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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

We show that and specify how space-time symmetries give rise to corresponding symmetries of the fermionic signature operator and generalized fermionic projector states., Comment: 21 pages, LaTeX, 1 figure, minor improvements (published version)

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether {\it regularity singularities} (points where the curvature is in $L^\infty$ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to $C^{1,1}$ by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to $C^{1,1}$ locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme., Comment: V5: Improved presentation, in particular, to Section 6. Results unchanged. V4: We shortened the presentation, added Def 3.1 and removed last section of previous version. V3: We extended results from connections of bounded variation to connections bounded in $L^\infty$, otherwise main results remain identical. V2: Result of Theorem 1.5 was extended from 2-D to spherically symmetric space-times

We give a non-perturbative construction of a distinguished state for the quantized Dirac field in Minkowski space in the presence of a time-dependent external field of the form of a plane electromagnetic wave. By explicit computation of the fermionic signature operator, it is shown that the Dirac operator has the strong mass oscillation property. We prove that the resulting fermionic projector state is a Hadamard state., Comment: 27 pages, LaTeX, 3 figures, minor improvements (published version)

We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical limit as $c\rightarrow \infty$. As our main result, we prove that the fluid pressure of solutions of these incompressible "relativistic" Euler equations satisfies an elliptic equation on each of the hypersurfaces orthogonal to the fluid four-velocity, which indicates infinite speed of propagation., Comment: 7 pages. Version 2: Improved wording and presentation

We introduce a damping term for the special relativistic Euler equations in $3$-D and show that the equations reduce to the non-relativistic damped Euler equations in the Newtonian limit. We then write the equations as a symmetric hyperbolic system for which local-in-time existence of smooth solutions can be shown., Comment: 11 pages

We address the Riemann and Cauchy problems for systems of $n$ conservation laws in $m$ unknowns which are subject to $m-n$ constraints ($m\geq n$). Such constrained systems generalize systems of conservation laws in standard form to include various examples of conservation laws in Physics and Engineering beyond gas dynamics, e.g., multi-phase flow in porous media. We prove local well-posedness of the Riemann problem and global existence of the Cauchy problem for initial data with sufficiently small total variation, in one spatial dimension. The key to our existence theory is to generalize the $m\times n$ systems of constrained conservation laws to $n\times n$ systems of conservation laws with states taking values in an $n$-dimensional manifold and to extend Lax's theory for local existence as well as Glimm's random choice method to our geometric framework. Our resulting existence theory allows for the accumulation function to be non-invertible across hypersurfaces., Comment: 48 pages. Version 2: Improvements of wording, specifically in the introduction. Restructuring of Introduction. Version 3: I improved wording; I moved Section 4 to the end; I added Example 2.4 and 2.5; I added references

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of points of shock wave collision in General Relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier RSPA-publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist. Thus, our result implies that regularity singularities, (a type of mild singularity introduced in our RSPA-paper), do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original RSPA-publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations remains open., Comment: 21 pages. In this paper we summarise the results and proofs in arXiv:1409.5060. The result here corrects the wrong conclusion in arXiv:1105.0798 and arXiv:1112.1803

It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term {\it regularity singularity} was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system of the $C^{1,1}$ atlas. An existence theory for shock wave solutions in $C^{0,1}$ admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but $C^{1,1}$ is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves within the larger $C^{1,1}$ atlas. To clarify this open problem, we identify new "Coriolis type" effects in the geometry of $C^{0,1}$ shock wave metrics and prove they are essential in the sense that they can never be made to vanish within the atlas of {\it smooth} coordinate transformations, the atlas usually assumed in classical differential geometry. Thus the problem of existence of regularity singularities is equivalent to the question as to whether or not these Coriolis type effects are essentially non-removable and `real', or merely coordinate effects that can be removed, (in analogy to classical Coriolis forces), by going to the less regular atlas of $C^{1,1}$ transformations. If essentially non-removable, it would argue strongly for a `real' new physical effect for General Relativity, providing a physical context to the open problem of regularity singularities., Comment: 29 pages. Version 2: Corrections of some typographical errors and improvements of wording. Results are unchanged

