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The vanishing (artificial) viscosity method has played a fundamental role in the theory of classical shock waves, both by providing a mollified limit that identifies the correct physical (Lax admissible) shock waves, and as a guiding principle in the design of numerical difference schemes for simulating shock waves. However, for relativistic fluid flow, the underlying dissipation mechanism based on the Euclidean Laplace operator violates Lorentz invariance (and hence the speed of light bound) -- the fundamental principle of Special Relativity. In this paper we introduce a simple dissipation mechanism for the relativistic Euler equations which is Lorentz invariant and consistent with the laws of Special Relativity. To establish basic consistency of the model for the study of shock waves, we prove existence and decay of Fourier Laplace mode solutions (implying dissipation), and we prove that 1-D shock profiles (viscous travelling wave approximations) exist if and only if the approximated shock waves are Lax admissible. Our analysis of shock profiles reveals an interesting simplification over classical artificial viscosity, leading to a one dimensional fixed point problem, due to the speed of light bound of Relativity., Comment: 18 pages

We introduce a natural, mathematically consistent definition of the essential (highest possible) regularity of an affine connection -- a geometric property independent of atlas -- together with a checkable necessary and sufficient condition for determining whether connections are at their essential regularity. This condition is based on the relative regularity between the connection and its Riemann curvature. Based on this, we prove that authors' theory of the RT-equations provides a computable procedure for constructing coordinate transformations which simultaneously lift connection and curvature components to essential regularity, and lift the atlas of coordinate charts to the regularity required to preserve the essential regularity of the connection. This provides a definitive theory for determining whether or not singularities in a geometry are essential or removable by coordinate transformation, together with an explicit procedure for lifting removable singularities to their essential regularity, both locally and globally. The theory applies to general affine connections with components in $L^p$, $p>n$, which naturally includes shock wave singularities in General Relativity as well as continuous metrics with infinite gradients, (both obstacles to numerical simulation), but not yet singularities at the lower regularity $p\leq n$ associated with discontinuous metric components, for example, the event horizon of a black hole.

A limiting case is considered in which the causal action principle for causal fermion systems describing Minkowski space gives rise to the linear Fock space dynamics of quantum electrodynamics. The quantum nature of the bosonic field is a consequence of the stochastic description of a multitude of fluctuating fields coupled to non-commuting operators, taking into account dephasing effects. The scaling of all error terms is specified. Our analysis leads to the concept of holographic mixing, which is introduced and explained in detail., Comment: 71 pages, LaTeX, 3 figures

We show that taking account of bounded curvature reduces the threshold regularity of connection coefficients required for existence and uniqueness of solutions to the geodesic equation, to $L^p_\text{loc}$, one derivative below the regularity $W^{1,p}_\text{loc}$ required if one does not take account of curvature, ($L_{loc}^p$ for existence, $W_{loc}^{1,p}$ for uniqueness). Our argument is based on authors' theory of the RT-equations for regularizing connections to optimal regularity by coordinate transformation. The incoming regularity is too low to formulate a weak version of the geodesic equation based on the standard method of multiplying by smooth test functions and integrating by parts, so alternatively, we define weak solutions by coordinate transformation and we give an explicit procedure for mollifying the original connection such that the correct weak solution is indeed a limit of smooth solutions of the mollified equations in the original coordinates. This is an example where limits under suitable mollification are more fundamental than a weak formulation, indicative of more complicated PDE's in which the standard weak formulation of the equations does not adequately rule out unphysical solutions. Our results apply to general second order ODE's in which the lack of regularity can be isolated in the connection coefficients. The results apply to General Relativity.

In this paper we propose a weaker version of Penrose's much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextentability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in $L^p$. Lipschitz continuity is the threshold regularity for causal structures, and curvature bounds rule out infinite tidal accelerations, arguing for physical significance of this weaker SCC conjecture. The main result of this paper, under the assumption that no extensions exist with higher connection regularity $W^{1,p}_\text{loc}$, proves in the affirmative this SCC conjecture with bounded curvature for $p$ sufficiently large, ($p>4$ with uniform bounds, $p>2$ without uniform bounds).

This thesis investigates share buybacks, specifically share buyback announcements. It addresses how to recognize such announcements, the excess return of share buybacks, and the prediction of returns after a share buyback announcement. We illustrate two NLP approaches for the automated detection of share buyback announcements. Even with very small amounts of training data, we can achieve an accuracy of up to 90%. This thesis utilizes these NLP methods to generate a large dataset consisting of 57,155 share buyback announcements. By analyzing this dataset, this thesis aims to show that most companies, which have a share buyback announced are underperforming the MSCI World. A minority of companies, however, significantly outperform the MSCI World. This significant overperformance leads to a net gain when looking at the averages of all companies. If the benchmark index is adjusted for the respective size of the companies, the average overperformance disappears, and the majority underperforms even greater. However, it was found that companies that announce a share buyback with a volume of at least 1% of their market cap, deliver, on average, a significant overperformance, even when using an adjusted benchmark. It was also found that companies that announce share buybacks in times of crisis emerge better than the overall market. Additionally, the generated dataset was used to train 72 machine learning models. Through this, it was able to find many strategies that could achieve an accuracy of up to 77% and generate great excess returns. A variety of performance indicators could be improved across six different time frames and a significant overperformance was identified. This was achieved by training several models for different tasks and time frames as well as combining these different models, generating significant improvement by fusing weak learners, in order to create one strong learner.

