text:
# Measuring poverty

Poverty is measured in different ways by different bodies, both governmental and nongovernmental. Measurements can be absolute, which references a single standard, or relative, which is dependent on context. Poverty is widely understood to be multidimensional, comprising social, natural and economic factors situated within wider socio-political processes. The capabilities approach argues that capturing the perceptions of poor people is fundamental to understanding poverty.[2]

Map of world poverty by country, showing percentage of population living on less than $1.25 per day. Information is based on different years (2000-2006) for different countries. Data is missing for countries colored grey. World map showing life expectancy. World map showing the HDI (Human Development Index) World map of income inequality Gini coefficients by country (as %). Based on World Bank data ranging from 1992 to 2020.[1] • Above 50 • Between 45 to 50 • Between 40 to 45 • Between 35 to 40 • Between 30 to 35 • Below 30 • No data The percentage of the world's population living on less than$1 per day has halved in 20 years. Most of this improvement has occurred in East and South Asia. The graph shows the 1981–2001 period.
Life expectancy has been increasing and converging for most of the world. Sub-Saharan Africa has recently seen a decline, partly related to the AIDS epidemic. The graph shows the 1950–2005 period.

The main poverty line used in the OECD and the European Union is a relative poverty measure based on 60% of the median household income. The United States uses an absolute poverty measure based on the U.S. Department of Agriculture's "economy food plan", adjusted for inflation. The World Bank also defines poverty in absolute terms. It defines extreme poverty as living on less than US$1.90 per day.[3] (PPP), and moderate poverty as less than$3.10 a day.

It has been estimated that in 2008, 1.4 billion people had consumption levels below US$1.25 a day and 2.7 billion lived on less than$2 a day.

## Absolute vs relative poverty

When measured, poverty may be absolute or relative. Absolute poverty refers to a set standard which is consistent over time and between countries. An example of an absolute measurement would be the percentage of the population eating less food than is required to sustain the human body (approximately 2000–2500 calories per day).

Relative poverty, in contrast, views poverty as socially defined and dependent on social context. One relative measurement would be to compare the total wealth of the poorest one-third of the population with the total wealth of the richest 1% of the population. In this case, the number of people counted as poor could increase while their income rises. There are several different income inequality metrics; one example is the Gini coefficient.

Although absolute poverty is more common in developing countries, poverty and inequality exist across the world.

## Measurements

The main poverty line used in the OECD and the European Union is a relative poverty measure based on "economic distance", a level of income usually set at 60% of the median household income.[4]

The United States, in contrast, uses an absolute poverty measure. The US poverty line was created in 1963–64 and was based on the dollar costs of the U.S. Department of Agriculture's "economy food plan" multiplied by a factor of three. The multiplier was based on research showing that food costs then accounted for about one-third of money income. This one-time calculation has since been annually updated for inflation.[5]

The U.S. line has been critiqued as being either too high or too low. For example, the Heritage Foundation, a conservative U.S. think tank, objects to the fact that, according to the U.S. Census Bureau, 46% of those defined as being in poverty in the U.S. own their own home (with the average poor person's home having three bedrooms, with one and a half baths, and a garage).[6] Others, such as economist Ellen Frank, argue that the poverty measure is too low as families spend much less of their total budget on food than they did when the measure was established in the 1950s. Further, federal poverty statistics do not account for the widely varying regional differences in non-food costs such as housing, transport, and utilities.[7]

Both absolute and relative poverty measures are usually based on a person's yearly income and frequently take no account of total wealth. Some people argue that this ignores a key component of economic well-being. Major developments and research in this area suggest that standard one dimensional measures of poverty, based mainly on wealth or calorie consumption, are seriously deficient. This is because poverty often involves being deprived on several fronts, which do not necessarily correlate well with wealth. Access to basic needs is an example of a measurement that does not include wealth. Access to basic needs that may be used in the measurement of poverty are clean water, food, shelter, and clothing.[8][9] It has been established that people may have enough income to satisfy basic needs, but not use it wisely. Similarly, extremely poor people may not be deprived if sufficiently strong social networks, or social service systems exist.[10] For deeper discussions, see also the Wikipedia article on Multidimensional poverty.

## Indicator income

Benefits:

1. Easy to measure (normally less incomes than expenditure)
2. Measures degree control over finance of household
3. Cheaper collecting of data

.. Help in the allocation of resources by the government

1. Probably income will be underreported
2. May be affected by short-term fluctuations (for example, the seasonal works), so reporting period might not catch right information
3. Some parts of income are hard to survey (self-home production or self-employment income)
4. Link between income and welfare can be confusing[11]

## Indicator expenditure

Benefits:

1. Report actual material standard of living.
2. For long term measuring is better and more precisely
3. Less understated so expenditures are easier to recall

1. Consumption of household could be unevenly (for example borrowing money)
2. Consumption choices couldn't be rational. People who spend less, could save or live humbly
3. Data could be damaged, because of untruly information
4. Difficult to measure durable goods in some period of research.[11]

