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We consider the problem of convective heat transfer across the laminar boundary-layer induced by an isothermal moving surface in a Newtonian fluid. In previous work (Barlow, Reinberger, and Weinstein, 2024, \textit{Physics of Fluids}, \textbf{36} (031703), 1-3) an exact power series solution was provided for the hydrodynamic flow, often referred to as the Sakiadis boundary layer. Here, we utilize this expression to develop an exact solution for the associated thermal boundary layer as characterized by the Prandtl number ($\Pr$) and local Reynolds number along the surface. To extract the location-dependent heat-transfer coefficient (expressed in dimensionless form as the Nusselt number), the dimensionless temperature gradient at the wall is required; this gradient is solely a function of $\Pr$, and is expressed as an integral of the exact boundary layer flow solution. We find that the exact solution for the temperature gradient is computationally unstable at large $\Pr$, and a large $\Pr$ expansion for the temperature gradient is obtained using Laplace's method. A composite solution is obtained that is accurate to $O(10^{-10})$. Although divergent, the classical power series solution for the Sakiadis boundary layer -- expanded about the wall -- may be used to obtain all higher-order corrections in the asymptotic expansion. We show that this result is connected to the physics of large Prandtl number flows where the thickness of the hydrodynamic boundary layer is much larger than that of the thermal boundary layer. The present model is valid for all Prandtl numbers and attractive for ease of use., Comment: 32 pages, 5 figures

It has recently been established [Naghshineh et al., IMA J. of Appl. Math., 88, 1 (2023)] that a convergent series solution may be obtained for the Sakiadis boundary layer problem once key parameters are determined iteratively using the series itself. Here, we provide exact and explicit analytical expressions for these parameters, including that associated with wall shear, thus completing the exact analytical solution. The complete exact analytical solution to the Sakiadis problem is summarized here for direct use.

The Sakiadis boundary layer induced by a moving wall in a semi-infinite fluid domain is a fundamental laminar flow field relevant to high speed coating processes. This work provides an analytical solution to the boundary layer problem for Ostwald-de Waele power law fluids via a power series expansion, and extends the approach taken for Newtonian fluids [Naghshineh et al., "On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs", IMA J. of Appl. Math., (2023)] in which variable substitutions (which naturally determine the gauge function in the power series) are chosen to be consistent with the large distance behavior away from the wall. Contrary to prior literature, the asymptotic behavior dictates that a solution only exists in the range of power law exponents, $\alpha$, lying in the range $0.5 < \alpha \leq 1$. An analytical solution is obtained in the range of approximately $0.74 \leq \alpha < 1$, using a convergent power series with an asymptotically motivated gauge function. For power laws corresponding to $0.5 < \alpha<0.74$, the gauge function becomes ill-defined over the full domain, and an approximate analytical solution is obtained using the method of asymptotic approximants [Barlow et al., "On the summation of divergent, truncated, and underspecified power series via asymptotic approximants", Q. J. Mech. Appl. Math., (2017)]. The approximant requires knowledge of two physical constants, which we compute a priori using a numerical shooting method on a finite domain. The utility of the power series solution is that it can be solved on the entire semi-infinite domain and--in contrast to a numerical solution--does not require a finite domain length approximation and subsequent domain length refinement.

Industrial coating processes create thin liquid films with tight thickness tolerances, and thus models that predict the response to inevitable disturbances are essential. The mathematical modeling complexities are reduced through linearization as even small thickness variations in films can render a product unsalable. The signaling problem, considered in this paper, is perhaps the simplest model that incorporates the effects of repetitive (oscillatory) disturbances that are initiated, for example, by room vibrations and pump drives. In prior work, Gordillo and P\'erez (Phys. Fluids 14, 2002) examined the structure of the signaling response for linear operators that admit exponentially growing or damped solutions, i.e., the medium is classified as unstable or stable via classical stability analysis. The signaling problem admits two portions of the solution, the transient behavior due to the start-up of the disturbance and the long-time behavior that is continually forced; the superposition reveals how the forced solution evolves through the passage of a transient. In this paper, we examine signaling for the linear operator examined by King et al. (King et al. 2016, Phys. Rev. Fluids 1(7)) that governs varicose waves in a thin liquid sheet and that can admit solutions having algebraic growth although the underlying medium is classified as being neutrally stable. Long-time asymptotic methods are used to extract critical velocities that partition the response into distinct regions having markedly different location-dependent responses, and the amplitudes of oscillatory responses in these regions are determined. Together, these characterize the magnitude and breadth of the solution response. Results indicate that the signaling response in neutrally stable flows (that admit algebraic growth) is significantly different from that in exponentially unstable systems.

