Textbook Formula Research Articles

Optical interpretation of linear-optics superradiance and subradiance

Super- and subradiance are usually described in the framework of Dicke collective states, which is an ``atomic picture'' in which the electromagnetic field only provides an effective interaction between the atoms. Here, we discuss a complementary picture, in which we describe the propagation and scattering of light in the atomic medium, which provides a complex susceptibility and scatterers. This ``optical picture'' is valid in the linear-optics regime for disordered samples and is mainly relevant at low density, when the susceptibility and scattering cross-section can be computed from simple textbook formulas. In this picture, superradiance is a dispersion effect due to a single scattering event dressed by an effective refractive index, whereas subradiance is due to multiple scattering. We present numerical and experimental data supporting our interpretation.

Probabilities, odds, and vigorish

This paper uses actual data on horse racing to illustrate probabilities, odds, and expected values, and offers cautionary remarks about applying textbook formulas to gambling on real-world sporting events.

Investigating Preservice Mathematics Teachers’ Definitions, Formulas, and Graphs of Directly and Inversely Proportional Relationships

This study investigates 48 preservice middle school teachers’ formal textbook definitions, formulas, and graphs of the directly and inversely proportional relationships. The connections among these three types of representations and how preservice teachers’ representations differ according to the two relationships are also examined. The data of the study included the preservice teachers’ responses to a paper-and-pencil test with two items. An explanatory research design model was followed when developing this study. The study findings indicated the preservice teachers’ over-attention to the simultaneous increases and/or decreases when representing the directly and inversely proportional relationships. Only four preservice teachers stated in their definitions that the ratio was constant in the directly proportional relationship. Whereas none of them stated constant product in their inversely proportional definition. Although the preservice teachers were slightly better at writing the direct proportion formula than the inverse proportion formula, this difference was much greater in terms of drawing the direct and inverse proportion graphs. Sixty percent of the preservice teachers drew their inverse proportion graphs as linear with negative slopes. Moreover, the analysis showed that the preservice teachers’ definitions were not well-linked with their graphs and formulas. On the other hand, the preservice teachers who were better at providing correct formulas were also better at drawing the correct graphs. Implications for teaching and future study suggestions are discussed.

A New Estimator for Standard Errors with Few Unbalanced Clusters

In linear regression analysis, the estimator of the variance of the estimator of the regression coefficients should take into account the clustered nature of the data, if present, since using the standard textbook formula will in that case lead to a severe downward bias in the standard errors. This idea of a cluster-robust variance estimator (CRVE) generalizes to clusters the classical heteroskedasticity-robust estimator. Its justification is asymptotic in the number of clusters. Although an improvement, a considerable bias could remain when the number of clusters is low, the more so when regressors are correlated within cluster. In order to address these issues, two improved methods were proposed; one method, which we call CR2VE, was based on biased reduced linearization, while the other, CR3VE, can be seen as a jackknife estimator. The latter is unbiased under very strict conditions, in particular equal cluster size. To relax this condition, we introduce in this paper CR3VE-λ, a generalization of CR3VE where the cluster size is allowed to vary freely between clusters. We illustrate the performance of CR3VE-λ through simulations and we show that, especially when cluster sizes vary widely, it can outperform the other commonly used estimators.

Divergences in anomalous dimension matrices of quarks at three loops: Explanation and simple solution

Three-loop counterterms for the Standard Model (SM) revealed that the matrix of anomalous dimensions ($\gamma$) of quarks is divergent in the $d \to 4$ limit unless a carefully chosen non-Hermitian square-root of $Z$ matrix is used in the textbook formula for $\gamma$. Here, an alternative prescription is given, which expresses $\gamma$ and $\beta$ functions directly in terms of counterterms (instead of $\sqrt{Z}$ and conventional `bare couplings') and produces finite results. In the SM, this prescription $automatically$ reproduces results obtained previously by adjusting $\sqrt{Z}$.

Edge corrections for parallel-plate capacitors

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

Treasury Bond Return Data Starting in 1962

Academics and research analysts in financial economics frequently use returns on government bonds for their empirical analyses. In the United States, government bonds are also called Treasury bonds. The Federal Reserve publishes the yield-to-maturity of Treasury bonds. However, the Treasury bond returns earned by investors are not publicly available. The purpose of this study is to provide these currently not publicly available return series and provide formulas such that these series can easily be updated by researchers. We use standard textbook formulas to convert the yield-to-maturity data to investor returns. The starting date of our series is January 1962, when end-of-month data on the yield-to-maturity become publicly available. We compare our newly created total return series with alternative series that can be purchased. Our return series are very close, suggesting that they are a high-quality public alternative to commercially available data.