## Other Measures of Household Welfare

Even if income and expenses are measured perfectly, none of these measures show well-being objectively. For example, it includes leisure, public goods, health care, education and even peace and security. One criterion is the consumption of calories per person. According to this criterion, we can see what percentage of the population suffers from hunger. Although the average limit is around 2,100 calories, it still varies from person to person. It depends on age, gender, etc. What percentage of income do we spend on food consumption? Research shows that the more developed a country, the smaller the budget we spend on food. The disadvantage of this scale is that people differ relatively in the quality of food and it is consumption per person. Measuring results rather than inputs. Rather than the amount of food, we should focus on the anthropometric state (underweight, etc.). This measurement really shows the quality of the household. However, it cannot be used for comparisons between states or continents, because nationalities differ in scale. The last idea of how to assess poverty is up to the decision among citizens. In Vietnam, for example, some villages are judged by the people themselves, who need help out of poverty. Although this simple solution works somewhere, it is very often distorted by various influences. In short, there is as yet no ideal measure for the well-being of the population. This is not an argument to end the measurement, but rather a warning to minimize errors and take into account as many factors as possible.[11]

## Definitions

The World Bank defines poverty in absolute terms. The bank defines extreme poverty as living on less than US$1.90 per day.[3] (PPP), and moderate poverty as less than$3.10 a day. It has been estimated that in 2008, 1.4 billion people had consumption levels below US$1.25 a day and 2.7 billion lived on less than$2 a day. The proportion of the developing world's population living in extreme economic poverty has fallen from 28 percent in 1990 to 21 percent in 2001. Much of the improvement has occurred in East and South Asia. In Sub-Saharan Africa GDP/capita shrank with 14 percent, and extreme poverty increased from 41 percent in 1981 to 46 percent in 2001. Other regions have seen little or no change. In the early 1990s the transition economies of Europe and Central Asia experienced a sharp drop in income. Poverty rates rose to 6 percent at the end of the decade before beginning to recede.[12] There are criticisms of these measurements.[13]

## Non-monetary indicators

Some economists, such as Guy Pfeffermann, say that other non-monetary indicators of "absolute poverty" are also improving. Life expectancy has greatly increased in the developing world since World War II and is starting to close the gap to the developed world where the improvement has been smaller. Even in Sub-Saharan Africa, the least developed region, life expectancy increased from 30 years before World War II to a peak of about 50 years — before the HIV pandemic and other diseases started to force it down to the current level of 47 years. Child mortality has decreased in every developing region of the world.[14] The proportion of the world's population living in countries where per-capita food supplies are less than 2,200 calories (9,200 kilojoules) per day decreased from 56% in the mid-1960s to below 10% by the 1990s. Between 1950 and 1999, global literacy increased from 52% to 81% of the world. Women made up much of the gap: Female literacy as a percentage of male literacy has increased from 59% in 1970 to 80% in 2000. The percentage of children not in the labor force has also risen to over 90% in 2000 from 76% in 1960. There are similar trends for electric power, cars, radios, and telephones per capita, as well as the proportion of the population with access to clean water.[15]

Headcount index (Po) is a widely-used measure, which simply indicates the proportion of the poor population. Although it does not indicate how poor the poor are.

Formula: ${\displaystyle Po={\frac {Np}{N}}}$ , where Np is the number of poor and N is the total population.

Example: If 10 people are poor in a survey that samples 1000 people, then Po = 10/1000 = 0.01 = 1%

${\displaystyle Po={\frac {1}{N}}\sum _{i=1}^{N}I(yi  , Here, I(·) is an indicator function that takes on a value of 1 if the bracketed expression is true, and 0 otherwise. So if expenditure (yi) is less than the poverty line (z), then I(·) equals 1 and the household would be counted as poor.

This index is easy to understand, but has few disadvantages. It does not show the poverty rate. In addition, the headcount index does not show how poor the poor are. Moreover, this estimate is made on households and not on individuals.[16]

## Poverty gap index

The Poverty gap index is the mean distance below the poverty line as a proportion of the poverty line where the mean is taken over the whole population, counting the non-poor as having zero poverty gap.

Using the index function, we have: ${\displaystyle Gi=(yi-z)\times I(yi , where define the poverty gap (Gi) as the poverty line (z) less actual income (yi) for poor individuals; the gap is considered to be zero for everyone else.

Could be rewrite: ${\displaystyle P1={\frac {1}{N}}\sum _{k=1}^{N}{\frac {Gi}{z}}}$

This method is only reasonable if the transfers could be made perfectly efficiently, what is unlikely.[16]

## Sen index

The Sen index connects the number of poor with the size of their poverty and the distribution of poverty in the sample.[16]

${\displaystyle Ps-Po(1-(1-G^{p}){\frac {U^{P}}{4}}}$

## Sen-Shorrocks-Thon Index

The Sen-Shorrocks-Thon index (sometimes referred to as SST index) is an improved version of the Sen index.[16]

${\displaystyle Psst=PoP_{1}^{p}(1+G^{p})}$

## Common survey problems

These surveys interpret data that has some common problems:

Random sample: All research is based on randomly selecting people into a sample, and each should have the same chance of being selected. Unfortunately, it is not easy for the sample not to be biased, because some groups of people are simply difficult to trace.

Sampling: We can see inequalities from experience and from surveys themselves. Surveys show us only an estimated formula. Another thing we should be interested in is how the sampling was performed. In areas with dense population, there is usually under-sampling.