This paper is dedicated to a detailed analysis and computation of quantum states of causal fermion systems. The mathematical core is to compute integrals over the unitary group asymptotically for a large dimension of the group, for various integrands with a specific scaling behavior in this dimension. It is shown that, in a well-defined limiting case, the localized refined pre-state is positive and allows for the description of general entangled states., Comment: 89 pages, LaTeX, 9 figures, small improvements (published version)

We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of General Relativity. Compactness is the essential tool of mathematical analysis for establishing validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with $L^p$ curvature. Solutions of the RT-equations furnish coordinate and gauge transformations which give a non-optimal connection a gain of one derivative over its Riemann curvature, (i.e., to optimal regularity). The RT-equations are elliptic regardless of metric signature, and regularize singularities in solutions of the hyperbolic Einstein equations. As an application, singularities at GR shock waves are removable, implying geodesic curves, locally inertial coordinates and the Newtonian limit all exist. By the extra derivative we extend Uhlenbeck compactness from Uhlenbeck's setting of vector bundles with compact gauge groups over Riemannian manifolds, to the case of compact and non-compact gauge groups over non-Riemannian manifolds. Our version of Uhlenbeck compactness can also be viewed as a "geometric" improvement of the Div-Curl Lemma, improving weak continuity of wedge products to strong convergence., Comment: Version 3: Section 7 added. Version 2: Minor rewordings in abstract and introduction

We extend authors' prior results on optimal regularity and Uhlenbeck compactness for affine connections to general connections on vector bundles. This is accomplished by deriving a vector bundle version of the RT-equations, and establishing a new existence theory for these equations. These new RT-equations, non-invariant elliptic equations, provide the gauge transformations which transform the fibre component of a non-optimal connection to optimal regularity, i.e., the connection is one derivative more regular than its curvature in $L^p$. The existence theory handles curvature regularity all the way down to, but not including, $L^1$. Taken together with the affine case, our results extend optimal regularity of Kazden-DeTurck and the compactness theorem of Uhlenbeck, applicable to Riemannian geometry and compact gauge groups, to general connections on vector bundles over non-Riemannian manifolds, allowing for both compact and non-compact gauge groups. In particular, this extends optimal regularity and Uhlenbeck compactness to Yang-Mills connections on vector bundles over Lorentzian manifolds as base space, the setting of General Relativity., Comment: Version 4: Improved local and (new) global results; curvature regularity down to L1. Version 3: More details of proof in Section 5. Inclusion of Theorems 2.6 and 2.7. Version 2: New title; improvements to presentation; slightly weaker regularity assumption for the optimal regularity result, and slightly stronger assumption for Uhlenbeck compactness; otherwise results unchanged

We describe the application of structured imaging with a single pixel camera to imaging through fog. We demonstrate the use of a high-pass filter on the detected bucket signals to suppress the effects of temporal variations of fog density and enable an effective reconstruction of the image. A quantitative analysis and comparison of several high-pass filters are demonstrated for the application. Both computational ghost imaging and compressive sensing techniques were used for image reconstruction and compressive sensing was observed to give a higher reconstructed image quality.

To further confirm the causality and stability of a second-order hyperbolic system of partial differential equations that models the relativistic dynamics of barotropic fluids with viscosity and heat conduction (H. Freist\"uhler and B. Temple, J. Math. Phys. 59 (2018)), this paper studies the Fourier-Laplace modes of this system and shows that all such modes, relative to arbitrary Lorentz frames, (a) decay with increasing time and (b) travel at subluminal speeds. Stability is also shown for the related model of non-barotropic fluids (H. Freist\"uhler and B. Temple. Proc. R. Soc. A 470 (2014) and Proc. R. Soc. A 473 (2017)). Even though these properties had been known for a while in the sense of numerical evidence, the fully analytical proofs for the subluminality of modes of arbitrary wave numbers in arbitrary frames given here appear to be the first regarding any five-field formulation of dissipative relativistic fluid dynamics., Comment: 19 pages

