Investigating non-Gaussianity in Cosmic Microwave Background temperature maps using spherical harmonic phases

In the era of precision cosmology, accurate estimation of cosmological parameters is based upon the implicit assumption of the Gaussian nature of Cosmic Microwave Background (CMB) radiation. Therefore, an important scientific question to ask is whether the observed CMB map is consistent with Gaussian prediction. In this work, we extend previous studies based on CMB spherical harmonic phases (SHP) to examine the validity of the hypothesis that the temperature field of the CMB is consistent with a Gaussian random field (GRF). The null hypothesis is that the corresponding CMB SHP are independent and identically distributed in terms of a uniform distribution in the interval [0, 2π] [1,2]. We devise a new model-independent method where we use ordered and non-parametric Rao's statistic, based on sample arc-lengths to comprehensively test uniformity and independence of SHP for a given ℓ mode and independence of nearby ℓ mode SHP. We performed our analysis on the scales limited by spherical harmonic modes ≤ 128, to restrict ourselves to signal-dominated regions. To find the non-uniform or dependent sets of SHP, we calculate the statistic for the data and 10000 Monte Carlo simulated uniformly random sets of SHP and use 0.05 and 0.001 α levels to distinguish between statistically significant and highly significant detections. We first establish the performance of our method using simulated Gaussian, non-Gaussian CMB temperature maps, along with observed non-Gaussian 100 and 143 GHz Planck channel maps. We find that our method, performs efficiently and accurately in detecting phase correlations generated in all of the non-Gaussian simulations and observed foreground contaminated 100 and 143 GHz Planck channel temperature maps. We apply our method on Planck satellite mission's final released CMB temperature anisotropy maps- COMMANDER, SMICA, NILC, and SEVEM along with WMAP 9 year released ILC map. We report that SHP corresponding to some of the m-modes are non-uniform, some of the ℓ mode SHP and neighboring mode pair SHP are correlated in cleaned CMB maps. The detection of non-uniformity or correlation in the SHP indicates the presence of non-Gaussian signals in the foreground minimized CMB maps.

Synthetic Aircraft Trajectories Generated with Multivariate Density Models

Aircraft trajectory generation is a high stakes problem with a wide scope of applications, including collision risk estimation, capacity management and airspace design. Most generation methods focus on optimizing a criterion under constraints to find an optimal path, or on predicting aircraft trajectories. Nevertheless, little in the way of contribution has been made in the field of the artificial generation of random sets of trajectories. This work proposes a new approach to model two-dimensional flows in order to build realistic artificial flight paths. The method has the advantage of being highly intuitive and explainable. Experiments were conducted on go-arounds at Zurich Airport, and the quality of the generated trajectories was evaluated with respect their shape and statistical distribution. The last part of the study explores strategies to extend the work to non-regularly shaped trajectories.

Two-Step Method for Assessing Similarity of Random Sets

The paper concerns a new statistical method for assessing dissimilarity of two random sets based on one realisation of each of them. The method focuses on shapes of the components of the random sets, namely on the curvature of their boundaries together with the ratios of their perimeters and areas. Theoretical background is introduced and then, the method is described, justified by a simulation study and applied to real data of two different types of tissue - mammary cancer and mastopathy.

Approaches to assessing the effectiveness of earthquake predictions using the LURR (load-unload response ratio) method on Sakhalin

A retrospective analysis of the seismicity of Sakhalin from 1997 to 2019 was performed to demonstrate the possibilities of the LURR technique recently in previous our work. The following results were obtained: 84 % of earthquakes (16 out of 19, with M ≥ 5) are predicted, 25% alarms (4 out of 15 predicted areas) were false. This paper proposes an analytical dependence to describe the forecast effectiveness (Ke) for this research. The extremes of Ke were found at the value of the alarm period of 12 and 24 months. At the same time, Ke is significantly higher for the alarm period of 24 months and decreases after a two-year alarm period. Another way to prove the results obtained is the random spatio-temporal distributions of the predicted objects (19 earthquakes with M ≥ 5). 10 such random sets have been assigned to 15 predicted areas, the result shows a significant advantage of a real sample over random ones, and also practically confirms the reliability of the algorithm for using the LURR technique. The methodology and results of this work can serve as practical recommendations for working with the LURR method for seismologists.

Topics in Algorithmic Randomness and Computability Theory

<p>This thesis establishes results in several different areas of computability theory. The first chapter is concerned with algorithmic randomness. A well-known approach to the definition of a random infinite binary sequence is via effective betting strategies. A betting strategy is called integer-valued if it can bet only in integer amounts. We consider integer-valued random sets, which are infinite binary sequences such that no effective integer-valued betting strategy wins arbitrarily much money betting on the bits of the sequence. This is a notion that is much weaker than those normally considered in algorithmic randomness. It is sufficiently weak to allow interesting interactions with topics from classical computability theory, such as genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random sets in terms of standard notions from computability theory. In the second chapter we extend the technique of forcing with bushy trees. We use this to construct an increasing ѡ-sequence 〈an〉 of Turing degrees which forms an initial segment of the Turing degrees, and such that each an₊₁ is diagonally noncomputable relative to an. This shows that the DNR₀ principle of reverse mathematics does not imply the existence of Turing incomparable degrees. In the final chapter, we introduce a new notion of genericity which we call ѡ-change genericity. This lies in between the well-studied notions of 1- and 2-genericity. We give several results about the computational power required to compute these generics, as well as other results which compare and contrast their behaviour with that of 1-generics.</p>

Topics in Algorithmic Randomness and Computability Theory

<p>This thesis establishes results in several different areas of computability theory. The first chapter is concerned with algorithmic randomness. A well-known approach to the definition of a random infinite binary sequence is via effective betting strategies. A betting strategy is called integer-valued if it can bet only in integer amounts. We consider integer-valued random sets, which are infinite binary sequences such that no effective integer-valued betting strategy wins arbitrarily much money betting on the bits of the sequence. This is a notion that is much weaker than those normally considered in algorithmic randomness. It is sufficiently weak to allow interesting interactions with topics from classical computability theory, such as genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random sets in terms of standard notions from computability theory. In the second chapter we extend the technique of forcing with bushy trees. We use this to construct an increasing ѡ-sequence 〈an〉 of Turing degrees which forms an initial segment of the Turing degrees, and such that each an₊₁ is diagonally noncomputable relative to an. This shows that the DNR₀ principle of reverse mathematics does not imply the existence of Turing incomparable degrees. In the final chapter, we introduce a new notion of genericity which we call ѡ-change genericity. This lies in between the well-studied notions of 1- and 2-genericity. We give several results about the computational power required to compute these generics, as well as other results which compare and contrast their behaviour with that of 1-generics.</p>

Set-Valued T-Translative Functions and Their Applications in Finance

A theory for set-valued functions is developed, which are translative with respect to a linear operator. It is shown that such functions cover a wide range of applications, from projections in Hilbert spaces, set-valued quantiles for vector-valued random variables, to scalar or set-valued risk measures in finance with defaultable or nondefaultable securities. Primal, dual, and scalar representation results are given, among them an infimal convolution representation, which is not so well known even in the scalar case. Along the way, new concepts of set-valued lower/upper expectations are introduced and dual representation results are formulated using such expectations. An extension to random sets is discussed at the end. The principal methodology consisted of applying the complete lattice framework of set optimization.