Goods Coverage and Valuation: To cover and refine the information, we should ask the sample more generally. Not only the issues of income and expenditure are enough, but also own consumption from the family farm. It is necessary to gather information on housing and components of durable consumption.

Variability and the Time Period of Measurement: Income, consumption and other factors change over time. In less developed countries, therefore, the focus is only on consumption, which is more stable from income.

Comparisons across Households at Similar Consumption Levels: Comparisons between households are difficult because households differ not only in the size of income and expenditure, but also in the environment, leisure, quality of the environment, etc.[11]

## Stats

Even if poverty may be lessening for the world as a whole, it continues to be an enormous problem:

• One third of deaths — some 18 million people a year or 50,000 per day — are due to poverty-related causes. That's 270 million people since 1990, the majority women and children, roughly equal to the population of the US.
• Every year nearly 11 million children die before their fifth birthday.
• In 2001, 1.1 billion people had consumption levels below $1 a day and 2.7 billion lived on less than$2 a day.
• 800 million people go to bed hungry every day.[17]

## Other factors

The World Bank's Voices of the Poor initiative,[18] based on research with over 20,000 poor people in 23 countries, identifies a range of factors that poor people consider elements of poverty. Most important are those necessary for material well-being, especially food. Many others relate to social rather than material issues.

• precarious livelihoods
• excluded locations
• gender relationships

## References

1. ^ "Gini index (World Bank estimate)". data.worldbank.org. Retrieved 2022-04-23.
2. ^ Sen, Amartya (1999). Development as Freedom. OUP. ISBN 0-19-289330-0.
3. ^ a b "New Data Show 1.4 Billion Live On Less Than US\$1.25 A Day, But Progress Against Poverty Remains Strong". worldbank.org/. The World Bank. Retrieved 13 March 2015.
4. ^ "Poverty and Inequality in the European Union - the Poverty Site". Archived from the original on 2007-11-23. Retrieved 2011-10-10.
5. ^ US Department of Human Services, FAQ Poverty Guidelines and Poverty
6. ^ Rector, Robert E. and Johnson, Kirk A., Understanding Poverty in America, Executive Summary, Heritage Foundation, January 15, 2004, No. 1713
7. ^ Frank, Ellen, "Dr. Dollar: How Is Poverty Defined in Government Statistics?" Dollars & Sense magazine, January/February 2006. Accessed April 13, 2008
8. ^ "Measuring Poverty - Poverty is defined by deprivation, and can be measured with economic or social indicators". boundless.com. Archived from the original on 2 April 2015. Retrieved 12 March 2015.
9. ^ Pogge, Thomas. "Measuring Poverty". thomaspogge.com/. Archived from the original on 29 January 2016. Retrieved 12 March 2015.
10. ^ Thorbecke, E., 2008. Multidimensional Poverty: Conceptual and Measurement Issues. In: Kakwani, N., Silber, J. (Eds.), The Many Dimensions of Poverty. Palgrave Macmillan, New York, pp. 3–20
11. ^ a b c d "Handbook on Poverty and Inequality" (PDF).{{cite web}}: CS1 maint: url-status (link)
12. ^ Worldbank.org reference
13. ^ TFF, 1999
14. ^ The Eight Losers of Globalization Archived 2008-05-01 at the Wayback Machine by Guy Pfeffermann. Pfeffermann, former Chief Economist of the International Finance Corporation, is the CEO of Global Business School Network. This is a non-profit organization supporting business and management education in the developing world. 2002.

It is an area where not only is there little or no consensus among disciplines, but where economists themselves have widely differing views. So, what can one say with a fair degree of certainty about growth and inequality in developing countries? Life expectancy at birth — the most basic and robust of all social indicators — has increased very considerably around the world.

15. ^ www.sciencedirect.org
16. ^ a b c d "Poverty Manual, Chapter 4" (PDF).{{cite web}}: CS1 maint: url-status (link)
17. ^ "millenniumcampaign.org". Archived from the original on 2006-12-09. Retrieved 2007-01-14.
18. ^ "Voices of the Poor". worldbank.org. The World Bank. Retrieved 13 March 2015.


text:
# Deriving the Doob Meyer decomposition of a Sub Martingale using Ito's

Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.:

$$X_t =B^6_t+2t$$

I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing process]+[m.g.]

Sadly I keep applying Ito's formula in the wrong way and end up with wrong answers. Can you please help me find the right Ito's formula to obtain the following result:

$dX_t= 6B_t^5 dB_t+ \frac{1}{2}(6)(5)B_t^4dt+2t$

$X_t =\int_0^t(15B_s^4+2)ds+ \int_0^t(6B_s^5)dB_s$

Thank you.

I've been using:

$f(B_t,t)=\frac{\partial f}{ \partial X_t} dB_t + \frac{\partial f}{ \partial t} dt - \frac {1}{2} \frac{\partial^2 f}{ \partial X_t^2} d <X>_t$

And I can't get the same results. I understand that the Ito's formula would need a function but I'm off track trying to identify the one needed in this case.