We resolve the problem of optimal regularity and Uhlenbeck compactness for affine connections in General Relativity and Mathematical Physics. First, we prove that any affine connection $\Gamma$, with components $\Gamma \in L^{2p}$ and components of its Riemann curvature ${\rm Riem}(\Gamma)$ in $L^p$, in some coordinate system, can be smoothed by coordinate transformation to optimal regularity, $\Gamma \in W^{1,p}$ (one derivative smoother than the curvature), $p> \max\{n/2,2\}$, dimension $n\geq 2$. For Lorentzian metrics in General Relativity this implies that shock wave solutions of the Einstein-Euler equations are non-singular -- geodesic curves, locally inertial coordinates and the Newtonian limit, all exist in a classical sense, and the Einstein equations hold in the strong sense. The proof is based on an $L^p$ existence theory for the Regularity Transformation (RT) equations, a system of elliptic partial differential equations (introduced by the authors) which determine the Jacobians of the regularizing coordinate transformations. Secondly, this existence theory gives the first extension of Uhlenbeck compactness from Riemannian metrics, to general affine connections bounded in $L^\infty$, with curvature in $L^{p}$, $p>n$, including semi-Riemannian metrics, and Lorentzian metric connections of relativistic Physics. We interpret this as a "geometric" improvement of the generalized Div-Curl Lemma. Our theory shows that Uhlenbeck compactness and optimal regularity are pure logical consequences of the rule which defines how connections transform from one coordinate system to another -- what one could take to be the "starting assumption of geometry"., Comment: V7: Simplification of proof in Section 12, using cutoff functions. V6: Minor improvements to Introduction and title. V5: Extension of main results from $L^\infty$ to $L^p$ curvature. New title. V3 & V4: Minor modifications to the write-up; main results unchanged. V2: Corollaries 2.2, 6.1 and 6.2 added; main results unchanged

We show that, when the integration time of the single photon detectors is longer than the correlation time of the biphoton, the attainable spatial resolution in ghost imaging with entangled signal idler pairs generated in type II spontaneous parametric down conversion is limited by the angular spread of single-frequency-signal idler pairs. If, however, the detector integration time is shorter than the biphoton correlation time, the transverse k-vectors of different spectral components combine coherently in the image, improving the spatial resolution., Comment: 16 pages, 15 figures, included in proceedings of Quantum Information Science, Sensing, and Computation XI, SPIE, 2019

We present authors' new theory of the RT-equations, nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections $\Gamma$ to optimal regularity, one derivative smoother than the Riemann curvature tensor ${\rm Riem}(\Gamma)$. As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at GR shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations, by application of elliptic regularity theory in $L^p$ spaces. The theory and results announced in this paper apply to arbitrary $L^\infty$ connections on the tangent bundle $T\mathcal{M}$ of arbitrary manifolds $\mathcal{M}$, including Lorentzian manifolds of General Relativity., Comment: This paper summarises the theory developed in arXiv:1805.01004, arXiv:1808.06455 and arXiv:1912.12997. V5: Improved wording and minor corrections in Section 7. V4: Cor 3.1 added. V3: Thms 2.6 - 2.8, addressing Uhlenbeck compactness and $L^\infty$ connections, added. Improved discussion section and introduction. V2: Improved presentation; Curvature bound added to Thm 2.4 and 2.5

Shock wave solutions of the Einstein equations have been constructed in coordinate systems in which the gravitational metric is only Lipschitz continuous, but the connection $\Gamma$ and curvature $Riem(\Gamma)$ are both in $L^{\infty}$. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. Here we address the mathematical problem as to whether the condition that $Riem(\Gamma)$ has the same regularity as $\Gamma$, is sufficient for the existence of a coordinate transformation which perfectly cancels out the jumps in the leading order derivatives of $\delta\Gamma$, thereby raising the regularity of the connection and the metric by one order--a subtle problem. We have now discovered, in a framework much more general than GR shock waves, that the regularization of non-optimal connections is determined by a nonlinear system of elliptic equations with matrix valued differential forms as unknowns, the Regularity Transformation equations, or RT-equations. In this paper we establish the first existence theory for the nonlinear RT-equations in the general case when $\Gamma, {\rm Riem}(\Gamma)\in W^{m,p}$, $m\geq1$, $n

A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open problem whether there always exist coordinate transformations which smooth a gravitational metric to optimal regularity, two full derivatives above the curvature tensor, or whether regularity singularities exist. We resolve this open problem above a threshold level of smoothness by proving in this paper that the existence of such coordinate transformations is equivalent to solving a system of nonlinear elliptic equations in the unknown Jacobian and transformed connection, both viewed as matrix valued differential forms. We name these the Regularity Transformation equations, or RT-equations. In a companion paper we prove existence of solutions to the RT-equations for connections $\Gamma\in W^{m,p},$ curvature ${\rm Riem}(\Gamma) \in W^{m,p}$, assuming $m\geq1$, $p>n$. Taken together, these results imply that there always exist coordinate transformations which smooth arbitrary connections to optimal regularity, (one derivative more regular than the curvature), and there are no regularity singularities, above the threshold $m\geq1$, $p>n$. Authors are currently working on extending these methods to the case of GR shock waves, when gravitational metrics are only Lipschitz continuous, ($m=0$, $p=\infty$), and optimal regularity is required to recover basic properties of spacetime., Comment: Version 5: Subsections are promoted to sections. Version 3 & 4: Improved wording, results unchanged. Version 2: We improved our main result, Theorem 1.1, by replacing the problematic boundary data (1.5) with linear data (eqn (1.5) in V.2), and we replaced eqns (1.1) - (1.2) of V.1 with the Poisson equation (3.53) of V.1. We modified the rest of the paper to adapt to our improved main result