text:
# GED Math : Multiplication

## Example Questions

← Previous 1 3 4 5

### Example Question #1 : Multiplication

Multiply:

Explanation:

Multiply one digit at a time:

The product is

### Example Question #1 : Multiplication

Which of the following statements demonstrates the commutative property of multiplication?

Explanation:

The commutative property of multiplication states that two numbers can be multiplied in either order to yield the same product. Of the four choices,

demonstrates this. It is the correct choice.

### Example Question #1 : Multiplication

Which statement demonstrates the associatve property of multiplication?

Explanation:

The associative property of multiplication states that when three numbers are multiplied, any two can be multiplied first and the same product will appear. Of the four choices,

demonstrates this.

### Example Question #4 : Multiplication

What property of arithmetic is symbolized below?

Commutative

Transitive

Distributive

Associative

Commutative

Explanation:

The symbols show that two numbers can be multiplied in either order to achieve the same result. This is the commutative property of multiplication.

### Example Question #61 : Basic Operations

Above is the menu for a coffee shop; there is no sales tax charged on purchases.

Myoshi wants to throw an informal party. She wants to serve one large cafe latte and two butter croissants to each guest, and she wants to keep expenses at or below $200. Counting herself, how many people, at most, will be at the party? Possible Answers: Correct answer: Explanation: Since one large cafe latte and two butter croissants will cost , this will be the amount Myoshi spends per person. She wants to spend at most$200, so, letting  stand for the number of people at the party, we can set up and solve this inequality:

There can be at most 23 people at Myoshi's party.

### Example Question #221 : Ged Math

Multiply:

Explanation:
When multiplying numbers that have more than one digit, you follow the order below:

1. Multiply each digit in the first number () by the second digit in the second number (the in )

2. Add a  in the units place under the  in

3. Multiply each digit in the first number () by the first digit in the second number  (the  in ) and put the product on the same line as the .

3. Add  and  to get the answer.

### Example Question #1 : Multiplication

Multiply the numbers:

Explanation:

Multiply the first number with the ones digit of the second number.

Multiply 54 with the tens digit of the second number.

Add a zero to the end of this number and add this number with the first number.

### Example Question #2 : Multiplication

Multiply the numbers:

Explanation:

Multiply the first number with the ones digit of 15.

Repeat the process with the tens digit of 15.

Add a zero to the end of this number and add this number with the first number.

### Example Question #1 : Multiplication

Multiply the numbers:

Explanation:

Multiply 26 with the ones digit of the second number.

Multiply 26 with the tens digit of the second number.

Add an extra zero to the end of this number and add this value with the first number.

### Example Question #1 : Multiplication

Multiply the following numbers:

Explanation:

Multiply the first number with the ones digit of the second number.

Repeat the process with the tens digit of the second number.

Add a zero to the end of this number and add this with the first number.


text:
# Dynamic Programming with 2 state variables

1. Nov 8, 2012

### physicsguy142

1. The problem statement, all variables and given/known data

Derive the Euler Equation of the dynamic programming problem:

$$max_{\{ z_t \}^\infty_{t=0}} \sum_{t=0}^{\infty} \delta^t f(x_t, y_t, z_t)$$

subject to:

$x_{t+1} = g_1(x_t, y_t, z_t), \ y_{t+1} = g_2(x_t, y_t, z_t), \ x_0 = x^0, y_0 = y^0$

and where $\delta <1$
2. Relevant equations

We can write the value function as:

$V(x,y) = max_z [f(x, y, z) + \delta V(g_1(x, y, z), g_2(x, y, z))]$

3. The attempt at a solution

The solution is characterized by 3 equations:

The first-order-condition at the optimal z* is:

$\frac{\partial f(x, y, z^*)}{\partial z} + \delta[\frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_1(x, y, z^*)}\frac{\partial g_1(x, y, z^*)}{\partial z} + \frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_2(x, y, z^*)}\frac{\partial g_2(x, y, z^*)}{\partial z}] = 0$

Differentiating the value function with respect to each state gives us:

$\frac{\partial V (x, y)}{\partial x} = \frac{\partial f(x, y, z^*)}{\partial x} + \delta[\frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_1(x, y, z^*)}\frac{\partial g_1(x, y, z^*)}{\partial x} + \frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_2(x, y, z^*)}\frac{\partial g_2(x, y, z^*)}{\partial x}]$

and

$\frac{\partial V (x, y)}{\partial y} = \frac{\partial f(x, y, z^*)}{\partial y} + \delta[\frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_1(x, y, z^*)}\frac{\partial g_1(x, y, z^*)}{\partial y} + \frac{\partial V (g_1(x, y, z^*), g_2(x, y, z^*))}{\partial g_2(x, y, z^*)}\frac{g_2(x, y, z^*)}{\partial y}]$

I think I should be able to derive an Euler Equation with the above 3 equations, but I'm not sure how to manipulate the equations to get a meaningful answer.

Last edited: Nov 8, 2012
2. Nov 8, 2012

### physicsguy142

bump. any help would be greatly appreciated!

3. Nov 8, 2012

### physicsguy142

bumping this on the off-chance somebody will be able to help out.

format: plain


text:
# GATE Mechanical 2014 Set 2 | Question: 19

A component can be produced by any of the four processes $I$, $II$, $III$ and $IV$. The fixed cost and the variable cost for each of the processes are listed below. The most economical process for producing a batch of $100$ pieces is

$$\begin{array}{|c|c|c|} \hline \textbf{Process} & \textbf{Fixed cost (in Rs.)} & \textbf{Variable cost per piece (in Rs.)} \\ \hline I & 20 & 3 \\ \hline II & 50 & 1 \\ \hline III & 40 & 2 \\ \hline IV & 10 & 4 \\ \hline \end{array}$$

1. $I$
2. $II$
3. $III$
4. $IV$

recategorized

## Related questions

Consider the two states of stress as shown in configurations I and II in the figure below. From the standpoint of distortion energy (von-Mises) criterion, which one of the following statements is true? $I$ yields after $II$ $II$ yields after $I$ Both yield simultaneously Nothing can be said about their relative yielding
A rectangular hole of size $100$ $mm \times 50$ $mm$ is to be made on a $5$ $mm$ thick sheet of steel having ultimate tensile strength and shear strength of $500$ $MPa$ and $300$ $MPa$, respectively. The hole is made by punching process. Neglecting the effect of clearance, the punching force (in $kN$) is $300$ $450$ $600$ $750$
A cantilever beam of length, $L$, with uniform cross-section and flexural rigidity, $EI$, is loaded uniformly by a vertical load, $w$ per unit length. The maximum vertical deflection of the beam is given by $\dfrac{wL^4}{8EI} \\$ $\dfrac{wL^4}{16EI} \\$ $\dfrac{wL^4}{4EI} \\$ $\dfrac{wL^4}{24EI}$
The flexural rigidity $(EI)$ of a cantilever beam is assumed to be constant over the length of the beam shown in figure. If a load $P$ and bending moment $PL/2$ are applied at the free end of the beam then the value of the slope at the free end is $\dfrac{1}{2}\dfrac{PL^2}{EI} \\$ $\dfrac{PL^2}{EI} \\$ $\dfrac{3}{2}\dfrac{PL^2}{EI} \\$ $\dfrac{5}{2}\dfrac{PL^2}{EI}$


text:
## The Annals of Probability

### A local CLT for convolution equations with an application to weakly self-avoiding random walks

#### Abstract

We prove error bounds in a central limit theorem for solutions of certain convolution equations. The main motivation for investigating these equations stems from applications to lace expansions, in particular to weakly self-avoiding random walks in high dimensions. As an application we treat such self-avoiding walks in continuous space. The bounds obtained are sharper than those obtained by other methods.

#### Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 206-234.