We investigate the specific form that the collapsed quantum state of a signal photon can take when its entangled idler is measured in an entangled ghost imaging configuration using a type II collinear phase matched spontaneous parametric downconversion (SPDC) interaction. Calculation of the correlated counting rate distributions in the ghost image plane and diffraction plane show that agreement between collapse and non-collapse models is obtained if the signal is assumed to collapse into a specific mixed state. However, if the signal is assumed to collapse into a pure state, significant differences arise between the predictions of the two collapse models., Comment: 25 pages, 23 figures

We present a detailed analysis of a previously published realization of Popper's experiment using entangled ghost imaging. Our analysis, which is based on optical diffraction integrals, shows that, for the configuration previously described, the transverse spread of an unmeasured particle (the signal photon here) does not increase in inverse proportion to the width of its virtual confinement when its entangled twin is confined in transverse dimension by a physical slit. Rather we show that the spread of the unmeasured particle carries no dependence on the width of its virtual confinement in the published configuration. Instead, it spreads geometrically at a rate determined by the numerical aperture of the ghost imaging system. We further propose an alternative configuration for which the spread of the unmeasured particle does increase in inverse proportion to the width of its virtual confinement. We discuss the relation of these results to the predictions of Popper.

We prove that spacetime is locally inertial at points of shock wave collision in General Relativity. The result applies for collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. We give a constructive proof that there exist coordinate transformations which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of such points of shock wave interaction, and a $C^{1,1}$ metric regularity suffices for locally inertial frames to exist. This result corrects an error in our earlier RSPA-publication, which led us to the wrong conclusion that such coordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist. Our result here proves that regularity singularities, (a type of mild singularity introduced in our RSPA-publication), do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes, and this generalizes Israel's famous 1966 result to the case of such shock wave interactions. The strategy of proof here is an extension of the strategy outlined in our RSPA-paper, but differs fundamentally from the method used by Israel. The question whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations still remains open., Comment: 79 pages. The result here corrects the wrong conclusion in arXiv:1105.0798 and arXiv:1112.1803. This paper contains the proofs of the results announced in arXiv:1506.04081. V2: Minor improvements of wording; correction of a minor error in Lemma 8.3. Main results are unchanged. V3: Improvements of wording

The previous functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds is extended to space-times of infinite lifetime. The construction is based on an analysis of families of solutions of the Dirac equation with a varying mass parameter. It makes use of the so-called mass oscillation property which implies that integrating over the mass parameter generates decay of the Dirac wave functions at infinity. We obtain a canonical decomposition of the solution space of the massive Dirac equation into two subspaces, independent of observers or the choice of coordinates. The constructions are illustrated in the examples of ultrastatic space-times and de Sitter space-time., Comment: 29 pages, LaTeX, statement of Proposition 3.4 modified, footnote on page 8 added

We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the "negative-energy solutions" of the Dirac equation and to the WKB approximation is explained and quantified by a detailed analysis of closed Friedmann-Robertson-Walker universes., Comment: 29 pages, LaTeX, 1 figure, minor improvements (published version)