• The representation you obtained $X_t = \int_0^t \left(2 + 15 B_s^4\right) \mathrm{d}s + 6 \int_0^t B_s^5 \mathrm{d}B_s$ is correct. – Sasha Jan 22 '15 at 21:44
• Thank you, that's good to know but I am also looking for the right Ito's formula to use to obtain that. Can you provide the version of the Ito's formula in detail please? – Clemente Cortile Jan 22 '15 at 21:47

Ito's Lemma: For suitable stochastic process $X_t(t, B_t)$,

$$dX_t{}={}\left(\dfrac{\partial}{\partial t}X_t{}+{}\mu_b\dfrac{\partial}{\partial b}X_t{}+{}\dfrac{1}{2}\sigma_b^2\dfrac{\partial^2}{{\partial b}^2}X_t\right)dt{}+{}\sigma_b\dfrac{\partial}{\partial b}X_tdB_t\,.$$ where $\mu_b{}={}0$ and $\sigma_b{}={}1$ for brownian motion, $B_t$ .

From the equation of $X_t$ given in the question, compute each of the non-zero terms as follows:

(i) $\,\,\,\,\,\,\, \dfrac{\partial}{\partial t}X_t{}={}2\,;$

(ii) $\,\,\,\,\,\,\, \dfrac{1}{2}\sigma_b^2\dfrac{\partial^2}{{\partial b}^2}X_t{}={}\dfrac{1}{2}6\cdot5\,B_t^4{}={}15B_t^4\,;$

(iii) $\,\,\,\,\,\,\, \sigma_b\dfrac{\partial}{\partial b}X_t{}={}6B_t^5$.

Direct substitution of these into the lemma gives the result:

$$dX_t{}={}\left(15B_t^4+2\right)dt{}+{}6B_t^5 dB_t\,.$$

• Thank you Ki3i, used this version of the formula a lot, and passed the class thanks to your answer. – Clemente Cortile Feb 5 '15 at 11:32
• @ClementeCortile, :) Well done. +1 for your success in applying what you learned here. Glad to have been of some help. – ki3i Feb 5 '15 at 13:43

format: plain


text:
## The neuromorphic wafer module¶

At the core of the BrainScaleS wafer-scale hardware system (see Figure 90) is an uncut wafer built from mixed-signal ASICs [1], named High Input Count Analog Neural Network chips (HICANNs), which provide a highly configurable substrate that physically emulates adaptively spiking neurons and dynamic synapses (Schemmel et al. (2010), Schemmel et al. (2008)). The intrinsic time constants of these VLSI model circuits are multiple orders of magnitude shorter than their biological counterparts. Consequently, the hardware model evolves with a speedup factor of $$10^3$$ up to $$10^5$$ compared to biological real time, the precise value depending on the configuration of the system. This speedup enables power-efficient computation as the energy consumption for synaptic transmissions is several orders of magnitude lower than in classically simulated neuronal networks.

In addition to a high-bandwidth asynchronous on-wafer event communication infrastructure, 48 FPGA [2] communication modules provide off-wafer connectivity (to other wafers and to host computers).

A full wafer system comprises 384 interconnectable HICANNs, each of which implements more than 114,688 programmable dynamic synapses and up to 512 neurons, resulting in a total of approximately 44 million synapses and up to 196,608 neurons per wafer module. The exact number of neurons depends on the configuration of the substrate, which allows to combine multiple neuron building blocks to increase the input count per cell.

Via the communication FPGAs the system can be configured and operated from a host computer. Each communication FPGA is connected to a dedicated area on the wafer which contains 8 HICANNs. This FPGA-HICANN link is used to configure the HICANNs as well as to transmit pulse events to and from the neural circuits on the wafer. The pulse communication between the on-wafer neurons is performed by a bus-like network directly on the wafer.

The system provides a high degree of configurability with respect to network architecture and neuron parameters:

• each neuron provides configurable AdEx neuron dynamics
• the synapses provide 4-bit weight resolution and STDP functionality
• the connection topology can be configured

For a detailed specification see the Platform specification.

## The full BrainScaleS system¶

The BrainScaleS system, shown in Figure 91, consists of 20 neuromorphic wafer modules together with support infrastructure and a conventional compute cluster used for controlling the wafer modules and for simulated environments.

Figure 92 provides a simplified overview of the BrainScaleS system. The support infrastructure is responsible for power supply, off-wafer communication and analog readout functionality. A dedicated Raspberry Pi monitors and controls all power links as well as other operating parameters of the wafer system. Analog readout (e.g., recording of membrane voltages) functionality is provided by a custom analog readout module (AnaRM). Several AnaRMs are handled by another dedicated control computer. The control/compute cluster orchestrates the configuration of the system and the execution of neuronal network experiments including all input and output data of the emulated network.

## Glossary¶

 [1] Application Specific Integrated Circuit
 [2] Field Programmable Gate Array


text:
# For any subset of associated points, is there a section whose support is the closure of that subset? (cf: Vakil 5.5.O)

Let $$A$$ be a Noetherian ring and $$M$$ a finitely generated module. Vakil's Exercise 5.5.O says

Show that those subsets of Spec(A) which are the support of an elements of M are precisely those subsets which are the closure of a subset of the associated points.

I wonder if

for any subset of associated points, there is a section whose support is the closure of that subset.

So far, I am able to show

(1)for any associated point $$p$$, there is a section supported precisely on the closure of $$p$$.