Dates
First available in Project Euclid: 2 February 2016

https://projecteuclid.org/euclid.aop/1454423039

Digital Object Identifier
doi:10.1214/14-AOP971

Mathematical Reviews number (MathSciNet)
MR3456336

Zentralblatt MATH identifier
1343.60009

#### Citation

Avena, Luca; Bolthausen, Erwin; Ritzmann, Christine. A local CLT for convolution equations with an application to weakly self-avoiding random walks. Ann. Probab. 44 (2016), no. 1, 206--234. doi:10.1214/14-AOP971. https://projecteuclid.org/euclid.aop/1454423039

#### References

• [1] Bauerschmidt, R., Duminil-Copin, H., Goodman, J. and Slade, G. (2012). Lectures on self-avoiding walks. In Probability and Statistical Physics in Two and More Dimensions. Clay Math. Proc. 15 395–467. Amer. Math. Soc., Providence, RI.
• [2] Bolthausen, E. and Ritzmann, Ch. (2001). A central limit theorem for convolution equations and weakly self-avoiding walks. Preprint. Available at arXiv:math.PR/0103218.
• [3] Brydges, D. and Spencer, T. (1985). Self-avoiding walk in $5$ or more dimensions. Comm. Math. Phys. 97 125–148.
• [4] Ritzmann, Ch. (2001). Strong pointwise estimates for the weakly self-avoiding walk. Ph.D. thesis, Univ. Zürich.
• [5] Slade, G. (2006). The Lace Expansion and Its Applications. Lecture Notes in Math. 1879. Springer, Berlin.
• [6] van der Hofstad, R., den Hollander, F. and Slade, G. (1998). A new inductive approach to the lace expansion for self-avoiding walks. Probab. Theory Related Fields 111 253–286.
• [7] van der Hofstad, R. and Slade, G. (2002). A generalised inductive approach to the lace expansion. Probab. Theory Related Fields 122 389–430.


text:
# Phenomenal World

Print

## Introduction

What happens when you give people cash? How do they use the money, and how does it change their lives? Every cash study on this list is different: the studies vary in intervention type, research design, location, size, disbursement amount, and effects measured. The interventions listed here include basic income and proxies—earned income tax credits, negative income tax credits, conditional cash transfers, and unconditional cash transfers. The variety present here prevents us from being able to make broad claims about the effects of universal basic income (UBI). But because of its variety, this review provides a sense of the scope of research in the field, capturing what kinds of research designs have been used, and what effects have been estimated, measured, and reported. The review also allows us to draw some revealing distinctions across experimental designs.

If you’re interested in creating a UBI policy, there are roughly three levels of effects (after ODI) that you can examine. On the micro level, you can scrutinize what happens to individuals and households that receive a transfer, e.g., how does children’s schooling change, how does consumption change? On the meso level, you can explore how groups of households or villages, community organizations, and community institutions change in response to a saturated transfer effect. The third level, macro, is the golden question of UBI research, but also the hardest to address: what does UBI do at the size of a province or nation? We’ve included the Levy Institute model as a starting point for macro-analysis. But this literature review mainly looks at what these studies reveal to be the micro and meso level effects, economic and social, including effects on schooling, health, crime, and labor markets.

What trends can we draw from these studies? This won’t be a surprise: overall, cash increases consumption.

Perhaps more surprising to UBI skeptics, the research overall does not show that cash transfers drastically lower labor supply by decreasing motivation to work. The negative effect of cash on labor supply is moderate if present at all. As a whole, the research also does not show that recipients of cash transfers are spending on temptation goods--rather, they’re putting the money into smoothing consumption, like increased food security and purchases of durable goods.

Beyond that, an intriguing metaresearch trend is how few UBI studies address labor market effects, and when they do, how broadly those studies define labor market effects. Most of the studies look at overall labor market equilibrium at most, and do not delve into a more nuanced analysis of, for example, number of hours worked, people moving in and out of full-time jobs and contract work, changing types of work, changing quality of life, and so on. Only one paper, on the conditional cash transfer program in Mexico, covers these areas with nuance. We hope to see more comprehensive research on labor effects of a basic income in the future.

As you read through this informal draft of what will become a formal academic review, we want to draw your attention to a few useful distinctions: differences between micro, meso, and macro level effects; differences in interventions; how different studies measure different effects; and the differences in the evidence from pilots in developing and developed world. Finally, there is a distinction between short-term cash interventions and long-term cash interventions—pilots that gave people money for a year, or decades, or indefinitely. We’ve included literature pertinent to both effects, but it’s important to note that this evidence is relevant to UBI, as ideally conceived, only partially, since a UBI would involve an indefinite disbursement of a transfer to cover minimum standards of living for a population.

The list is iterative and will be updated periodically as new research is published. Each section is organized chronologically, with the most recently reviewed articles at the top.

## Universal Basic Income

Universal basic income is one of several terms used to describe a system of regular and unconditional direct cash payments for all individuals in a given country or region. There are three important criteria which a basic income usually satisfies. First, it is given universally within a predefined community, which means treatment is not means-tested and is not meant for only a subsection of the community selected based on certain criteria. Second, the cash payments must be large enough to support a decent standard of living in the pre-defined community. Third, it usually constitutes a long-term stream of transfers to support the living needs of its recipients over a sizeable duration.

In recent years, a wide range of public, non-profit, and private actors have brought renewed attention to this concept, particularly its viability in the United States and other developed countries. The impact of a universal basic income is multi-dimensional—it can affect one’s consumption, work, physical and mental health, education and social relationships. It also has wide-reaching macroeconomic effects, potentially impacting economic growth, labor market outcomes and public finance.

We present here a list of papers and articles which touch on the various effects of and considerations behind designing and implementing a universal basic income. The pilots discussed in these paper usually fulfill (or come close to fulfilling) all three criteria of being a universal basic income. They vary by location and size, but taken together, these papers present some of the preliminary reported effects of a basic income.