We examine the classical problem of the height of a static liquid interface that forms on the outside of a solid vertical cylinder in an unbounded stagnant pool exposed to air. Gravitational and surface tension effects compete to affect the interface shape as characterized by the Bond number, $B = \rho g R^{2}/\sigma$, where $\rho$ is fluid density, $g$ is the gravitational constant, $R$ is the radius of the cylinder, and $\sigma$ is surface tension. Here, we provide a convergent power series solution for interface shapes that rise above the horizontal pool as a function of Bond number. The power series solution is expressed in terms of a pre-factor that matches the large distance asymptotic behavior -- the modified Bessel function of zeroth order -- and an Euler transformation that moves the influence of convergence-limiting singularities out of the physical domain. The power series solution is validated through comparison with a numerical solution, and the $B\rightarrow0$ matched asymptotic solutions of Lo (1983, J. Fluid Mech., 132, p.65-78). For a benchmark static contact angle of $45$ degrees, the power series approach exceeds the accuracy of matched asymptotic solutions for $B>0.01$; this lower limit on $B$ arises from round-off error in the computation of the series coefficients in double precision arithmetic, and is reduced as the contact angle is increased., Comment: 20 pages

This work considers two linear operators which yield wave modes that are classified as neutrally stable, yet have responses that grow or decay in time. Previously, King et al. (Phys. Rev. Fluids, 1, 2016, 073604:1-19) and Huber et al. (IMA J. Appl. Math., 85, 2020, 309-340) examined the one-dimensional (1D) wave propagation governed by these operators. Here, we extend the linear operators to two spatial dimensions (2D) and examine the resulting solutions. We find that the increase of dimension leads to long-time behaviour where the magnitude is reduced by a factor of $t^{\frac{-1}{2}}$ from the 1D solutions. Thus, regions of the solution which grew algebraically as $t^{\frac{1}{2}}$ in 1D now are algebraically neutral in 2D, whereas regions which decay (algebraically or exponentially) in 1D now decay more quickly in 2D. Additionally, we find that these two linear operators admit long-time solutions that are functions of the same similarity variable that contracts space and time.

We examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviors in semi-infinite domains. The first problem is that of the "Sakiadis" boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air-liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution -- given as distance from the wall as function of meniscus height -- has long been known (Batchelor, 1967). Here, we provide an explicit solution as meniscus height vs. distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that -- in both problems -- the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviors to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviors can be useful when proposing variable transformations to overcome power series divergence.

We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair\'en, V. L\'opez, and L. Conde, Am. J. Phys 56 (1), (1988), pp. 57-61], the series itself -- as well as the optimal location about which an expansion should be chosen to assure series convergence and maximize the domain of convergence -- was not examined, and is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the physical problem in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. In constructing the series solution, we re-derive the coefficients using an alternative approach that generalizes to other nonlinear problems of mathematical physics. Additionally, we provide an exact resummation of the pendulum series -- Motivated by the asymptotic approximant method given in [Barlow et al., Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] -- that accelerates the series' convergence uniformly from the top to the bottom of the trajectory. We also provide an accelerated exact resummation of the infinite series representation for the elliptic integral used in calculating the period of a pendulum's trajectory. This allows one to preserve analyticity in the use of the period to extend the pendulum series for all time via symmetry., Comment: The only change in this version is the removal of LaTeX characters in the abstract