(2) If $$m ∈ M$$, show that the support of $$m$$ is the closure of those associated points at which $$m$$ has nonzero germ.

The thing I am asking is more general than (2), I think.

We want to show that

for any subset of associated points, there is a section whose support is the closure of that subset.

Step 1: restricting to minimal elements does not change the closure.

As you have shown (I believe), if $$\mathfrak{p}=\mathrm{ann}(m)$$ is an associated prime, then $$\mathrm{supp}(m)=\overline{\mathfrak{p}}$$. Given any (necessarily finite) subset $$\{\mathfrak{p}_1, \dots, \mathfrak{p}_r\}$$ of $$\mathrm{Ass}(M)$$, say $$\mathfrak{p}_i = \mathrm{ann}(m_i)$$ for $$1\leq i\leq r$$, we claim that there is an element $$m\in M$$ whose support equals $$\bigcup_{i=1}^r \overline{\mathfrak{p}_i}$$. Note that if the minimal elements in $$\{\mathfrak{p}_1, \dots, \mathfrak{p}_r\}$$ are $$\{\mathfrak{p}_1, \dots, \mathfrak{p}_s\}$$, then $$\bigcup_{i=1}^s \overline{\mathfrak{p}_i} = \bigcup_{i=1}^r \overline{\mathfrak{p}_i}$$. So we may assume that each element of $$\{\mathfrak{p}_1, \dots, \mathfrak{p}_r\}$$ is minimal.

I claim that $$m:=\sum_{i=1}^r m_i$$ will be the desired element.

Step 2: inside the closure.

Indeed, at $$\mathfrak{p}_i$$ we have $$m_i\neq 0\in M_{\mathfrak{p}_i}$$. By minimality, for any other $$j\neq i$$, there exists $$f_j \in \mathfrak{p}_j\setminus \mathfrak{p}_i$$ and hence $$m_j=0 \in M_{\mathfrak{p}_i}$$. This shows that $$m = m_i \neq 0 \in M_{\mathfrak{p}_i}$$.

If $$\mathfrak{q}\in \overline{\mathfrak{p}_i}$$ for some $$i$$, then we have the localization homomorphism $$M_{\mathfrak{q}} \to M_{\mathfrak{p}_i}$$ under which $$m\mapsto m\neq 0$$, so $$m\neq 0 \in M_{\mathfrak{q}}$$.

Hence, $$\mathrm{supp}(m) \supseteq \bigcup_i \overline{\mathfrak{p}_i}$$.

Step 3: outside the closure.

Conversely, at any prime $$\mathfrak{q} \notin \bigcup_i \overline{\mathfrak{p}_i}$$, we have $$\mathfrak{q} \not\supset \mathfrak{p}_i$$ for all $$i$$. Then, there are $$f_i\in \mathfrak{p}_i\setminus \mathfrak{q}$$ for all $$i$$, so $$m_i=0\in M_\mathfrak{q}$$ for all $$i$$, which means that $$m=0\in M_\mathfrak{q}$$ and $$\mathrm{supp}(m) = \bigcup_i \overline{\mathfrak{p}_i}$$.


text:
# The number of football players is 4 times the number of basketball players, and the number of baseball players is 9 more than basketball players. If the total number of players is 93 and each one plays a single sport, how many are on each team?

Sep 28, 2015

$56$ football players
$14$ basketball players
$23$ baseball players

#### Explanation:

Define:
$\textcolor{w h i t e}{\text{XXX}} f :$number of football players
$\textcolor{w h i t e}{\text{XXX}} b :$number of basketball players
$\textcolor{w h i t e}{\text{XXX}} d :$number of baseball players

We are told:
[1]color(white)("XXX"color(red)(f=4b)
[2]$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{d = b + 9}$
[3]$\textcolor{w h i t e}{\text{XXX}} f + b + d = 93$

Substituting (from [1]) $\textcolor{red}{4 b}$ for $\textcolor{red}{f}$ and (from [2]) $\textcolor{b l u e}{b + 9}$ for $\textcolor{b l u e}{d}$ in [3]
[4]$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{4 b} + b + \textcolor{b l u e}{b + 9} = 93$

Simplifying
[5]$\textcolor{w h i t e}{\text{XXX}} 6 b + 9 = 93$
[6]$\textcolor{w h i t e}{\text{XXX}} 6 b = 84$
[7]$\textcolor{w h i t e}{\text{XXX}} b = 14$

Substituting $14$ for $b$ in [2]
[8]$\textcolor{w h i t e}{\text{XXX}} d = 14 + 9 = 23$

Substituting $14$ for $b$ in [1]
[9]$\textcolor{w h i t e}{\text{XXX}} f = 4 \cdot 14 = 56$

Sep 28, 2015

56 football players, 14 basketball players, and 23 baseball players.

#### Explanation:

Let the number of football players be x
Let the number of basketball players be y
Let the number of baseball players be z

Now rewrite all the sentences in algebraic form in terms of x, y and z. Doing so we get :

$x = 4 y$
$z = y + 9$
$x + y + z = 93$

Now we can substitute both x and z (which we have in terms of y) into the last equation and then solve for y. This yields

$4 y + y + \left(y + 9\right) = 93$

$\therefore 6 y = 84 \implies y = 14$ and so there 14 basketball players.