A proof that a new kind of non-removable {\it "regularity singularity"} forms when two shock waves collide within the theory of General Relativity, was first announced in ProcRoySoc A \cite{ReintjesTemple}. In the present paper we give complete proofs of the claims in \cite{ReintjesTemple} and extend the results on the regularity of the Einstein curvature tensor to the full Riemann curvature tensor. The main result is that, in a neighborhood of a point where two shock waves collide in a spherically symmetric spacetime, the gravitational metric tensor cannot be lifted from C0,1 to C1 within the class of C1,1 coordinate transformations. This contrasts Israel's celebrated theorem \cite{Israel}, which states that around each point on a {\it single} shock surface there exist a coordinate system in which the metric is C1,1 regular. Moreover, at points of shock wave interaction, delta function sources exist in the second derivatives of the gravitational metric tensor in all coordinate systems of the C1,1-atlas, but due to cancellation, the Einstein and Riemann curvature tensor remain sup-norm bounded. We conclude that points of shock wave interaction are a new kind of spacetime singularity, (which we name "regularity singularity"), singularities that can form from the evolution of smooth initial data for perfect fluids and that lie in physical spacetime, but at such points {\it locally inertial} coordinates fail to exist., Comment: This article has been withdrawn since the main result is wrong due to an computational error. See arXiv:1409.5060 for a correction of this error and a proof of the opposite statement. This article has been withdrawn, since the main result and conclusion of this paper are wrong due to an arithmetic error. This error is corrected in arXiv:1409.5060, where we in fact prove an result opposite to the incorrect one in this paper. Namely, we give a constructive proof that there exist coordinate transformations which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighbourhood of points of shock wave interaction, between shocks from different families. The result in arXiv:1409.5060 thus shows that regularity singularities do not exist at such points of shock interaction, contradicting the wrong result here

We show that the regularity of the gravitational metric tensor in spherically symmetric spacetimes cannot be lifted from $C^{0,1}$ to $C^{1,1}$ within the class of $C^{1,1}$ coordinate transformations in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel's Theorem which states that such coordinate transformations always exist in a neighborhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of singularity for perfect fluids evolving in spacetime, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the spacetime is not locally Minkowskian under any coordinate transformation. In particular, at such singularities, delta function sources in the second derivatives of the gravitational metric tensor exist in all coordinate systems of the $C^{1,1}$ atlas, but due to cancelation, the curvature tensor remains uniformly bounded., Comment: This article has been withdrawn since the main result is wrong due to an computational error. See arXiv:1506.04081 and arXiv:1409.5060 for a correction of this error and a proof of the opposite statement

We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a space-time normalization integral. This analysis allows us to introduce the fermionic projector in a closed Friedmann-Robertson-Walker geometry and to specify its global normalization as well as its local form., Comment: 22 pages, LaTeX, sign error in equation (3.7) corrected

We propose a ``channelization'' architecture to achieve wide-band electromagnetically induced transparency (EIT) and ultra-slow light propagation in atomic Rb-87 vapors. EIT and slow light are achieved by shining a strong, resonant ``pump'' laser on the atomic medium, which allows slow and unattenuated propagation of a weaker ``signal'' beam, but only when a two-photon resonance condition is satisfied. Our wideband architecture is accomplished by dispersing a wideband signal spatially, transverse to the propagation direction, prior to entering the atomic cell. When particular Zeeman sub-levels are used in the EIT system, then one can introduce a magnetic field with a linear gradient such that the two-photon resonance condition is satisfied for each individual frequency component. Because slow light is a group velocity effect, utilizing differential phase shifts across the spectrum of a light pulse, one must then introduce a slight mismatch from perfect resonance to induce a delay. We present a model which accounts for diffusion of the atoms in the varying magnetic field as well as interaction with levels outside the ideal three-level system on which EIT is based. We find the maximum delay-bandwidth product decreases with bandwidth, and that delay-bandwidth product ~1 should be achievable with bandwidth ~50 MHz (~5 ns delay). This is a large improvement over the ~1 MHz bandwidths in conventional slow light systems and could be of use in signal processing applications., Comment: Published in SPIE Proceedings, Photonics West 2005 (San Jose, CA, Jan. 22-27, 2005)

This reported research centered on development of the concept of a translating computer interface by which the networking of heterogeneous interactive information systems may be achieved during the period in which information retrieval system and network standards are evolving. The particular concepts and techniques investigated are the virtual system concept, a common command language, a master index and thesaurus, and a common bibliographic data structure. In addition to the theoretical study of the problem, an experimental interface has been developed that connects the MEDLINE and Interex retrieval system via ARPANET communication links and that performs some of the networking functions of the virtual system. (Author/WH)