Planar flow casting (PFC) is a method that can be used to make thin, long, and wide metallic alloy foils by extruding molten liquid through a thin and wide nozzle and immediately quenching on a moving roller. The quenching rates are high enough that amorphous metallic glasses may be formed which have many desirable properties for a wide variety of applications. This paper reviews how PFC processes were developed, examines the typical operability range of PFC, and reviews the defects that commonly form. The geometrical similarities between PFC and slot coating process are apparent, and this paper highlights differences between the operability ranges of both processes., Comment: 19 pages, 14 figures

The interface shape of a fluid in rigid body rotation about its axis and partially filling the container is often the subject of a homework problem in the first graduate fluids class. In that problem, surface tension is neglected, the interface shape is parabolic and the contact angle boundary condition is not satisfied in general. When surface tension is accounted for, the shapes exhibit much richer dependencies as a function of rotation velocity. We analyze steady interface shapes in rotating right-circular cylindrical containers under rigid body rotation in zero gravity. We pay especial attention to shapes near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation at certain specific rotational speeds. We examine geometries where the container is axially infinite and derive properties of their solutions. We then examine in detail two special cases of menisci in a cylindrical container: a meniscus spanning the cross section; and a meniscus forming a bubble. In each case we develop exact solutions for the respective axial lengths as infinite series in powers of appropriate rotation parameters; and we find the respective asymptotic behaviors as the shapes approach their critical configuration. Finally we apply the method of asymptotic approximants to yield analytical expressions for the axial lengths of menisci over the whole range of rotation speeds. In this application, the analytical solution is employed to examine errors introduced by the assumption that the interface is a right circular cylinder; this assumption is key to the spinning bubble method used to measure surface tension., Comment: 24 pages, 14 figures

The inertial collapse of two interacting and non-translating spherical bubbles of equal size is considered. The exact analytic solution to the nonlinear ordinary differential equation that governs the bubble radii during collapse is first obtained via a slowly converging power series. An asymptotic approximant is then constructed that accelerates convergence of the series and imposes the asymptotic collapse behavior when the radii are small. The solution generalizes the classical 1917 Rayleigh problem of single bubble collapse, as this configuration is recovered when the distance between the bubble centers far exceeds that of their radii., Comment: 14 pages, 3 figures

We consider the time-dependent response of a gravitationally-thinning inviscid liquid sheet (a coating curtain) leaving a vertical slot to sinusoidal ambient pressure disturbances. The theoretical investigation employs the hyperbolic partial differential equation developed by \cite{weinp1}. The response of the curtain is characterized by the slot Weber number, $W_{e_0} = \rho q V/2\sigma$, where $V$ is the speed of the curtain at the slot, $q$ is the volumetric flow rate per unit width, $\sigma$ is the surface tension, and $\rho$ is the fluid density. Flow disturbances travel along characteristics with speeds relative to the curtain of $\pm \sqrt{uV/W_{e_0}}$, where $u = \sqrt{V^2 + 2gx}$ is the curtain speed at a distance $x$ downstream from the slot. When the flow is subcritical ($W_{e_0} < 1$), upstream traveling disturbances near the slot affect the curtain centerline, and the slope of the curtain centerline at the slot oscillates with an amplitude that is a function of $W_{e_0}$. In contrast, all disturbances travel downstream in supercritical curtains ($W_{e_0} > 1$) and the slope of the curtain at the slot is vertical. Here, we specifically examine the curtain response under supercritical and subcritical flow conditions near $W_{e_0} = 1$ to deduce whether there is a substantial change in the overall shape and magnitude of the curtain responses. Despite the local differences in the curtain solution near the slot, we find that subcritical and supercritical curtains have similar responses for all imposed sinusoidal frequencies.