Now substitute the value of y back into the first 1 equations to determine x and z.
$\therefore x = 4 \times 14 = 56 \mathmr{and} z = 14 + 9 = 23$
This means there are then 56 football players and 23 baseball players.


text:
# Math Help - Energy

1. ## Energy

Each second, 1250 m3 of water passes over a waterfall 115 m high. Three-fourths of the kinetic energy gained by the water in falling is transferred to electrical energy by a hydroelectric generator. At what rate does the generator produce electrical energy? (The mass of 1 m3 of water is 1000 kg.)
I know that $E=E_p+E_k=mgd+.5mv^2$. At the end of the fall there is no more potential energy so all I need is $E_k=.75* (.5mv^2)$.

To get v I used $v=\sqrt{2gd}$.

So the final equation is $E_k=.75*(mgd)$.

I plugged in the numbers and got 1056562500W. According to loncapa (online system) this is wrong. Can anyone see something I did incorrectly?

2. Originally Posted by WhoCares357
I know that $E=E_p+E_k=mgd+.5mv^2$. At the end of the fall there is no more potential energy so all I need is $E_k=.75* (.5mv^2)$.

To get v I used $v=\sqrt{2gd}$.

So the final equation is $E_k=.75*(mgd)$.

I plugged in the numbers and got 1056562500W. According to loncapa (online system) this is wrong. Can anyone see something I did incorrectly?
significant figures, maybe ?

1.06 GW


text:
5

# If p is prime then aP = a (mod p) for all a ez_(6 points) Useto show that a and a5 have the same units digit for any integer...

## Question

###### If p is prime then aP = a (mod p) for all a ez_(6 points) Useto show that a and a5 have the same units digit for any integer

If p is prime then aP = a (mod p) for all a ez_ (6 points) Use to show that a and a5 have the same units digit for any integer