• Using data from the 2017 Current Population Survey, Hoynes and Rothstein (2018) calculate that funding a complete UBI program would amount to around double the aggregate cost of present income transfers in the US. They argue that replacing existing anti-poverty programs with a UBI is likely to have negative impacts. However, existing literature is unable to assess the impact of a truly universal UBI, nor does it adequately examine the long term effects resulting from human capital development.
• In an August 2018 paper, Maximilian Kasy (2018) argues that a UBI holds economic, moral, and political advantages over existing subsidies like Earned Income Tax Credits (EITC). Economically, a UBI would produce a more equal income distribution by balancing labor market distortions resulting from EITC incentives to find work. It would also rectify the wage repression caused by these same incentives. Morally, a UBI increases individual liberty as well as equality of opportunity. People would face reduced barriers to exiting abusive relationships on which they economically depend, and unpaid care workers, artists, and community activists would receive remuneration. Politically, a UBI would see broader democratic support than means tested benefits. Kasy argues that a viable political case for a UBI must move beyond distinctions between the “deserving” and “undeserving” poor, replacing them instead with a shared vision of political interests.
• Abhijit Banerjee, Paul Niehaus, and Tavneet Suri (2018) summarize findings on the potential consequences of a UBI for developing countries. By alleviating credit, insurance, and psychological constraints to economic activity, Banerjee et. al argue that a UBI may generate economic growth. They also suggest that a UBI may overcome two drawbacks of targeted transfers: their constraining effect on political pressure for the preservation and expansion of benefit schemes, and the possible disincentivizing effects they could hold for labor markets.
• In a 92-page digital book, Karl Widerquist (2018) deals with problems of communication and interest inherent to policymaking in general, and UBI research in particular. UBI experiments are fundamentally compromised by the fact that researchers are unable ensure recipients receive the net benefit they would be left with after paying taxes. Added to this are challenges arising from community effects, long term effects, the hawthorne effect, the streetlight effect, and difficulty distinguishing between consequences arising from the size and type of program. The caveats to research findings are lost on policy makers and the public, who seek conclusive answers to fundamental questions regarding UBI consequences and implementation. In light of these imperfections, Widerquist recommends that researchers recognize that their work is embedded within a broader public discussion and focus on communicating what they can definitively contribute, while addressing the broader moral and political questions inherent to UBI debates.
• In this IMF working paper, Francese and Prady (2018) develop a two dimensional model which incorporates measures of UBI generosity, coverage, and progressivity. Their analysis illuminates three trade-offs policymakers face when considering UBI implementation: effective coverage of poor households risks leakages to richer households, generosity of transfers may produce work disincentives, and the fiscal cost of a UBI may detract resources from alternative programs. Using a series of static simulations, the report concludes that a UBI calibrated at 25% of net median income would produce a 5 point reduction in the gini coefficient and significantly reduce poverty in both developed and developing countries, but at a high fiscal cost.
• Earlier IMF research by Coady and Prady (2018) uses India as a case study to test the tradeoffs associated with UBI adoption. They demonstrate that replacing the 2011 Public Distribution System (which provides low income families with subsidies for wheat, sugar, and kerosene) with a UBI was found to increase coverage of low income households, but at the expense of leakage to some higher income households. A significant number of current PDS recipients would lose from the transition. However, replacing existing inefficient energy subsidies with a UBI was clearly found to have distributional gains and increase incentives for improving energy efficiency.
• In the Journal of Economic Perspectives, Hanna and Olken (2018) consider the potential for a UBI program to help governments of developing countries overcome targeting challenges resulting from the size of the informal sector. They investigate the empirical trade-offs of a UBI using data from Indonesia and Peru. Their research reveals a tradeoff between eliminating exclusion error and effectively meeting the needs of low income individuals. They conclude that targeting methods, while imperfect, can deliver more to recipients on an as-needed basis than universal programs. The welfare gains generated by targeted programs offset the losses in horizontal equity resulting from a portion of low income families failing to receive support.
• In this (2018) ILO Working Paper, Ortiz et. al argue that regressive UBI policies involving the elimination of other social safety nets are likely to generate greater inequalities and consequently fail to comply with ILO standards. The report finds that a progressive UBI which complies with ILO standards for income security would range in cost from 17.9% of GDP in the Middle East to about 25% in Asia, Europe, and Latin America. Sub-Saharan Africa constitutes an outlier to the general findings, with a UBI expected to cost about 50.3% of GDP. The authors argue that funding a UBI in this region is largely a question of priority-- the cost of the program for all low income countries would constitute only 0.68% of global GDP.
• Ravallion (2018) critically reviews both guaranteed income and guaranteed work from a policy perspective. With versions of the two rights-based policies in India as a case study, he argues that cost, targeting, corruption, and context variation across settings are central restrictions on unequivocal support of either policy in developing countries. In the case of India, Ravallion offers evidence that the country’s employment guarantee programs have been less effective at poverty alleviation than one might expect from a universal cash transfer program. He argues that the question of whether universal transfer programs will reduce poverty more than workfare or finely-targeted transfers is an empirical one that will vary across contexts, but suggests that the benefits of universal programs are worth consideration in certain settings.
• The Alaska Permanent Fund has paid a yearly dividend to state residents since 1982. To study the impact of the fund on labor market outcomes, Jones and Marinescu (2018) compare Alaska with a synthetic control constructed from a weighted average of control states. They find that universal and permanent cash transfers do not significantly decrease aggregate employment. One possible explanation is that general equilibrium effects will offset the tendency for cash transfers to decrease labor supply.
• Schulz (2017) argues that a universal basic income can help combat historical structural inequalities inherited by modern women by freeing women-led households from the vulnerability of poverty and their subsequent marginalization. The authors suggest that an additional source of income can also help balance unequal power relationships with male relatives and partners.
• Using the Levy Institute macroeconomic model, Nikiforos, Steinbaum, and Zezza (2017) show that enacting a UBI and paying for it with increases to the federal debt would grow the economy. The model also shows economic growth if the policy were paid for with increased taxes on households.
• Based on a review of the empirical results from U.S. and Canadian negative income tax experiments, the Alaska Permanent Fund Dividend, the Eastern Band of Cherokees casino dividend program, and other assorted studies on unconditional cash transfer programs, Marinescu (2017) finds no impact or a moderate decrease in labor participation. She finds additional effects in other quality-of-life benefits, such as improved mental and physical health, educational outcomes, parenting, and reduced criminal activity.
• A model developed by the District of Columbia’s City Council in a policy and impact statement on basic income offered contrasting results. According to their simulation model, providing a minimum income that is equal to 100% of the Federal Poverty Level is detrimental to the District’s economic growth and employment. The effect is worsened when the minimum income provided is raised to 450% of the Federal Poverty Level. They recommend further study on how such a policy intervention can be funded.
• Salehi-Isfahani and Mostafavi-Dehzooei (2018) study the effects of large scale cash transfer program in Iran starting in 2011, which distributed a fixed amount (amounting to 29% of the median per capita household income) of cash to all Iranians (with a 95% take-up). Using panel data and a difference-in-differences model, they find no evidence that cash transfers reduced labor supply, except among youth (20-29 years), where they find a negative effect. They also find that service sector workers increased their hours of work.
• Calnitsky and Latner (2017) analyze the labor market effects of the Manitoba Basic Annual Income Experiment (Mincome) (1975-77) and report that the moderately sized basic income generated an 11.3 percentage point reduction in labor market participation, however this was largely driven by young and single-member household participants. They also note 30% of this reduction can be attributed to “social interaction” or “community context” effects.
• Colombino (2015) argues that an unconditional basic income can redistribute the gains from automation and globalization, build a buffer against shocks and systemic risks, and generate positive labor supply incentives amongst the poor. While Colombino suggests a UBI is simple, he notes that financing it likely requires higher taxes, hence tax systems need to be carefully designed to avoid canceling the potential benefits of unconditional basic income through efficiency losses.
• Assessing a study involving the distribution of a monthly basic income grant of NAD100 to around 1000 citizens in Namibia, Rigmar (2013) reports that the study lacked transparency and rigorous reporting, and the results raise concerns on the validity of the reported findings. Therefore, the experiment could not address questions about basic income in a political decision making context.
• According to Forget, Peden and Strobel (2013), empirical evidence and social theories suggest that cash transfers not only affect individual decision-making but also impact how households interact within a community, hence it is important to understand the social impact of basic income when addressing questions on how best to design cash transfer schemes.
• In the Manitoba basic annual income field experiment (Mincome) from 1974 to 1979, families with no income received 60% of the Statistics Canada Low-Income Cut-Off (LICO). Forget (2011) reports this resulted in an 8.5 percent reduction in hospitalization rates and reduced contact with physicians, showing that a relatively modest guaranteed annual income can improve population health and bring significant health system savings.
• Akee et al (2010) assessed the effects of a universal cash transfer administered by Eastern Band Cherokee, and highlight that regular unconditional cash transfers led to the children of recipients achieving higher levels of education in their young adulthood and a lower incidence of criminality for minor offenses, likely attributable to increased stability and support in the home.

## Earned Income, Earned Income Tax Credit, and Negative Income Tax

A negative income tax (NIT) is a progressive tax where households whose income is below a predetermined level receive an income supplement. An earned income tax credit is a refundable tax credit for workers of low to middle income, which has a similar effect of creating a progressive tax system. These differ from a basic income as they are not distributed universally within a predefined community, and the transfer amounts depend on the income level of the household.

We include earned income studies in this review as the effects of a lump-sum increase in income can be relevant to understanding the effects of a basic income. One of the primary concerns economists have regarding a NIT is that it is distortionary and affects recipients’ incentive to work. Since unemployed individuals are ineligible for benefits, an NIT is thought to be better than a Basic Income as it still incentivizes households to work. On the other hand, some economists think that a NIT will reduce labor supply, since the magnitude of the benefit distributed varies negatively with the income level of the household. As such, many studies have been conducted to analyze what exactly is the impact of an NIT on labor supply and employment outcomes. Other studies have also documented economic, health and social effects of an NIT which policymakers should consider.