Dipolic Flows Relevant to Aquifer Storage and Recovery: Strack’s Sink Solution Revisited

Steady, 2-,3-D Darcian flows generated by a dipole (a pair of horizontal or vertical injection–abstraction wells closely placed one above another), with circulation of fresh water inside an interface confined lens or “bubble” underneath an impermeable caprock, surrounded by a static saline groundwater, are analytically studied. For 2-D dipole, the complex potential domain is a plane with a horizontal cut. This domain is conformally mapped onto a reference half-plane where the Keldysh–Sedov formula is used to obtain the complex physical coordinates. Explicit closed-form expressions for the vase-shaped interface, flow net, isohypses, magnitudes of the Darcian velocity and Riesenkampf’s resultant force are obtained, depending on the dipole moment, its position with respect to the caprock, and the ratio of densities of the two fluids. It is shown that for sufficiently small injection-pumping rates the fresh water “vase” separates from the caprock and becomes a circle, inside which streamlines are Newtons’ loops of monodiametral degenerate hyperbolae (cubics). Two numerical codes, MT3DMS and SEAWAT, are also used for delineation of isoconcentric lines, which qualitatively corroborate the analytical solutions in delineation of the “bubble” in the part where the sharp interface model predicts stable free boundaries and evidencing “dimples” on the boundary of the “bubble” where the saline water overlies the fresh one. For 3-D dipole not bounded by the caprock, the analytical fresh water “bubble” is a sphere and solution follows, mutatis mutandis, from the textbook formulae for flow of an ideal fluid past an impermeable sphere. The Stokes streamlines inside the sphere are sixtics; isotachs are plotted in an axial section. Stability of the soil matrix near the wells is also discussed.

Reciprocity and Open-Circuit Voltage in Solar Cells

The equation for open-circuit voltage of a solar cell based on optoelectronic reciprocity is combined with the standard textbook formula to obtain a general result that includes photon recycling as well as losses by carrier transport. It is shown that, in indirect-gap semiconductors, the general expression reduces to the “conventional” and “optoelectronic” expressions in the limits of low and high radiative efficiency.

A Polywell Fusion Reactor Designed for Net Power Generation

A brief history of Polywell progress is recounted. The present PIC simulation explains why the most recent Polywell fusion reactor failed to produce fusion energy. Synchronized variations of multiple parameters would require DC power supplies, not available in historic model testing. Even with DC power, the simulation showed that the trapping of cold electrons would ruin plasma stability during start-up. A theoretical solution to this trapping problem was found in Russian literature describing diocotron-pumping of electrons out of a plasma trap at Kharkov Institute. In Polywell, diocotron-pumping required matching the depth of the potential-well to the electron-beam current falling on a special aperture installed in one of the electromagnets. With diocotron-pumping the reactor was simulated to reach steady-state, maximum-power operation in a few milliseconds of simulated time. These improvements, validated in simulating small-scale DD reactors, were scaled up by a factor of 30 to simulate a large, net-power reactor burning p + 11B fuel. Power-balance was estimated from a textbook formula for fusion power density by numerically integrating the power density. Unity power-balance required the size of the p + 11B reactor to be somewhat larger than ITER.

Quantile Forecasting in Operational Planning and Inventory Management – an Initial Empirical Verification

In the paper we present our initial results of an empirical verification of different methodologies of quantile forecasting used in operational management to calculate the re-order point or order-up-to level as well as the optimal order quantity according to the newsvendor model. The comparison encompasses 26 procedures including quantile regression, the basic bootstrap method and popular textbook formulas. Our results, obtained on the base of 30 time series concerning such diversified phenomena as supermarket sales, passenger transport and water and gas demand, point to the usefulness of regression medians, regression quantiles, bootstrap methods and the procedures available in the SAP ERP system.

High-quality-factor planar optical cavities with laterally stopped, slowed, or reversed light.

In a planar optical cavity, the resonance frequencies increase as a function of in-plane wavevector according to a standard textbook formula. This has well-known consequences in many different areas of optics, from the shifts of etalon peaks at non-normal angles, to the properties of transverse modes in laser diodes, to the effective mass of microcavity photons, and so on. However, this standard formula is valid only when the reflection phase of each cavity mirror is approximately independent of angle. There is a certain type of mirror-a subwavelength dielectric grating near a guided mode resonance-with not only a strongly angle-dependent reflection phase, but also very high reflectance and low losses. Simulations show that by using such mirrors, high-quality-factor planar cavities can be designed that break all these textbook rules, leading to resonant modes that are slow, stopped or even backward-propagating in the in-plane direction. In particular, we demonstrate experimentally high-Q planar cavities whose resonance frequency is independent of in-plane wavevector-i.e., the resonant modes have zero in-plane group velocity, for one polarization but both in-plane directions. We discuss potential applications in various fields including lasers, quantum optics, and exciton-polariton condensation.

The Hidden Connections between Different Perpetuity Growth Models

Perpetuity growth models are very popular throughout academia and practice. Given their parsimonious nature and their often implicit assumptions, it is not surprising that they can yield rather different results. First, the Miller and Modigliani (1961) model, coined as the investment opportunity approach, reformulates the Gordon and Shapiro (1956) model. Both are based on marginal rates of return. Second, we establish that steady state is a sufficient condition to align models employing marginal rates of return with models employing average rates of return, mostly stated as the “textbook formula”.