## Answers

#### Similar Solved Questions

5 answers
##### Find 841 derivative of the function2g(x) 4 ~Ix
Find 841 derivative of the function 2 g(x) 4 ~Ix...
5 answers
##### Consider the following functions. Vx-9 g(x) Sketch the region bounded by the graphs of the functions.Flndthe arerthe rglon
Consider the following functions. Vx-9 g(x) Sketch the region bounded by the graphs of the functions. Flndthe arer the rglon...
5 answers
##### SW Airlines is studying its service from AtlantaChicago. Historical data shows that the mean number of passengers per flight is 120. If the number of passengers flying are normally distributed, with variance of 100 passengers, What is the probability that a randomly selected flight will carry between 11O to 125 passengers' What is the probability that a typical flight will carry less than 105 passengers? (Show your answers graphically) What the probability the airlines will carry 12 pa
SW Airlines is studying its service from Atlanta Chicago. Historical data shows that the mean number of passengers per flight is 120. If the number of passengers flying are normally distributed, with variance of 100 passengers, What is the probability that a randomly selected flight will carry ...
5 answers
##### 3 Find the mass of the 3-D solid which is inside the sphere x2 + y2 + z2 = 9 and inside the cylinder x2 + y2 = 4,ifthe density is 8(x,Y,2) = 9-x2-y2
3 Find the mass of the 3-D solid which is inside the sphere x2 + y2 + z2 = 9 and inside the cylinder x2 + y2 = 4,ifthe density is 8(x,Y,2) = 9-x2-y2...
5 answers
##### QUESTION31esicla #PP chartha tano: Mamenn [ethc {Snare Rjn G17 prorcin: on tha vuticle Rab GDP ptoreint thewerkke RsbG1P protcins 8nthe VesicleDlcin-KnechPSARZQUESTION 32Jna: ElareAneeIPooulei Prepino ounnt Dmtcteili In Turcan4mjdll fandwich 5p uslly troron ToethIna thtndImaliotabl WetolieDcodutOcahesmal Jendetr- veleâ‚¬onlnln membran Prore nfanJspt nfjtotbQUESTION 3]Deol Upulaico cem rurete Ln Po7an 8r certteaTtmbtictHfidu drrnedom unehd Lrelollarnnn DeTordttrzDunneee Imuln-mdrednireuieKrmulFb
QUESTION31 esicla #PP chartha tano: Mamenn [ethc {Snare Rjn G17 prorcin: on tha vuticle Rab GDP ptoreint thewerkke RsbG1P protcins 8nthe Vesicle Dlcin- Knech PSARZ QUESTION 32 Jna: Elare Anee IPooulei Prepino ounnt Dmtcteili In Turcan 4mjdll fandwich 5p uslly troron Toeth Ina thtnd Imaliot abl Wetol...
5 answers
##### Write the complex number in polar form: Express the argument in degrees, rounded to the nearest tenth; if necessary:12+16i20 (cos 233.1 + isin 233.1020 (cos 126.95 +isin 126.98)20 (cos 53.15 +isin 53.19)20 (cos 306.99 + i sin 306.98
Write the complex number in polar form: Express the argument in degrees, rounded to the nearest tenth; if necessary: 12+16i 20 (cos 233.1 + isin 233.10 20 (cos 126.95 +isin 126.98) 20 (cos 53.15 +isin 53.19) 20 (cos 306.99 + i sin 306.98...
5 answers
##### Q.2. A puma jumps from a rock of height m, If the puma jumps off the rock with an initial horizontal velocity of 6 m/s to a distance of 3 from the rock along the ground; find Y.
Q.2. A puma jumps from a rock of height m, If the puma jumps off the rock with an initial horizontal velocity of 6 m/s to a distance of 3 from the rock along the ground; find Y....
5 answers
##### Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the X-axis_V# 2sin > + 1, 2 =0, ! = 0, I =T37 + 47 + 8707 +4x37 + 83
Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the X-axis_ V# 2sin > + 1, 2 =0, ! = 0, I =T 37 + 4 7 + 87 07 +4x 37 + 83...
5 answers
##### Recall the ring homomorphism p: Qlr] Q defined by p(f(c)) = f(3). ker(y) . Recall that every fiber of is translate of I. That is, Let for each a â‚¬ Q there exists a 9(r) â‚¬ Ql] such that p-'(a) = I+ g(r) (a) Show that the translations of by two different polynomials can be the same fiber of p. That is, show that there exist polynomials f(r) + g(r) such that I + f(r) = I+g(1). (b) Show that the translations of by two different constant polynomials cannot be the same: That is, show that, i
Recall the ring homomorphism p: Qlr] Q defined by p(f(c)) = f(3). ker(y) . Recall that every fiber of is translate of I. That is, Let for each a â‚¬ Q there exists a 9(r) â‚¬ Ql] such that p-'(a) = I+ g(r) (a) Show that the translations of by two different polynomials can be the same f...
1 answers
##### Draw charge-separated resonance structures to give a representation of the bonding in $\mathrm{PF}_{5}$ such that the octet rule is strictly obeyed.
Draw charge-separated resonance structures to give a representation of the bonding in $\mathrm{PF}_{5}$ such that the octet rule is strictly obeyed....
5 answers
##### A rod of negligible mass is pivoted a 2 point that is off-center . S0 that length / is different from length &> The figures above show [Wo cascs in which masses are suspended from the ends of the rod each case the unenoum MAA balanced by known mass M or M> so that the rod remains horizontal . What is the value of m in terms ot the knoun masses' (4)M M; (B) 'M; MJ) (C)M M; (DJ VM,M;
A rod of negligible mass is pivoted a 2 point that is off-center . S0 that length / is different from length &> The figures above show [Wo cascs in which masses are suspended from the ends of the rod each case the unenoum MAA balanced by known mass M or M> so that the rod remains horizonta...
5 answers
##### Suppose it takes you 45.0 s to travel 10Om down a long straight line hallway toward your departure gate at an airport: At the 100m mark; you realize you missed the restroom, and you return back 25.0 m along the same hallway; taking 10.0 s to make the return trip: a) What is the magnitude of the average velocity for your trip? b) What is the average speed for your trip?
Suppose it takes you 45.0 s to travel 10Om down a long straight line hallway toward your departure gate at an airport: At the 100m mark; you realize you missed the restroom, and you return back 25.0 m along the same hallway; taking 10.0 s to make the return trip: a) What is the magnitude of the aver...
1 answers
##### The $x^{\prime} y^{\prime}$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system. The coordinates of a point in the $x y$ -coordinate system are given. Find the coordinates of the point in the rotated coordinate system. $$\theta=90^{\circ},(0,3)$$
The $x^{\prime} y^{\prime}$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system. The coordinates of a point in the $x y$ -coordinate system are given. Find the coordinates of the point in the rotated coordinate system. $$\theta=90^{\circ},(0,3)$$...
1 answers
5 answers
##### Tbe boting point of pure ethanol (C H,OH, MM = 46.07 'C.A solution is made using 100.0 glmol) is 78.4 9 of glucose (MM = g of ethanol (the solvent) and 18.1 solution, 180.16 g/mol) What is the boiling point of this in 'C? (Kb of ethanol is 1.22 Clm)
Tbe boting point of pure ethanol (C H,OH, MM = 46.07 'C.A solution is made using 100.0 glmol) is 78.4 9 of glucose (MM = g of ethanol (the solvent) and 18.1 solution, 180.16 g/mol) What is the boiling point of this in 'C? (Kb of ethanol is 1.22 Clm)...
5 answers
##### Give the solution in obtaining the volume of two buffers usednamely NaH2PO4 and Na2HPO4 pka=6.7pH=6.5 conc. of base= 0.2 Mconc.of acid= 0.2 Mtotal volume= 20mL
give the solution in obtaining the volume of two buffers used namely NaH2PO4 and Na2HPO4 pka=6.7 pH=6.5 conc. of base= 0.2 M conc.of acid= 0.2 M total volume= 20mL...
5 answers
##### The folowing observations have been arranged ascending order: 12, 16. 17, 19, 3,27, 37, 38, lnthe medkin the datu 23 hat the vahie 0l 4The following observations have been arranged in ascending 12, 16, 17, 19,*,X + 3, 27, 37, 38, 40 If the median of the data 23.5, what IS the value of x?Hoin150 . ing trann running thid tate & 60 anynour Euat t0 00r+If Loga 4 0.5, then 0' IS equal to
The folowing observations have been arranged ascending order: 12, 16. 17, 19, 3,27, 37, 38, lnthe medkin the datu 23 hat the vahie 0l 4 The following observations have been arranged in ascending 12, 16, 17, 19,*,X + 3, 27, 37, 38, 40 If the median of the data 23.5, what IS the value of x? Hoin 150 ....
5 answers
##### Let G undirected graph with n 11 vertices and edges Edges {(i1 j1), (im jm)}: Prove that can always color its vertices in 11 colors such at least %1m edges don connect vertices of the same color.
Let G undirected graph with n 11 vertices and edges Edges {(i1 j1), (im jm)}: Prove that can always color its vertices in 11 colors such at least %1m edges don connect vertices of the same color....
5 answers
##### Supposeoxygen-16 nuclide transforms Into nitrogen-|3 nuclide by absorblng proton and emltting an alpha particleComplete the nuclear chemical equation below that describes this nuclear reaction;0b"N +
Suppose oxygen-16 nuclide transforms Into nitrogen-|3 nuclide by absorblng proton and emltting an alpha particle Complete the nuclear chemical equation below that describes this nuclear reaction; 0b "N +...
5 answers
##### Consider the probability function. (Give your answers correct to two decimal places- R(x) = 0.2, for X = 0, 1, 2, 3, 4(a) Find the mean.(b) Find the standard deviation.
Consider the probability function. (Give your answers correct to two decimal places- R(x) = 0.2, for X = 0, 1, 2, 3, 4 (a) Find the mean. (b) Find the standard deviation....

