Foam-like phantoms for comparing tomography algorithms

Tomographic algorithms are often compared by evaluating them on certain benchmark datasets. For fair comparison, these datasets should ideally (i) be challenging to reconstruct, (ii) be representative of typical tomographic experiments, (iii) be flexible to allow for different acquisition modes, and (iv) include enough samples to allow for comparison of data-driven algorithms. Current approaches often satisfy only some of these requirements, but not all. For example, real-world datasets are typically challenging and representative of a category of experimental examples, but are restricted to the acquisition mode that was used in the experiment and are often limited in the number of samples. Mathematical phantoms are often flexible and can sometimes produce enough samples for data-driven approaches, but can be relatively easy to reconstruct and are often not representative of typical scanned objects. In this paper, we present a family of foam-like mathematical phantoms that aims to satisfy all four requirements simultaneously. The phantoms consist of foam-like structures with more than 100000 features, making them challenging to reconstruct and representative of common tomography samples. Because the phantoms are computer-generated, varying acquisition modes and experimental conditions can be simulated. An effectively unlimited number of random variations of the phantoms can be generated, making them suitable for data-driven approaches. We give a formal mathematical definition of the foam-like phantoms, and explain how they can be generated and used in virtual tomographic experiments in a computationally efficient way. In addition, several 4D extensions of the 3D phantoms are given, enabling comparisons of algorithms for dynamic tomography. Finally, example phantoms and tomographic datasets are given, showing that the phantoms can be effectively used to make fair and informative comparisons between tomography algorithms.

An Analysis of Zeno’s Paradox and Non-measurable Sets Based on Dialectical Infinity

The Problem of Continuity and Discreteness is the basic problem of philosophy and mathematics. For a long time, there is no clear understanding of this problem, which leads to the stagnation of the problem, because the essence of the problem is a problem of finity and infinity. The essence of the philosophical thought on which the mathematical definition of “line segment is composed of dots” is the idea of actual infinity, and geometric dot is equivalent to algebraic zero in terms of measure properties. In view of the above contradictions, this paper presents two solutions satisfying both the philosophical and mathematical circles based on the view of dialectical infinity, and the authors make a deep analysis of Zeno’s paradox and the non-measurable set based on both solutions.

Mathematical Incorrectnes of So Called Higuchi‘S Fractal Dimension

So called Higuchi’s method of fractal dimension estimation is widely used and the term Higuchi’s fractal dimension even occurs in many publications. This paper deals with this method from mathematical point of view. Terms distance and dimension and its basic properties are explained and Higuchi’s dimension according the original source is defined. Definition of Higuchi’s dimension was comparated with mathematical definition of the distance and dimension. It is showed, that the definition of the Higuchi’s dimension does not satisfy axioms of distance and dimension. So called Higuchi’s method and Higuchi’s dimension are mathematically incorrect. Therefore, all results achieved by this method are scientifically unreliable.

Vectorial integer bootstrapping: flexible integer estimation with application to GNSS

In this contribution, we extend the principle of integer bootstrapping (IB) to a vectorial form (VIB). The mathematical definition of the class of VIB-estimators is introduced together with their pull-in regions and other properties such as probability bounds and success rate approximations. The vectorial formulation allows sequential block-by-block processing of the ambiguities based on a user-chosen partitioning. In this way, flexibility is created, where for specific choices of partitioning, tailored VIB-estimators can be designed. This wide range of possibilities is discussed, supported by numerical simulations and analytical examples. Further guidelines are provided, as well as the possible extension to other classes of estimators.

Mathematics education undergraduates’ personal definitions of the notion of angle of contiguity in Kinematics

The concept of a mathematical definition causes severe difficulties among students during problem solving and proving activities. Students’ difficulties with the use of mathematical definitions often arise from the fact that students are often given those definitions instead of constructing them. With the aim of developing an understanding of the kinds of student teachers evoked concept images of the notion of angle of contiguity, a qualitative case study was conducted at one state university in Zimbabwe. Purposive sampling was used to select 28 mathematics undergraduate student teachers who responded to a test item. Qualitative data analysis was guided by ideas drawn from the theoretical framework of Abstraction in Context and idea of imperative features of a mathematical definition. Student teachers written responses revealed that student teachers personal concept definitions consisted of ambiguous and irrelevant formulations that did not capture the essence of the idea of the angle of contiguity. In some cases their responses were not consistent with the definition of the angle of contiguity. Although there were a few instances of adequate descriptions of the concept, (8 out of 32) these and the inadequate descriptors elicited can contribute significantly towards efforts intended to improve mathematics instruction. Improved mathematics instruction will lead to enhanced conceptualizations of mathematics concepts.

INTEREST RATES SENSITIVITY ARBITRAGE – THEORY AND PRACTICAL ASSESMENT FOR FINANCIAL MARKET TRADING

Purpose – Nowadays popular algorithmic trading uses many strategies which are algoritmizable and promise profitability. This research assess if it is possible successfully use interest rates sensitivity arbitrage in bond portfolio (also known as convexity arbitrage) in financial praxis. This arbitrage is sparsely described in literature and an assessment about its practical success is missing. Research methodology – Methodology steps: mathematical definition of given arbitrage; construction of sufficient portfolio; backtesting on USD zero-coupon curves. Portfolio of two bonds is constructed (theoretically and practically) to have the same Macaulay duration and price, but a different convexity at certain YTM point. Therefore, being long the first bond while shorting the second (of higher convexity) would result in a market-directional bet for parallel zero-coupon yield curve shifts. Findings – To construct practically the portfolio which is sufficient for the convexity arbitrage could be unrealistic on markets with low liquidity; the presumptions necessary to practically succeed are not fulfilled enough to ensure the arbitrage is profitable. Research limitations – The backtesting is limited to USD market, testing other markets is recommended, but different result is not expected. Practical implications – The research helps practitioners considering this strategy for its implementation to algorithmic trading. Originality/Value – New important results for financial practitioners; states that practical and profitable utilization of convexity arbitrage is unrealizable and save costs during implementation of the strategy.