Variance matters (in stochastic dividend discount models)

Stochastic dividend discount models (Hurley and Johnson in Financ Anal J 50–54. http://www.jstor.org/stable/4479761 , 1994, J Portf Manag 27–31. doi: 10.3905/jpm.1998.409658 , 1998; Yao in J Portf Manag 99–103. doi: 10.3905/jpm.1997.409618 , 1997) present expressions for the expected value of stock prices when future dividends, periodically received by shareholders as a reward for their risky investment, evolve through time in a Markovian setting by the means of a discretely distributed random rate of growth. Such result extends and makes more flexible the classical textbook formula for stock prices known as Gordon model. This paper introduces a closed-form expression for the variance of random stock prices, determines how their variance is affected by the variance of the dividend rate of growth, establishes that, in this framework, the dividend process is non-stationary, and perform a simple econometric analysis applying real market data.

Computing present values: Capital budgeting done correctly

This paper shows that the standard textbook formula for computing the present value of a future random cash flow – the discounted expected value – is formally incorrect and can generate significant errors when used to compute present values. The correct present value method is provided as well as a simple adjustment to the textbook formula which can be used to obtain an approximation to the correct value.

Computing Present Values: Capital Budgeting Done Correctly

This paper shows that the standard textbook formula for computing the present value of a future random cash flow - the discounted expected value - is formally incorrect and can generate significant errors when used to compute present values. The correct present value method is provided as well as a simple adjustment to the textbook formula which can be used to obtain an approximation to the correct value.

On the safety of Gomory cut generators

Gomory mixed-integer cuts are one of the key components in Branch-and-Cut solvers for mixed-integer linear programs. The textbook formula for generating these cuts is not used directly in open-source and commercial software that work in finite precision: Additional steps are performed to avoid the generation of invalid cuts due to the limited numerical precision of the computations. This paper studies the impact of some of these steps on the safety of Gomory mixed-integer cut generators. As the generation of invalid cuts is a relatively rare event, the experimental design for this study is particularly important. We propose an experimental setup that allows statistically significant comparisons of generators. We also propose a parameter optimization algorithm and use it to find a Gomory mixed-integer cut generator that is as safe as a benchmark cut generator from a commercial solver even though it generates many more cuts.

Semiclassical formula for quantum tunneling in asymmetric double-well potentials

Despite quantum tunneling has been studied since the advent of quantum mechanics, the literature appears to contain no simple (textbook) formula for tunneling in generic asymmetric double-well potentials. In the regime of strong localization, we derive an succinct analytical formula based on the WKB semi-classical approach. Two different examples of asymmetric potentials are discussed: when the two localized levels are degenerate or not. For the first case, we also discuss a time-dependent problem showing quantum Zeno effect.

The Cost of Debt Capital Revisited

We propose a method to consistently estimate the cost of debt in a continuous-time framework with an infinite time horizon. The approach builds on the EBIT-based model of Goldstein et al. (2001). The model is capable of splitting the observed yield spread of a corporate bond into the risk premium, which adds to the expected return of bondholders, and the default premium, which accounts for expected losses. The model can easily be calibrated for non-public firms. The model-implied cost of debt proves to be very insensitive with respect to non-observable parameters. Analyzing the weighted average cost of capital (WACC) in the model, we demonstrate potential errors from using the textbook formula for the WACC.

Analytic treatment of nuclear spin–lattice relaxation for diffusion in a cone model

We consider nuclear spin–lattice relaxation rate resulted from a diffusion equation for rotational wobbling in a cone. We show that the widespread point of view that there are no analytical expressions for correlation functions for wobbling in a cone model is invalid and prove that nuclear spin–lattice relaxation in this model is exactly tractable and amenable to full analytical description. The mechanism of relaxation is assumed to be due to dipole–dipole interaction of nuclear spins and is treated within the framework of the standard Bloemberger, Purcell, Pound-Solomon scheme. We consider the general case of arbitrary orientation of the cone axis relative the magnetic field. The BPP-Solomon scheme is shown to remain valid for systems with the distribution of the cone axes depending only on the tilt relative the magnetic field but otherwise being isotropic. We consider the case of random isotropic orientation of cone axes relative the magnetic field taking place in powders. Also we consider the cases of their predominant orientation along or opposite the magnetic field and that of their predominant orientation transverse to the magnetic field which may be relevant for, e.g., liquid crystals. Besides we treat in details the model case of the cone axis directed along the magnetic field. The latter provides direct comparison of the limiting case of our formulas with the textbook formulas for free isotropic rotational diffusion. The dependence of the spin–lattice relaxation rate on the cone half-width yields results similar to those predicted by the model-free approach.