On the Summation Σx^n

The symbol Σxn will be used to denote the sum of the n th powers of the first x positive integers, that is, ln + 2n + 3n +... +xn, n integral. It is well known that this sum is equivalent to a polynomial of degree (n+l), having no constant term

Two Theorems on Annuities

In each of the special cases h + k = n - 1, and n annual payments are made. The values of h and k are so chosen that the present comes at the desired point - for the amount of an annuity at the time of the last payment; for the present value of an annuity one year before the first payment; for the amount of an annuity-due one year after the last payment; for the present value of an annuity-due at the time of the first payment; and for the present value of a deferred annuity r + 1 years before the first payment

A New Interpolation Formula

113. A new interpolation If we wish to interpolate values of a function between two given values y1, and y2, we may employ a polynomial in x1. When the only conditions to be satisfied are that the function take the value y1 when x=l, and the value y2 when x=2, the polynomial of minimum degree is of the form a0+a1 (x-1). Interpolation by means of this polynomial is that ordinarily employed when using trigonometric and logarithmic tables, and is spoken of as interpolation by first differences. If, however, additional conditions are imposed the degree of the polynomial will increase one unit for each condition. For example, if the interpolated values between y1 and y2 are dependent upon the conditions that the function takes the value y0 when x=0, and the value y3 when x=3, then the interpolation polynomial is of the form a0+a1 (x-1) + a2 (x-1)2 + a3 (x-1)3

To My Sweet Little Southern Gal

https://digitalcommons.library.umaine.edu/mmb-vp/6672/thumbnail.jp

The Cost of Climate Policy in the United States

We consider the cost of meeting emissions reduction targets consistent with a G8 proposal of a 50 percent global reduction in emissions by 2050, and an Obama Administration proposal of an 80 percent reduction over this period. We apply the MIT Emissions Prediction and Policy Analysis (EPPA), modeling these two policy scenarios if met by applying a national cap-and-trade system, and compare results with an earlier EPPA analysis of reductions of this stringency. We also test results to alternative assumptions about program coverage, banking behavior, and cost of technology in the electric power sector. Two main messages emerge from the exercise. First, technology uncertainties have a huge effect on the generation mix but only a moderate effect on the emissions price and welfare cost of achieving the assumed targets. Measured in terms of changes in economic welfare, the economic cost of 80 percent reduction by 2050 is in the range of 2 to 3% by 2050, with CO2 prices between $48 and$67 in 2015 rising to between $190 and$266 by 2050. Second, implementation matters. When an idealized economy-wide cap-and-trade is replaced by coverage omitting some sectors, or if the credibility of long-term target is weak (limiting banking behavior) prices and welfare costs change substantially.Massachusetts Institute of Technology. Center for Energy and Environmental Policy Research

Assessment of U.S. Cap-and-Trade Proposals

The MIT Emissions Prediction and Policy Analysis model is applied to synthetic policies that match key attributes of a set of cap-and-trade proposals being considered by the U.S. Congress in spring 2007. The bills fall into two groups: one specifies emissions reductions of 50% to 80% below 1990 levels by 2050; the other establishes a tightening target for emissions intensity and stipulates a time-path for a "safety valve" limit on the emission price that approximately stabilizes U.S. emissions at the 2008 level. Initial period prices are estimated between $7 and$50 per ton CO2-e with these prices rising by a factor of four by 2050. Welfare costs vary from near zero to less than 0.5% at the start, rising in the most stringent case to near 2% in 2050. If allowances were auctioned these proposals could produce revenue between $100 billion and$500 billion per year depending on the case. Outcomes from U.S. policies depend on mitigation effort abroads, and simulations are provided to illuminate terms-of-trade effects that influence the emissions prices and welfare effects, and even the environmental effectiveness, of U.S. actions. Sensitivity tests also are provided of several of key design features. Finally, the U.S. proposals, and the assumptions about effort elsewhere, are extended to 2100 to allow exploration of the potential role of these bills in the longer-term challenge of reducing climate change risk. Simulations show that the 50% to 80% targets are consistent with global goals of atmospheric stabilization at 450 to 550 ppmv CO2 but only if other nations, including the developing countries, follow suit.