An analytic solution is obtained to the SEIR Epidemic Model. The solution is created by constructing a single second-order nonlinear differential equation in $\ln S$ and analytically continuing its divergent power series solution such that it matches the correct long-time exponential damping of the epidemic model. This is achieved through an asymptotic approximant (Barlow et. al, 2017, Q. Jl Mech. Appl. Math, 70 (1), 21-48) in the form of a modified symmetric Pad\'e approximant that incorporates this damping. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic., Comment: original version had substantial text overlap with arXiv:2004.07833; this is now less so

An accurate closed-form solution is obtained to the SIR Epidemic Model through the use of Asymptotic Approximants (Barlow et. al, 2017, Q. Jl Mech. Appl. Math, 70 (1), 21-48). The solution is created by analytically continuing the divergent power series solution such that it matches the long-time asymptotic behavior of the epidemic model. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic., Comment: Updates: Fixed a typo in previous version's equation 6c. Included link to Python code

We demonstrate that the asymptotic approximant applied to the Blasius boundary layer flow over a flat plat (Barlow et al., 2017 Q. J. Mech. Appl. Math., 70(1): 21-48) yields accurate analytic closed-form solutions to the Falkner-Skan boundary layer equation for flow over a wedge having angle $\beta\pi/2$ to the horizontal. A wide range of wedge angles satisfying $\beta\in[-0.198837735, 1]$ are considered, and the previously established non-unique solutions for $\beta<0$ having positive and negative shear rates along the wedge are accurately represented. The approximant is used to determine the singularities in the complex plane that prescribe the radius of convergence of the power series solution to the Falkner-Skan equation. An attractive feature of the approximant is that it may be constructed quickly by recursion compared with traditional Pad\'e approximants that require a matrix inversion. The accuracy of the approximant is verified by numerical solutions, and benchmark numerical values are obtained that characterize the asymptotic behavior of the Falkner-Skan solution at large distances from the wedge.

This paper reports a breakdown in linear stability theory under conditions of neutral stability that is deduced by an examination of exponential modes of the form $h\approx {{e}^{i(kx-\omega t)}}$, where $h$ is a response to a disturbance, $k$ is a real wavenumber, and $\omega(k)$ is a wavelength-dependent complex frequency. In a previous paper, King et al (Stability of algebraically unstable dispersive flows, \textit{Phys. Rev. Fluids}, 1(073604), 2016) demonstrates that when Im$[\omega(k)]$=0 for all $k$, it is possible for a system response to grow or damp algebraically as $h\approx {{t}^{s}}$ where $s$ is a fractional power. The growth is deduced through an asymptotic analysis of the Fourier integral that inherently invokes the superposition of an infinite number of modes. In this paper, the more typical case associated with the transition from stability to instability is examined in which Im$[\omega(k)]$=0 for a single mode (i.e., for one value of $k$) at neutral stability. Two partial differential equation systems are examined, one that has been constructed to elucidate key features of the stability threshold, and a second that models the well-studied problem of rectilinear Newtonian flow down an inclined plane. In both cases, algebraic growth/decay is deduced at the neutral stability boundary, and the propagation features of the responses are examined.

We successfully model the behavior of two-spin systems using neural networks known as conditional Restricted Boltzmann Machines (cRBMs) which encode physical information in the properties of a thermal ensemble akin to an Ising model. The result gives local "hidden" variable models for product states and entangled states, including the singlet state used in the EPR-Bohm experiment. Bell's theorem is circumvented because the state of the system is dependent not only on the preparation but also on the measurement setup (the detector settings). Though at first glance counterintuitive, the apparent "retrocausality" in these models has a historical precedent in the absorber theory of Wheeler and Feynman and an intuitive analog in the simple AC circuit of an electric guitar., Comment: 18 pages, many colorful figures. Comments welcome. Version 2: title changed, author's contact email address corrected, additional references added, training technique for Boltzmann machine clarified, other minor changes for clarity

Highly accurate closed-form expressions that describe the full trajectory of photons propagating in the equatorial plane of a Kerr black hole are obtained using asymptotic approximants. This work extends a prior study of the overall bending angle for photons (Barlow, et al. 2017, Class. Quantum Grav., 34, 135017). The expressions obtained provide accurate trajectory predictions for arbitrary spin and impact parameters, and provide significant time advantages compared with numerical evaluation of the elliptic integrals that describe photon trajectories. To construct approximants, asymptotic expansions for photon deflection are required in various limits. To this end, complete expansions are derived for the azimuthal angle as a function of radial distance from the black hole in the far-distance and closest-approach (pericenter) limits, and new coefficients are reported for the bending angle in the weak-field limit (large impact parameter)., Comment: 29 pages, 11 figures