• The MDRC (2018) presents findings from a three year RCT of the Paycheck Plus program in New York City. The study found that Paycheck Plus reduced severe poverty, marginally increased employment among women and disadvantaged men, increased tax filing rates, and increased child support payments among noncustodial parents. Participants also reported reduced depression and/or anxiety. Analysis of these findings is expected to continue upon the termination of a similar RCT in Atlanta.
• Bastian and Jones (2018) reconsider the cost of EITC in light of the behavioral changes it elicits in recipients. Using thirty years’ worth of data on EITC policy expansions, they conclude that EITC is one of the least expensive anti-poverty programs in the United States. For every 350 dollars in EITC spending, total government revenues increased by 303 dollars, compensating for 87% of the program cost through positive spillover effects. Expansions in EITC were found to increase average annual earnings and labor supply, reflecting an overall supply elasticity of 0.33. EITC expansions were also found to increase payroll and sales taxes paid and reduce dependence on public assistance.
• Using U.S. administrative prison release records released between 2000 and 2014, Agan and Makowsky (2018) use a difference-in-differences strategy to identify the effect of minimum wage increases and EITCs on recidivism. In their analysis on EITC, the availability of state top-ups to the federal EITC is associated with a 1.6 percentage point lower rate of recidivism amongst women, while having no significant effect on men.
• Bastian and Michelmore (2018) examine the long-run impact of an EITC on children who were exposed to EITC variations by using EITC exposure as an instrument for family income. With data from the 1968 to 2013 waves of the Panel Study of Income Dynamics they find that an exogenous shock in the form of increased pre-tax income induced by a change in EITC policy increased educational attainment and employment in adulthood. Their reduced-form estimate suggest that an increase of $1000 in EITC exposure increases high school graduation and college graduation by 1.3% and 4.2% respectively for children exposed to the increase when they were between the ages of 13 to 18. It also raises the probability of them being employed by 1.0%. However, EITC exposure had no significant effects on children below the age of 13 conditional upon later exposure. • The EITC has been found to reduce poverty rates as well as improve health and education outcomes. However, for families with two or fewer children, the maximum credit distributed to them has remained the same in inflation-adjusted terms since 1996. As such, Hoynes et al (2017) propose to increase the EITC credit by 10% while maintaining its current structure so as to increase wages which have been stagnating, further increase employment, and bring about 600,000 people out of poverty. • Wiederspan et al (2015) argue that a negative income tax can help fill the gaps in current federal means-tested assistance programs. While past NIT experiments have led to a reduction in labor supply, the effects were not as large as often speculated, and time saved from work was spent on childcare and looking for work. Moreover, those experiments did not measure long-term, general equilibrium effects. The authors show that an NIT could be funded using expenditures from existing programs which it could replace, and if the marginal tax rate was chosen carefully. Finally, they assert that a NIT could be made more popular by appealing to moral values rather than focusing on the technical policy details. • The Earned Income Tax Credit (EITC) in the U.S. serves to offset the payroll taxes and supplement the wages of low-income workers, with payments concentrated in February and March when tax refunds are received. A 2008 study by Goodman-Bacon and McGranahan found that in 2004, the recipients concentrated most of their household spending in vehicle purchases and transportation spending in response to their EITC payments. • Based on a survey on five NIT experiments in Canada and the U.S. from 1969 to 1980, Widerquist (2005) concludes that labor market effects of negative income tax experiments are often overstated and misinterpreted, since without an estimate of the labor market demand response, researchers cannot estimate the overall market response. • Robins’s (1985) comparison of four negative income tax experiments sponsored by the US federal government between 1968 and 1982 found that a negative income tax led to a fall in labor supply by about the equivalent of two weeks of full-time employment for husbands, three weeks for wives and single female heads, and four weeks for youth. • A negative income tax led to an increase in the average reading achievement of children in grades 4 through 6, but did not have any observable beneficial effects for older children, according to this 1979 study. ## Cash Transfers, Wealth Transfers, and Unearned Income This category lists papers which look at the impact of cash transfer programs which were excluded from the first section due to the nature (smaller amounts of disbursement) and size (smaller sample size) of the experiments. These cash transfers are usually implemented by governments, NGOs or charities. A plethora of studies has been conducted to analyze how cash transfer programs affect economic, social, employment and health-related outcomes of households, and how various aspects of cash transfer programs, such as the program duration, magnitude of cash transfers and frequency of transfers can alter these effects. Some programs provide cash transfers with no strings attached, while others come with conditions that require recipients to take certain actions which will improve other aspects of their lives. In such programs, cash may be seen as an incentive or an enabler for recipients. Some studies also look at how cash transfer programs may have differential effects depending who in the household is a recipient of the transfer. These are all important considerations that need to be taken into account when designing a cash transfer program. • In a World Bank Policy Research Working Paper, Baird et. al (2018) examine the intergenerational effects of Conditional and Unconditional Cash Transfers (CCTs and UCTs). Between 2007 and 2012, they conducted a series of interviews with women aged 13-22 in Zomba, Malawi. Study participants received CCTs until 2009, and subsequent interviews allowed the authors to trace the impact of cash transfers on their lives as they established families and transitioned into adulthood. Their research indicates that the significant decline in HIV prevalence, teen pregnancy, and early marriage in UCT recipients did not endure two years after disbursements ended. Out-of-school recipients of CCTs, however, did exhibit higher educational attainment and a reduction in the number of births, despite showing no long term gains in health, labor market, and empowerment. • Bougen et. al (2018) test the ability of Randomized Control Trials (RCTs) to provide evidence for long term impacts in development economics. Among the 170 RCTs which test the impact of both CCTs ad UCTs, they found that only 18 are designed to enable long term research: 6 which have already done so, and 12 which hold the potential for follow up investigations. The remainder of cash transfer RCTs exhibited design or data limitations which prevent continuing research and analysis. The authors argue establishing uniform data collection and tracking protocols across multiple studies may generate comparable estimates on their long term effects, thereby improving external validity. • Wydick (2018) develops a model indicating when cash transfers are likely to significantly and sustainably increase the income of recipients. He argues that small cash transfers will increase the recipient's present consumption, while generating a continuous dependence on charity. Intermediate levels of cash transfers are likely to increase present consumption while facilitating saving for future consumption. Finally, large cash transfers are expected to generate investments in future productivity that can be expected to transform future income and labor activity. He concludes that large cash transfers which account for deviations in cognitive, behavioural, and aspirational phenomena, and which are situated within a system of complementary policies, programs, and interventions, are likely to be the most effective. • Akee et. al (2018) use data from the Great Smoky Mountains Study (GSMS) to examine how UCTs impact voting behavior across two generations. They find that the exogenous UCTs which were administered increased the likelihood that children from low income backgrounds will vote upon adulthood, but had no effect on the voting behaviors of parents, regardless of their socioeconomic background. They suggest that the results are likely driven by higher educational attainment and social capital acquisition, rather than the transmission of voting behavior from parents to children. • Maboshe and Woolard (2018) use a 2014-15 household survey to examine the progressivity of direct taxes and cash transfer programs in South Africa. Both programs are found to be highly progressive, with the top three deciles bearing the overwhelming weight of the tax burden, and cash transfers benefiting the most vulnerable households. Households headed by women, located in rural areas, and with children received significantly more than their counterparts. The South African tax and transfer program contributed to a 0.06 reduction in the Gini coefficient and a two thirds reduction in the rate of inequality. • Barham, Macours, and Maluccio (2018) examine the impact of CCT on the long term nutrition, education, and reproductive health of teenage girls. They rely on data compiled from a randomized CCT program in Nicaragua, which divided participants into two treatment groups, one of which began receiving benefits three years prior to the other. They measure outcomes ten years after the program began, and five years after participants in the late treatment group stopped receiving transfers. Girls randomly exposed to CCT in the early treatment group had higher earnings than those who began receiving transfers three years later. They also reached sexual maturity later, had a lower BMI, and exhibited lower overall fertility rates when they reached 18-21 years of age. • Using a two stage randomization design in a randomized controlled trial implemented by GiveDirectly, Haushofer and Shapiro (2013) found that unconditional cash transfers in Kenya led to a rise in food, medical and education expenditures, and a fall in expenditures on alcohol and tobacco. Cash transfers also resulted in large increases in the recipients' psychological well-being. Their research also suggests that households face savings constraints, since one-off lump-sum transfers tended to be spent on durables. • In the same experiment, Haushofer and Shapiro (2018) assess the economic and psychological outcomes three years after the beginning of the GiveDirectly cash transfer program in Kenya. The authors report that recipient households had 40% more assets than non-recipients in the same village, but there were no statistically significant effects in other outcomes. They also note spillover effects, wherein households affected by the spillovers have lower consumption and food security than pure control households. For detailed comments and discussions on this paper’s findings, click the following links from GiveDirectly, the World Bank, and CGDEV’s blog. • Kugler and Rojas (2018) assess the effect of the exposure to PROSPERA, a conditional cash transfer program, in Mexico on three education variables: educational attainment, high school education and at least one year of tertiary education, and five employment variables: probability of being employed, the number of hours worked per week, employment quality, likelihood of having an employment contract, the likelihood of having non-wage benefits, and hourly wages. After controlling for multiple hypothesis testing, the authors find significant and positive effects in all cases. The positive effects are, however, weaker for women and children of illiterate women. • Noting that there are several potential channels through which a cash transfer program can affect labor supply apart from the standard income effect, Baird et al (2018) review a range of studies on conditional and unconditional cash transfers. They conclude that unconditional cash transfer programs, cash transfer programs with