Randomized controlled trial of a good practice approach to treatment of childhood obesity in Malaysia: Malaysian childhood obesity treatment trial (MASCOT)

Context. Few randomized controlled trials (RCTs) of interventions for the treatment of childhood obesity have taken place outside the Western world. Aim. To test whether a good practice intervention for the treatment of childhood obesity would have a greater impact on weight status and other outcomes than a control condition in Kuala Lumpur, Malaysia. Methods. Assessor-blinded RCT of a treatment intervention in 107 obese 7- to 11-year olds. The intervention was relatively low intensity (8 hours contact over 26 weeks, group based), aiming to change child sedentary behavior, physical activity, and diet using behavior change counselling. Outcomes were measured at baseline and six months after the start of the intervention. Primary outcome was BMI z-score, other outcomes were weight change, health-related quality of life (Peds QL), objectively measured physical activity and sedentary behavior (Actigraph accelerometry over 5 days). Results. The intervention had no significant effect on BMI z score relative to control. Weight gain was reduced significantly in the intervention group compared to the control group (+1.5 kg vs. +3.5 kg, respectively, t-test p < 0.01). Changes in health-related quality of life and objectively measured physical activity and sedentary behavior favored the intervention group. Conclusions. Treatment was associated with reduced rate of weight gain, and improvements in physical activity and quality of life. More substantial benefits may require longer term and more intensive interventions which aim for more substantive lifestyle changes

Analysis of U.S. Greenhouse Gas Tax Proposals

The U.S. Congress is considering a set of bills designed to limit the nation's greenhouse gas (GHG) emissions. This paper complements the analysis by Paltsev et al. (2007) of cap-and-trade bills and applies the MIT Emissions Prediction and Policy Analysis (EPPA) model to carry out an analysis of the tax proposals. Several lessons emerge from this analysis. First, a low starting tax rate combined with a low rate of growth in the tax rate will not reduce emissions significantly. Second, the costs of GHG reductions are reduced with the inclusion of non-CO2 gases in the carbon tax scheme. Third, welfare costs of the policies can be affected by the rate of growth of the tax, even after controlling for cumulative emissions. Fourth, a carbon tax -- like any form of carbon pricing -- is regressive. However, general equilibrium considerations suggest that the short-run measured regressivity may be overstated. Additionally, the regressivity can be offset with a carefully designed rebate of some or all of the revenue. Finally, the carbon tax bills that have been proposed or submitted are for the most part comparable to many of the carbon cap-and-trade proposals that have been suggested. Thus the choice between a carbon tax and cap-and-trade system can be made on the basis of considerations other than their effectiveness at reducing emissions over some control period.

Optimal combination of signals from co-located gravitational wave interferometers for use in searches for a stochastic background

This article derives an optimal (i.e., unbiased, minimum variance) estimator for the pseudo-detector strain for a pair of co-located gravitational wave interferometers (such as the pair of LIGO interferometers at its Hanford Observatory), allowing for possible instrumental correlations between the two detectors. The technique is robust and does not involve any assumptions or approximations regarding the relative strength of gravitational wave signals in the detector pair with respect to other sources of correlated instrumental or environmental noise. An expression is given for the effective power spectral density of the combined noise in the pseudo-detector. This can then be introduced into the standard optimal Wiener filter used to cross-correlate detector data streams in order to obtain an optimal estimate of the stochastic gravitational wave background. In addition, a dual to the optimal estimate of strain is derived. This dual is constructed to contain no gravitational wave signature and can thus be used as on "off-source" measurement to test algorithms used in the "on-source" observation.Comment: 14 pages, 4 figures, submitted to Physical Review D Resubmitted after editing paper in response to referee comments. Removed appendices A, B and edited text accordingly. Improved legibility of figures. Corrected several references. Corrected reference to science run number (S1 vs. S2) in text and figure caption