We construct a hidden variable model for the EPR correlations using a Restricted Boltzmann Machine. The model reproduces the expected correlations and thus violates the Bell inequality, as required by Bell's theorem. Unlike most hidden-variable models, this model does not violate the $locality$ assumption in Bell's argument. Rather, it violates $measurement$ $independence$, albeit in a decidedly non-conspiratorial way., Comment: 5 pages, 3 figures. Comments welcome. References added in v2

A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.

An accurate closed-form expression is provided to predict the bending angle of light as a function of impact parameter for equatorial orbits around Kerr black holes of arbitrary spin. This expression is constructed by assuring that the weak- and strong-deflection limits are explicitly satisfied while maintaining accuracy at intermediate values of impact parameter via the method of asymptotic approximants (Barlow et al, 2016 Q. J. Mech. Appl. Math., doi=10.1093/qjmam/hbw014). To this end, the strong deflection limit for a prograde orbit around an extremal black hole is examined, and the full non-vanishing asymptotic behavior is determined. The derived approximant may be an attractive alternative to computationally expensive elliptical integrals used in black hole simulations., Comment: 16 pages, 7 figures, submitted to Classical and Quantum Gravity

Noise is widely understood to be something that interferes with a signal or process. Thus, it is generally thought to be destructive, obscuring signals and interfering with function. However, early in the 20th century, mechanical engineers found that mechanisms inducing additional vibration in mechanical systems could prevent sticking and hysteresis. This so-called "dither" noise was later introduced in an entirely different context at the advent of digital information transmission and recording in the early 1960s. Ironically, the addition of noise allows one to preserve information that would otherwise be lost when the signal or image is digitized. As we shall see, the benefits of added noise in these contexts are closely related to the phenomenon which has come to be known as stochastic resonance, the original version of which appealed to noise to explain how small periodic fluctuations in the eccentricity of the earth's orbit might be amplified in such a way as to bring about the observed periodic transitions in climate from ice age to temperate age and back. These noise-induced transitions have since been invoked to explain a wide array of biological phenomena, including the foraging and tracking behavior of ants. Many biological phenomena, from foraging to gene expression, are noisy, involving an element of randomness. In this paper, we illustrate the general principles behind dithering and stochastic resonance using examples from image processing, and then show how the constructive use of noise can carry over to systems found in nature., Comment: 21 pages, 9 figures, chapter to appear in_From Matter to Life: Informationa nd Causality_ by Walker, Davies, and Ellis (Cambridge University Press)

From classical mechanics to quantum field theory, the physical facts at one point in space are held to be independent of those at other points in space. I propose that we can usefully challenge this orthodoxy in order to explain otherwise puzzling correlations at both cosmological and microscopic scales., Comment: Submission to the 2012 FQXi essay contest "Which of Our Basic Physical Assumptions Are Wrong?". 11 pages including figures and references. arXiv admin note: text overlap with arXiv:0812.0349

It is a commonplace to note that in a world governed by special or general relativity, an observer has access only to data within her past lightcone (if that). The significance of this for prediction, and thus for confirmation, does not however seem to have been appreciated. In this paper I show that what we regard as our most well-confirmed relativistic theory, Maxwell's theory of electromagnetism, is not at all well-confirmed in the absence of an additional assumption, the assumption that all fields have sources in their past. I conclude that we have reason to believe that there is a lawlike time-asymmetry in the world.

It has been claimed that decoherence of open quantum systems explains the tendency of macroscopic systems to exhibit quasiclassical behavior. We show that quasiclassicality is in fact an unremarkable property, characterizing generic subsystems of environments even in the absence of dynamical decoherence. It is suggested that decoherence is best regarded as explaining the persistence of true classicality, rather than the emergence, rather than the emergence of quasiclassicality., Comment: 6 pages

The possibility of physics in multiple time dimensions is investigated. Drawing on recent work by Walter Craig and myself, I show that, contrary to conventional wisdom, there is a well-posed initial value problem--deterministic, stable evolution--for theories in multiple time dimensions. Though similar in many ways to ordinary, single-time theories, multi-time theories have some rather intriguing properties which suggest new directions for the understanding of fundamental physics.