conditions unrelated to employment and remittances tend to have limited to no impact on adult labor. On the other hand, cash transfers for job search assistance and small business start-up usually increase labor supply likely by alleviating liquidity and risk constraints, although the effects through these channels cannot be isolated from those due to increases in human capital as well as other general equilibrium effects. • Akee et al (2018) examine the effects of cash transfers from the distribution of casino profits amongst adult Eastern Cherokee tribe members on child development. Using data from the Great Smoky Mountains Study of Youth, they find that an increase in unearned household income led to large positive effects on children’s emotional and behavioral health and positive personality trait development, especially amongst children who did not do as well as their peers in these aspects prior to the treatment. They also find that improved parental behavior and familial relationships was likely to be an important channel through which financial well-being may impact child development. • Angelucci et al (2017) study both expected income shocks brought about by the PROSPERA program and unexpected income shocks due to health and employment shocks. They find that an unexpected negative shock in income reduces labor supply, food security, mental health and healthful behavior, but does not affect parenting, cognition, risk and time preferences, and aspirations about children's' educational attainment. On the other hand, an expected positive shock caused by a conditional cash transfer does not lead to any change in these outcomes apart from increasing disposable income. • Banerjee et al. (2017) studied data collated from seven CT/UCT/in-kind transfer studies in Africa and Asia and showed that cash transfer programs have no observable impact on one's propensity to work or the overall number of hours worked for both men and women. • Walker (2017) estimates the effect of exogenous positive income shocks to household income in Kenya on informal taxes and public goods provision. Using surveys of village leaders, and panel and experimental household data from a sample of participants who received a large one-time unconditional cash transfer in GiveDirectly’s RCT, Walker estimates regression models at household, village and sub-location level and reports no significant effects on the amount paid in formal taxes, nor on the informal tax rate. He notes a 2.4 percentage point increase in county tax payment which he relates to the fact that recipient households are more likely to be self-employed. There is also no attributable effect on public goods provision or the quality of public good provision, indicating no short run effects of the income changes on public good provision or informal taxation systems that can be attributable to the exogenous income shock. • Evans and Popova (2017) review the effects of cash transfers on temptation goods, defined as expenditures on alcohol and tobacco. Compiling 44 estimates from 19 studies on conditional and unconditional cash transfers in low- and middle-income countries from 1997-2014, they find with almost no exceptions either an insignificant effect or a significant negative effect of cash transfers on temptation goods. When they differentiate only the estimates from studies which had a randomized design, no estimate was reported to show a positive and significant effect on expenditures on temptation goods. • A systematic review of 201 studies by the Overseas Development Institute (2016) on both conditional and unconditional cash transfer published between 2000 and 2015 finds links to the following: reductions in monetary poverty, increases in total and food expenditure, increases in school attendance, increases in the uptake of health services, and increases in savings, livestock and agricultural ownership. The effect of cash transfers on employment outcomes remains unclear. Higher transfer levels are associated with larger impacts. Longer duration of exposure to cash transfers are linked to greater improvements in outcomes. Cash transfers conditional on certain behaviors more positively affect the outcome related to the behavior than in unconditional cash transfers. • Results from New York City’s conditional cash transfer program (2016) for families that received over$8700 on average over a three-year period show that the transfers reduced financial hardship, increased graduation rates for reading-proficient 9th graders who entered high school, and increased households' use of preventive dental services. It did not affect school outcomes for elementary and middle school students, the use of preventive medical services, and parents’ employment in or earnings from jobs covered by the unemployment insurance system.
• Baird et al (2016) investigates the medium-term effects of a two-year, two-treatment arms cash transfer program to adolescent females in Malawi, where one treated group received cash transfers conditional upon school attendance while the other received equal-sized unconditional cash transfers. They found that while unconditional cash transfers led to reduced teen marriages, live births and HIV infections as well as improvements in psychological wellbeing at the end of the program, these effects were not sustained after two years. Their husbands had relatively lower cognitive ability, although their children born during the program had improved HAZ scores. On the other hand, those that received conditional cash transfers saw sustained positive effects on school attainment, teenage marriage and pregnancy incidences, and desired fertility. They were also more likely to be matched with husbands who had completed secondary school. There are however no gains in other outcomes such as earnings, consumption, health, empowerment and subjective well-beings.
• Between 2010 and 2011, the Zambian government launched two unconditional cash transfer pilot programmes, described in this 2016 paper from Handa et al. The Child Grant Program (CGP) targeted all households with a child under three years old in three poor rural districts, while the Multiple Category Targeted Programme (MCP) was implemented in two rural districts and targeted vulnerable households such as those with a disabled family member, or those with a female or elderly head caring for orphans. Bimonthly transfers were made to each household irrespective of size, and this entailed a transfer of USD 24 per month in Purchasing Power Parity terms. ITT estimates made 24 months and 36 months into the study in nine domains, namely total consumption, food security, asset holdings, subjective well-being (relative poverty), income and revenues, finance and debt, material needs, schooling and young child anthropometry, show that that both the CGP and the MCP had statistically significant effects in almost all domains except for schooling and young child anthropometry for the CGP, and incomes and revenues after 24 months for the MCP (effects on young child nutrition were not measured for the MCP). A sizeable multiplier effect is observed given that households spent 59% more than the value of the transfer.
• In a large-scale unconditional cash transfer program in Indonesia, Bazzi et al (2015) found that timely receipts of cash transfers do not yield expenditure changes relative to non-recipients, while delaying the delivery of the transfers reduces expenditures by 7.5%, suggesting that the impact of cash transfers depends on the timing of transfers.
• Analyzing the long-run impact of the Mother's Pension program, a government-sponsored welfare program in the US (1911-1935), Aizer et al. (2014) found that male children of successful applicants lived one year longer, received one-third more years of schooling, were less likely to be underweight, and had higher income in adulthood compared to those of rejected mothers.
• Kenya's Cash Transfer for Orphans and Vulnerable Children Programme (2014) involve a flat monthly transfer of KSH 1500 to ultra-poor families with orphans and vulnerable children aged 17 years and younger. The transfers increased food consumption coming from home production, accumulation of productive assets, and formation of non-farm enterprises, especially amongst females. It also provided more flexibility to families in terms of labor allocation decisions and reduced child labor, especially for boys. However, it had little impact on crop production.
• Cunha (2014) studied a Mexican food assistance program aimed at promoting better nutrition, and found that in-kind food transfers did not lead to meaningful differential improvements in overall consumption and health outcomes as compared to the provision of cash transfers, as the cash was largely spent on nutritious food as well.
• McEwen and Stewart (2014) study the effects of the Canadian child allowance program that started in 1945. They highlight that significant correlation between lower income and worse child outcomes (social, cognitive, behavioral, emotional) can be explained by factors usually associated with a lower income, such as lower educational attainment, less employment, and living in a risky neighbourhood. A household's income itself has a relatively small influence on child outcomes after considering these factors. This suggests that raising a poor household's income alone may not have a sizeable impact on child outcomes.
• According to Akee et al. (2013), a cash transfer experiment in North Carolina (US) shows that, controlling for initial health conditions, household cash transfers lead to a larger increase in weight for children from the initially poorest households relative to those from richer households.
• Studying the 1832 Cherokee Land Lottery, in which winners were randomly selected, Bleakley and Ferrie (2013) found that winners were on average richer in the long-run, but this was due to a movement of the winners in the middle of the wealth distribution to the upper tail of the wealth distribution. At the same time, winners in the lower tail of the wealth distribution were unaffected (no movement), hence exacerbating inequality.
• Taylor et al (2013) examine the general equilibrium effects of a cash transfer for orphans and vulnerable children (CT-OVC) in Kenya. Using four administrative and private data sources, they simulate the estimates of income multipliers and highlight significant spillover effects of the CT-OVC transfers in local economies. Local income multipliers are significantly greater than 1 in nominal terms, even when capital and labor constraints limit the local supply response. They note most of the effects are among ineligible households, indicating that the program enables them to access more capital and labor to increase their production in response to local demand. They note variations in impacts across regions in the income multiplier, and highlight differences in capital, liquidity and labor market constraints.
• DFID (2011) reports that there is substantial evidence that cash transfers can reduce inequality, boost movement out of poverty for working age populations, and lead to sizable gains in access to health and education indicators (with less marked success in the final outcomes of these two sectors). The authors report indicative and deterministic factors showing that cash transfers can improve household productivity, income generation, gender equality, empowerment of the poor, and local market development. They also report that there is little evidence of negative labor market effects and negative effects on fertility. While acknowledging the limited evidence on aggregate growth, the authors suggest that cash transfers can lead to long term growth through improvements in human capital.
• Ardington, Case, Hosegood (2009) found that in South Africa, large cash transfers in the form of pension payments to the elderly led to increased employment amongst prime-aged adults through labor migration, as a pension increases household resources which can be used to support migrants until they become self-sufficient. The finding implies that pensioners are able to care for small children while prime-aged adults look for work.
• A study by Imbens, Rubin, Sacerdote (2001) assessing lottery winners in Massachusetts in the 1980s finds that unearned income reduces labor earnings, with a marginal propensity to consume leisure of approximately 11 percent, and with larger effects for individuals between 55 and 65 years old. The savings rate for unearned income is about 16 percent for those who received half their income and increases with the proportion of the income received.