Research in networking of heterogeneous interactive bibliographic retrieval systems is being conducted which centers on the concept of a virtual retrieval system. Such a virtual system would be created through a translating computer interface that would provide access to the different retrieval systems and data bases in a uniform and convenient way, even for the inexperienced user. An experimental interface, called CONIT, has been built to test the virtual system concept. Initial evaluation of CONIT, which connects four retrieval systems, suggests that the virtual system approach could be cost effective. Particular attention was focused on the requirements for a common command language, ease of use, and message interpretation and protocols in a networked interface. (Author/JY)

A primary goal of this project was to develop an interface that would provide direct access for inexperienced users to existing online bibliographic information retrieval networks. The experiment tested the concept of a virtual-system mode of access to a network of heterogeneous interactive retrieval systems and databases. An experimental translating computer interface named CONIT, that enables the virtual-system mode, was developed as a research test vehicle. The interface was designed to make the basic functions of three different bibliographic retrieval systems easy to use, even by inexperienced end users, by providing a simplified common command language coupled with extensive online instruction. Analysis of controlled experiments with end users indicates the probable success of operational interfaces using the virtual-system principle and other techniques demonstrated in the experimental interface. The research has also suggested that certain techniques implementable on an interface could enhance retrieval effectiveness for a wide class of users by aiding them in the development of search strategies. A project bibliography and references are included. (Author/RAA)

Examines hypothesis that it is technologically and economically feasible to move content of documents electronically among library network rather than documents themselves or photocopies thereof. Comparisons on basis of response-to-request time, quality of reproduced copy, cost factors, a three-node network, and economic study are highlighted. Six references are listed. (EJS)

A translating-computer-interface approach to providing a common, or virtual-system, mode of access to a network of heterogeneous online bibliographic retrieval systems has been investigated. Enhanced access to such systems by end users has been demonstrated through test usage of an experimental interface. A table-driven, rule-based message interpreter that provides a dynamic and flexible means for handling the automation of the interconnection of computer and human retrieval protocols has been found useful in implementing the interface. Experimental analysis has demonstrated the success of various computer-assisted instructional techniques as well as indicated basic tradeoffs and limitations in their implementation in working systems. Further development of interface systems appears promising and possibilities for further research have been outlined in a number of areas. (Author)

An outline of the principal features of the Petrocarb Pneumatic Feeding System is given. Early development and various commercial applications are included. It is concluded that the Petrocarb Injection System is capable of feeding dry solids into most of the processes being developed for utilizing coal.

Two third order processes, stimulated Raman scattering and self-focusing, with picosecond pulses are studied. In the case of transient stimulated scattering, the gain is reduced from the steady state value, and qualitatively new features, such as shortening and delay of the Stokes pulse relative to the laser pulse, appear. These predictions are extended to realistic laser pulses, and experiments confirm all of the theoretical predictions. The self-focusing and frequency broadening of picosecond pulses is studied in the absence of stimulated Raman scattering in several materials with large orientational Kerr constants. Measurements of the relaxation time indicate that the orientational Kerr effect is important in the self-focusing of picosecond pulses. Self-focused filaments are observed to propagate with constant diameters over a distance greater than 10 cm, but disappear before the end of a 20 cm cell. The filaments radiate light continuously along their path and the spectrum of the light shows that the frequency content extends symmetrically for several hundred wave numbers on either side of the laser frequency.

Single mode locked Nd-glass laser pulse time synchronization with Q switched ruby laser

Radar-type instrument to measure electromagnetic backscatter

Transient stimulated rotational and vibrational Raman scattering in gases using mode locked ruby laser

Airborne equipment for measuring X band scattering of earth surfaces

Flight tests of radar scattering coefficient measuring instrument