle and deterministic methods. Two applications were developed, one based on C and implemented on the embedded device, while the other application was made in Matlab. Findings Research shows that ultrasonic transmission tomography provides an effective analysis of tested objects in closed tanks. Research limitations/implications In the presented technique, the use of ultrasonic absorption wave has been limited. Nevertheless, the effectiveness of such a solution has been confirmed. Practical implications The presented solution can be used for research and monitoring of technological processes. Originality/value Author’s tomographic system consisting of a measuring system and image reconstruction algorithms.

Levenberg–Marquardt method for unconstrained optimization

We propose and study the Levenberg–Marquardt method globalized by means of linesearch for unconstrained optimization problems with possibly nonisolated solutions. It is well-recognized that this method is an efficient tool for solving systems of nonlinear equations, especially in the presence of singular and even nonisolated solutions. Customary globalization strategies for the Levenberg–Marquardt method rely on linesearch for the squared Euclidean residual of the equation being solved. In case of unconstrained optimization problem, this equation is formed by putting the gradient of the objective function equal to zero, according to the Fermat principle. However, these globalization strategies are not very adequate in the context of optimization problems, as the corresponding algorithms do not have “preferences” for convergence to minimizers, maximizers, or any other stationary points. To that end, in this work we considers a different technique for globalizing convergence of the Levenberg–Marquardt method, employing linesearch for the objective function of the original problem. We demonstrate that the proposed algorithm possesses reasonable global convergence properties, and preserves high convergence rate of the Levenberg–Marquardt method under weak assumptions.

Kepler’s Ellipse Generated by the Trigonometrically Organized Gravitons

Johannes Kepler made his great breakthrough when he discovered the elliptical path of the planet Mars around the Sun located in one focus of that ellipse (on the 11th October in 1605 in a letter to Fabricius). The first generation of researchers in the 17th century intensively discussed about the possible mechanism needed for the generation of that elliptical orbit and about the function of the empty focus of that ellipse. First generations of researchers proposed an interplay between attractive and repulsive forces that might guide the planet on its elliptical orbit. Isaac Newton made a giant mathematical progress in his Principia and introduced the concept of the attractive gravitational force between the Sun and planets. However, Newton did not propose a possible mechanism behind this attractive force. Albert Einstein in 1915 left the concept of attractive and repulsive forces and introduced his Theory based on the elastic spacetime. In his concept gravity itself became fictitious force and the attraction is explained via the elastic spacetime. In our proposed model we try to re-open the discussion of Old Masters on the existence of attractive and repulsive forces. The guiding principle for our trigonometrical model is the generation of the ellipse discovered by one of the last ancient Greek mathematicians – Anthemius of Tralles – who generated the ellipse by the so-called gardener’s method (one string and two pins fixed to the foci of that ellipse). Frans van Schooten in 1657 invented a series of original simple mechanisms for generating ellipses, hyperbolas, and parabolas. Schooten’s antiparallelogram might simulate the interplay of attractive and repulsive forces creating the elliptical path. We propose a model with trigonometrically organized Solar and planet gravitons. In this model the Solar and planet gravitons are reflected and refracted in predetermined directions so that their joint momentum transferred on the planet atoms guides the planet on an elliptical path around the Sun. At this stage we cannot directly measure the gravitons but we can use the analogy with behavior of photons. We propose to observe paths of photons emitted from one focus of the ellipse towards the QUARTER-silvered elliptical mirror. 1/4 of photons will be reflected towards to the second empty focus and the ¾ of photons might be reflected and refracted into the trigonometrically expected directions. (Until now we have experimental data only for the FULLY–silvered elliptical mirrors). The observed behavior of photons with the quarter-silvered elliptic mirror might support this concept or to exclude this model as a wrong model. The quantitative values of attractive and repulsive forces could be found from the well-known geometrical properties of the ellipse. The characteristic lengths of distances will be inserted into the great formula of Isaac Newton - the inverse square law. (In order to explain some orbit anomalies, we can use Paul Gerber’s formula derived for the Pierre Fermat principle). We have found that the Kant’s ellipse rotating on the Keppler’s ellipse might express the co-operation of attractive and repulsive forces to guide the planet on its elliptic path. Finally, we have derived a new formula inspired by Bradwardine - Newton - Tan - Milgrom that might contribute to the MOND gravitational model. We have found that the Kepler ellipse is the very elegant curve that might still keep some hidden secrets waiting for our future research.