## Conclusion

There is only a limited amount that studies on small-scale cash transfer interventions can tell us about UBI. However, one thing we can learn from these studies is that there is little empirical evidence to suggest the materialization of the harms or dangers usually associated with cash transfers. In other words, we don’t see entire populations quitting work or using the cash transfers on temptation goods. This possibly provides a reason for sustained interest in basic income research.

Based on our review of the literature, it would be useful for future research efforts to examine labor effects and community effects more closely. Most of the conditional cash transfers and cash transfers studies conduct sampling on the individual level, and it is hard to find studies which report on community effects and changes in interpersonal relationships. We would also like to see more post-intervention studies about what happens to recipients at the end of the disbursement periods, and in particular whether the changes in their behavior are sustained. We are also excited to see more papers on the long-term impacts of cash transfers—on the effects on distribution of wealth and intergenerational effects in households, for example. We want to note our appreciation for the papers on the study in Kenya, which cover a whole gamut of effects--the 2018 paper examines spillover effects, which is both rare and important in this discourse.

Finally, we see a need for pilots to be carried out on a much larger scale, in order to get at the population-wide effects of a UBI. A large-scale pilot will be useful in helping to generate empirical macroeconomic analysis, as well as reveal new effects at the micro- and meso- economic levels. After all, a UBI is not just a research study, but an institution. It is well worth considering what it would look like to implement such an institution.

Thank you to Karla Ganley, Lisa Hattori, Mario Saraiva, Claudia Vargas, Ivana Vasic-Lalovic, and Eirene Wang of Columbia SIPA for their help compiling these materials. We’d also like to acknowledge the World Bank Development Impact blog for inspiring the format of this post.

 Title Cash and Income Studies: A Literature Review of Theory and Evidence Authors Date 2019-02-04 Collection Reviews Filed Under

Sign up for the JFI Letter, a weekly digest of compelling research across the social sciences.


text:
### Home > GB8I > Chapter 7 Unit 8 > Lesson INT1: 7.1.6 > Problem7-73

7-73.

The Micky Mac apple plantation has a very large orchard of apple trees. They have Macintosh apple trees, which yield an average of $16\text{,}000$ pounds per acre, and Jonathan apple trees, which yield an average of $14\text{,}500$ pounds per acre. The Micky Mac plantation has $M$ acres of Macintosh apple trees and $J$ acres of Jonathan apple trees.

1. Last year, the Micky Mac plantation yielded a total of $120\text{,}000$ pounds of apples. Write an equation that relates $M$ and $J$ to the total yield.

How many pounds of Macintosh apples can be produced in $M$ acres?
How many pounds of Jonathan apples can be produced in $J$ acres?
How does this relate to $120\text{,}000$ lbs in the problem?

$16\text{,}000 M + 14\text{,}500 J =120\text{,}000$

2. Macintosh apples sell for $$650$ per metric ton and Jonathan apples sell for$$700$ per metric ton. If the plantation produces apples worth \$$1\text{,}070\text{,}000$ then write an equation relating $M$ and $J$. Note: One metric ton equals approximately $2200$ pounds.

If an orchard of Macintosh apples yields $16\text{,}000$ lbs of apples an acre, how many tons is it?
If an orchard of Jonathan apples yields $14\text{,}500$ lbs of apples an acre, how many tons is it?
Remember $2200\text{ lbs} = 1\text{ metric ton}$.

$650(7.27 M) + 700(6.59 J) = 1\text{,}070\text{,}000$ or $4725.50 M + 4613.00 J = 1\text{,}070\text{,}000$


text:
# Homework Help: Rearranging (1+x)e^-x = 0.5

1. Jan 6, 2008

### Xeract

Sorry if this is a very simple question, I am trying to rearrange (1+x)e^-x = 0.5 for x, and just can't seem to get my head around it. Any tips would be greatly appreciated.

2. Jan 6, 2008

### chaoseverlasting

Take log on both sides and go from there.

3. Jan 6, 2008

### rocomath

Take the natural log of both sides and then use the properties of log to split the left side.

$$\ln(a/b)=\ln a - \ln b$$

And remember that $$\ln e = 1 = \mbox{cancels each other out}$$

Last edited: Jan 6, 2008
4. Jan 6, 2008

### Xeract

This is where I've got to, thanks for the responses:

ln((1+x)/e^x) = ln0.5
ln(1+x)-x =ln0.5

Last edited: Jan 6, 2008
5. Jan 6, 2008

### rocomath

I can't seem to go anywhere from your last step, is the problem written down correctly?