Bell's theorem is purported to demonstrate the impossibility of a local "hidden variable" theory underpinning quantum mechanics. It relies on the well-known assumption of `locality', and also on a little-examined assumption called `statistical independence' (SI). Violations of this assumption have variously been thought to suggest "backward causation", a "conspiracy" on the part of nature, or the denial of "free will". It will be shown here that these are spurious worries, and that denial of SI simply implies nonlocal correlation between spacelike degrees of freedom. Lorentz-invariant theories in which SI does not hold are easily constructed: two are exhibited here. It is conjectured, on this basis, that quantum-mechanical phenomena may be modeled by a local theory after all., Comment: 15 pages, 9 figures (some color). Changes in version 2 include (1) a correction to the abstract: Bell's theorem is now said to demonstrate the impossibility of a local hidden variable theory, (2) the "non-spacelike" initial value problem is now the "mixed" initial value problem, (3)the figures should reproduce more clearly. Comments welcome

We study the initial value problem for the wave equation and the ultrahyperbolic equation for data posed on initial surface of mixed signature (both spacelike and timelike). Under a nonlocal constraint, we show that the Cauchy problem on codimension-one hypersurfaces has global unique solutions in the Sobolev spaces $H^{m}$, thus it is well-posed. In contrast, we show that the initial value problem on higher codimension hypersurfaces is ill-posed, at least when specifying a finite number of derivatives of the data, due to the failure of uniqueness. This is in contrast to a uniqueness result which Courant and Hilbert deduce from Asgeirsson's mean value theorem, for which we give an independent derivation. The proofs use Fourier synthesis and the Holmgren-John uniqueness theorem.

We consider the claim that decoherence explains the emergence of classicality in quantum systems, and conclude that it does not. We show that, given a randomly chosen universe composed of a variety of subsystems, some of which are macroscopic and subject to decoherence-inducing interactions, and some of which are microscopic, the macroscopic subsystems will not display any distinctively classical behavior. Therefore, a universe in which macroscopic and microscopic do display distinct behavior must be in a very special, highly nongeneric quantum state., Comment: Changed to emphasize the role of stability as a criterion for classicality, rather than diagonalization of the reduced density matrix. The conclusion is restated to emphasize that, irrespective of the dynamics, it is only for very special global quantum states that macroscopic objects will display behavior distinct from microscopic objects

Anthropic arguments in multiverse cosmology and string theory rely on the weak anthropic principle (WAP). We show that the principle, though ultimately a tautology, is nevertheless ambiguous. It can be reformulated in one of two unambiguous ways, which we refer to as WAP_1 and WAP_2. We show that WAP_2, the version most commonly used in anthropic reasoning, makes no physical predictions unless supplemented by a further assumption of "typicality", and we argue that this assumption is both misguided and unjustified. WAP_1, however, requires no such supplementation; it directly implies that any theory that assigns a non-zero probability to our universe predicts that we will observe our universe with probability one. We argue, therefore, that WAP_1 is preferable, and note that it has the benefit of avoiding the inductive overreach characteristic of much anthropic reasoning., Comment: 7 pages. Expanded discussion of selection effects and some minor clarifications, as published

The quantization of time-reparametrization invariant systems such as general relativity is plagued by an ambiguity relating to the role of time in the theory. If one parametrizes observables by the (unobservable) time, and then relies on the existence of an approximate "clock" degree of freedom to give physical meaning to the observables, one finds multiple quantum states that yield the same predictions yet interfere with each other., Comment: 3 pages, LaTeX. Requires mprocl.sty, available at ftp://shemesh.fiz.huji.ac.il/pub/mprocl.sty . To be published in the proceedings of the Eighth Marcel Grossmann Meeting on General Relativity. (Jerusalem, June, 1997)