6. Jan 6, 2008

### malawi_glenn

do you have to solve this analytically?

7. Jan 6, 2008

### nicksauce

It's certainly impossible to solve analytically.

8. Jan 6, 2008

### malawi_glenn

yes, thats why I asked the OP ;)

9. Jan 6, 2008

### HallsofIvy

In general, a problem involving "x" both in an exponent and not can only be solved using the "Lambert W function" which is defined as the inverse function to f(x)= xex- and may require manipulation of the equation to put it in that form.

10. Jan 16, 2008

### llemes4011

I don't know if i'm doing this right, but here are my thoughts. if you factor out the first side you get xe^x+e^x=0.5 Then if you divide both sides by x you get e^x+e^x=0.5/x add your like terms on the left and you get 2e^x=0.5/x (i think that's right)
See if you can get it from there
((i just realized that this was from over a week ago AFTER i finished lol))

11. Jan 16, 2008

### rocomath

That's wrong, read all the posts before yours.

12. Jan 16, 2008

### llemes4011

sorry, i also didn't read the original question right *sigh*

13. Jan 16, 2008

### rocomath

It's ok, no worries.


text:
## Tuesday, September 17, 2013

### my favorite surprise

I’m excited about tomorrow. Tomorrow in my probability class, we’re going to start discussing Bayes’ Formula. This is the main thing I remember about my college probability class. While we have already seen some surprising results in particular cases where the rules of probability are applied, this is, to me, the first truly surprising general result. It changes everything I think about probability.

Here’s my motivating example: suppose we have before us an urn that contains five blue balls and five red balls. We draw two balls, in order, and record their colors. (To be clear, this is sampling without replacement.) What is the probability that the first ball is red? “Well,” you say, “evidently the likelihood of that is 1/2, because half of the balls are red.” “Very well,” I say, “then what is the probability that the first ball is red, assuming that the second ball is blue?” “What does that have to do with it?” you ask. “When the second ball is drawn, the first one has already been chosen, so how could the second ball turning up blue have anything to do with the probability that the first ball is red?”

Let’s throw some formulas in here. Suppose $E$ and $F$ are two events in the sample space $S$ of an experiment. (I discussed the definitions of these terms in my previous post.) The conditional probability of $E$ given $F$, written $P(E\mid F)$, is the quotient $\dfrac{P(E \cap F)}{P(F)}$, meaning, loosing speaking, that we consider all the ways both $E$ and $F$ can occur (weighted by their individual probabilities), and think of this as just a subset of the outcomes where $F$ occurs (rather than all of $S$). “Sensible enough,” (I hope) I hear you say. Now, you will hopefully also agree that we can split $F$ into two parts: the one that intersects $E$ and the one that does not, i.e., $F = (F \cap E) \cup (F \cap E^c)$. “Aren’t you overcomplicating things?” you demur. “Just wait,” I plead. Because the events $F \cap E$ and $F \cap E^c$ are mutually exclusive (i.e., disjoint), and so we have $P(F) = P(F \cap E) + P(F \cap E^c)$. Interesting, no? So we can write $P(E \mid F) = \frac{P(E \cap F)}{P(E \cap F) + P(E^c \cap F)}$ (using the fact that $E \cap F = F \cap E$). And now perhaps it seems like this manipulation isn’t so weird, because in our “motivating case”, each of the terms in that expression isn’t so hard to compute, and in fact one of them appears twice!

So what happens? Let’s return to our urn and say $E$ is the event “the first ball is red”, while $F$ is the event “the second ball is blue”. Then $P(E \cap F) = \big(\frac{5}{10}\big)\big(\frac{5}{9}\big)$ and $P(E^c \cap F) = \big(\frac{5}{10}\big)\big(\frac{4}{9}\big)$, so $P(E \mid F) = \frac{\big(\frac{5}{10}\big)\big(\frac{5}{9}\big)}{\big(\frac{5}{10}\big)\big(\frac{5}{9}\big)+\big(\frac{5}{10}\big)\big(\frac{4}{9}\big)} = \frac{25}{25+20} = \frac{5}{9}.$ Since 5/9 > 1/2, it is more likely that the first ball was red if we know that the second ball is blue! (Surprised? Think about what happens if there are only two balls to begin with, one blue and one red. Once that’s sunk in, try the above again starting with $m$ blue and $n$ red balls in the urn.)

So far I’m cool with everything that’s happened. The realization that later events provide information about earlier ones is a bit of a jolt, but not so far-fetched after a little reflection. Bayes, however, endeavors to turn our minds further inside-out. We just need one new idea, just as simple as everything we’ve done to this point: the equation for conditional probability can be rewritten as $P(E \cap F) = P(F) \cdot P(E \mid F)$. And of course, because $E \cap F = F \cap E$, we could just as well write $P(E \cap F) = P(E) \cdot P(F \mid E)$. Now, as before, let’s split $F$ into $E \cap F$ and $E^c \cap F$. Using our most recent observation, we have $P(E \mid F) = \frac{P(E) \cdot P(F \mid E)}{P(E) \cdot P(F \mid E) + P(E^c) \cdot P(F \mid E^c)}.$ “Now why on Earth…?” you splutter, to which I reply, “Because sometimes the knowledge you have is more suited to computing the conditional probabilities on the right than finding the one on the left directly from the definition.”

Here’s a classic example. Suppose there is an uncommon illness that occurs in the general population with probability 0.005 (half of percent). Suppose further that there is a medical test for this affliction that is 99% accurate. That is, 99% percent of the time the test is used on a sick patient, the test returns positive, and 99% of the time it is used on a healthy patient, it returns negative. You are concerned that you might have this illness, and so you have the test. It comes back positive. What is the probability that you have the illness?

Do you see where this is going? You’re interested (well, we both are, really, because I care about your health) in the event $E$ “I have the illness.” The information we have, though, is that the event $F$ “the test came back positive” occurred. And what we know about the test is how its results depend on the patient being sick or well. That is, we know $P(F \mid E)$ and $P(F \mid E^c)$, and fortunately we also know $P(E)$ (ergo we also know $P(E^c) = 1 - P(E)$). We can compute the likelihood of your being ill as $P(E \mid F) = \frac{(0.005)(.99)}{(0.005)(0.99) + (0.995)(0.01)} \approx 0.3322.$ Far from it being a certainty that you have this particular illness, your chances are better than 2 in 3 that you don’t! Even if the illness were twice as common and occurred in 1% of the population, your chances of being sick are only 1 in 2 after the test comes back positive. (Notice that this probability—this conditional probability—depends not only on the efficacy of the test, but also on the prevalence of the illness.)

And that’s my favorite surprise in probability.

(If you haven’t read it yet, you should go look at the article Steven Strogatz wrote for the New York Times about Bayes’ Theorem, in which he makes it seem—somewhat—less surprising.)


text:
# Hom(CoherentSheaf,CoherentSheaf) -- global Hom

## Synopsis

• Function: Hom
• Usage:
Hom(F,G)
• Inputs:
• F,
• G, , both should be sheaves on a projective variety or scheme X = Proj R
• Outputs:
• , over the coefficient field of R

## Description

If F or G is a sheaf of rings, it is regarded as a sheaf of modules in the evident way.

 i1 : R = QQ[a..d]; i2 : P3 = Proj R o2 = P3 o2 : ProjectiveVariety i3 : I = monomialCurveIdeal(R,{1,3,4}) 3 2 2 2 3 2 o3 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c) o3 : Ideal of R i4 : G = sheaf module I o4 = image | bc-ad c3-bd2 ac2-b2d b3-a2c | 1 o4 : coherent sheaf on P3, subsheaf of OO P3 i5 : Hom(OO_P3,G(3)) 7 o5 = QQ o5 : QQ-module, free i6 : HH^0(G(3)) 7 o6 = QQ o6 : QQ-module, free


text:
# Tachyonics and General Relativity

79pages on
this wiki

The Tachyonics Operator Provides Grand Unified Field

I attempt here to convey more of my theory on Superluminal Gravitation, mostly in words rather than mathematical symbols; rendering many of the equations I employ into word-form. In some cases, however, it is impossible or impractical to put everything that completes a given description of a complicated equation into words, so I include print and online references as needed, so readers can look up the equations in standard form.

Einstein's theory of General Relativity (GR) is a geometric theory, based primarily on Riemannian Geometry. It describes the curvature of some region of space (the warping of space-time) brought about by the presence of matter in that region. Note, however, that the field equations of GR are entirely geometric in character, and do not specify the quantum nature of gravity. A quantum theory of gravity must be attached, in order to describe gravity at both microcosmic and macrocosmic distances.

In the field equations of GR, the tensor that gives the values corresponding to curvature of space associated with the presence of matter in a particular region of space, is called the "Einstein Tensor" (Kay, 119). It is a 2nd-order mixed tensor defined as the difference between two other tensors. One is a mixed tensor determined as the product of a Ricci Tensor (of second kind) and the contravariant 2nd-order Riemannian Metric Tensor. The other is one-half the result from applying the second-order mixed Kronecker Delta to the invariant radius-of-curvature, also called the "Ricci Scalar". [Note: The divergence of the Einstein Tensor is zero at all points determined using a Riemannian metric. (Kay, 120)] The equation expressing this difference states: the Einstein Tensor equals the product of the Ricci Tensor and the Riemannian Tensor minus one-half the result of applying the Kronecker Delta to the Ricci Scalar. (Kay, 119)

The Ricci Tensor of second kind is the 2nd-order covariant contraction of the 4rd-order mixed form (1st-order contra-variant, 3rd-order covariant) defined using operators called "Christoffel Symbols" and some generalized coordinates (Cartesian, curvilinear, etc.).

A Christoffel symbol is not a tensor (despite looking like one), but is an operator that yields a tensor when applied to a scalar, or to a vector or other actual tensor. In this case, Christoffel symbols are used to define the 4th-order Ricci Tensor as an operator defined as the sum of two differences; one is a function of coordinates, and the other involves two two-symbol products. The two functions of coordinates are operators obtained as partial-derivatives with respect to two generalized coordinates of one Christoffel symbol each. [Handle the subscripts and superscripts of the tensors and coordinates carefully.] The equation states: the 4th-order mixed Ricci Tensor is equal to ((the partial-derivative, with respect to the first generalized coordinate-set, of one Christoffel symbol)) minus ((another such partial-derivative, with respect to the other coordinate-set)) plus ((the product of two Christoffel symbols)) minus ((the product of two more symbols)), with subscripts and superscripts observing a required arrangement. [Reference: Kay, page 101, and Example 9.5, page 119 (on proving Bianci's Second Identity).] [Online, search "Ricci Tensor", "Christoffel Symbols", and "Riemannian Metric".]

The general Christoffel symbol used as the template for all of those used above is like a mixed tensor (1st-order contra-variant and 2nd-order covariant), defined as the product of its 3rd-order covariant form and the 2nd-order contra-variant conjugate of a Riemannian metric-tensor, whose general form is itself defined using a Jacobian matrix of elements obtained as the partial-derivatives of the given generalized coordinates with respect to any other corresponding system of generalized coordinates.

We use a Riemannian metric-tensor to define the applied space-time metric, which is the differential of the arc-length parameter, according to the formula: the signed square of the applied metric is equal to the product of the Riemannian metric-tensor and the two differentials of the two generalized coordinate-sets.

To put this in simpler terms, use Euclidean 3-space, which reduces the Riemannian metric-tensor to a Kronecker delta, so it takes only the two values; 1 when the delta's two indices are equal, or 0 when they are not equal. Then the square of the metric equals the sum of the squares of the differentials of the Cartesian coordinates, x, y and z.

The arc-length parameter is defined as a function of a curvature parameter, such as time. If it is applied to time, it is then defined as a definite integral (with respect to a dummy coordinate variable), evaluated between two time values. This integral is taken of the square-root of a purely multiplicative expression involving the 2nd-order covariant Riemannian metric-tensor, the standard derivatives (with respect to the same dummy variable) of the two generalized coordinate-sets, and a sign operator that can only be positive or negative unity, depending on whether the same multiplicative expression without it is greater-than or equal to zero for the positive case, or less than zero for the negative case. A given curve can be obtained, therefore, by treating the coordinate variables as functions of positive-real time.

Notice, so far, that nothing in these formulas indicates the quantum nature of the field, for which the radius-of-curvature is assumed to be centered on the center-of-mass in the region of space under study. Neither do they concern themselves with the nature of the mass; it could be a particle or a galaxy. They do not even mandate that an actual field of force is present; despite being categorized under the heading of "field equations" of GR.

Of prime importance to my theory on Superluminal Gravitation is the scalar radius-of-curvature, called the "Ricci Scalar". I assign the path of a radiated GET particle to be along this invariant. But the equation that defines this particular scalar is much more complicated than any ordinary radius-of-curvature, and is too complicated to relate here (would take too much explanation to be practical in words alone). Suffice it to say that it is a function of the Riemannian metric-tensor and the generalized coordinates.

Specifically, I use the form derived by D.C. Kay, in his work on Tensor Calculus (Kay, 106), where he examines the symmetry of a Ricci tensor of first kind, after defining it as a contraction of a Riemann tensor of second kind.

Ordinarily, the Ricci Scalar simply works as the radius-of-curvature for the path of an object moving in the region of space being considered. But it can also be specified as orienting the path of a point-like tachyon radiating outward from the source-mass, and passing through the said object. And being point-like by definition, it is easily defined using Quantum Mechanics (QM) -- which amounts to finding a way to make GR compatible with QM, wherein the forms of all existing formulations are preserved.

Note that a GET particle is point-like, waveless, and spin-less, as far as an observer in generalized bradyonic coordinate-systems with metrics larger than the Planck Length would be concerned. This reduces the GET Schroedinger equation to a linear reference (the equation of a line). Indeed, I propose that the GET epitomizes the very concept of a "point" in space, and that its path represents a perfectly-straight Cartesian "line" in space (i.e., a GET particle provides a physics-based definition for classical points and lines).

If, then, these GET particles (radiating spontaneously in all directions from the source-mass) impart some of their forward momentum to the objects through which they pass unswervingly on their way to an infinite distance away, we have a model explaining the Newtonian law of universal gravitation; F = G(Mm)/(r^2) , where G is Newton's constant, and r is the distance between the source-mass M and the object's mass m.

This is accomplished by showing that the GET particles collectively establish negative radiation-pressure in space, due to their reversed causality, where the contributions Fi of the individual GET particles sum-up to the classical force value; F = F1 + F2 + F3 + ... .

It also leaves the field equations of GR intact, and is consistent with the prevailing view that GR is essentially equivalent to Newtonian Gravity at the so-called "weak-field limit" of GR (although there is a debate as to whether this is actually the case).

In any event, the important proposal here is that using GET particles to explain quantum gravity (in the way I described) makes GR compatible with QM, and therefore serves as a basis for expressing gravity in a quantum-mechanical gauge-field format that unifies it with the other forces of nature (already empirically described in such a format).

In a simplistic way, therefore, we can denote a Grand Unified Field (GUF) representation as being the additive combination of the standard model of elementary particle physics (sans gravity), denoted as a basic Unified Field (UF), and the gauge-field model of gravity using GET particles as the mediators, and denoted using the Tachyonics Operator, I , to designate an imaginary gravitational field IG, so the unification amounts to a super-complex number, involving bradyonic quantities in the "real" component (without gravity), and tachyonic quantities (gravity-specific) in the "imaginary" component;

GUF = UF + IG .

Reference: Kay, David C. Tensor Calculus; Schaum's Outlines. McGraw-Hill, 1988.