-- 0.025320--


text:
# Why don't we need to do the error calculations for all the repeated measurements?

This is from John Taylor- "An Introduction to Error Analysis"
The passage asks us to calculate the uncertainty of refractive index n using Snell's Law
$$\sin i=n \sin r$$
The fractional uncertainty in $n$ is the quadratic sum of those in $\sin i$ and $\sin r$:
$$\frac{\delta n}{n}=\sqrt{\left(\frac{\delta \sin i}{\sin i} \right)^2+\left(\frac{\delta \sin r}{\sin r}\right)^2}$$
The calculation is then carried out as shown in the table below

And then in the last paragraph, the author mentioned that

"If you repeat an experiment like this one several times, the error calculations can become tedious if you do them for each repetition. You should recognize, however, that you almost never need to do the error calculations for all the repetitions; if you find the uncertainties in $n$ corresponding to the smallest and largest values of $i$ (and possibly a few intermediate values), then these uncertainties suffice for most purposes."

I don't understand why don't we need to do the error calculations for all the repetitions, since the $\frac{\delta n}{n}$ values look quite different for each repetition in the table? And what does he mean by "if you find the uncertainties in $n$ corresponding to the smallest and largest values of $i$"?

Really appreciate the help!

I think what the author means is that if you had values $i=30^\circ$ and $i=19^\circ$ then you could make an estimate of the error as being about $7\%$ as midway between $9\%$ and $5\%$.

The mention of intermediate values is there to make the interpolation of the error of intermediate values more accurate.

Note that in this example you could make the error calculation easier by noting that $\delta(\sin i)\approx \delta i \cos i$ and so $\dfrac{\delta ( \sin i)}{\sin i }\approx \delta i \cot i$ where $\delta i$ is in radians.

There is no reason you shouldn't do the error calculation for each measurement - especially if you enter the measurements into a spreadsheet.

However, I think all the book is getting at is that there is a clear relationship between the value of the angle and the size of its contribution to the error in $n$.

If $y = \sin \theta$, and there is a (small) error in $\theta$, then it is the case that $$\frac{\Delta y}{y} \simeq \frac{\cos \theta\ \Delta\theta}{\sin \theta} = \frac{\Delta\theta}{\tan \theta}$$

Thus as $\theta$ increases, the percentage error decreases, as you see in the table.

Or, wrapping up the whole expression for $n$ and assuming $i$ and $r$ are measured with the same precision $\Delta \theta$ $$\frac{\Delta n}{n}\simeq \Delta\theta \left( \frac{\cos^2 i}{\sin^2 i} + \frac{\cos^2 r}{\sin^2 r}\right)^{1/2}$$

A bit of trigonometry and use of Snell's law$^{*}$ shows you this is a decreasing function of $i$. $$\frac{\Delta n}{n}\simeq \Delta\theta\left(2\cot^2 i +\frac{n^2-1}{\sin^2 i}\right)^{1/2}$$ Of course, we don't know $n$ when we start the experiment, so all this shows is that the percentage error in $n$ should get smaller smoothly as $i$ increases.

$^{*}$ Replace $\sin^2 r$ with $\sin^2 i/n^2$ and $\cos^2 r$ with $1 - \sin^2 i/n^2$. $$\frac{\Delta n}{n}\simeq \Delta\theta \left( \frac{1-\sin^2 i}{\sin^2 i} + \frac{1-\sin^2 i/n^2}{\sin^2 i/n^2}\right)^{1/2}$$ $$\frac{\Delta n}{n}\simeq \Delta\theta \left( \frac{1-\sin^2 i +n^2 -\sin^2 i}{\sin^2 i}\right)^{1/2}$$ $$\frac{\Delta n}{n}\simeq \Delta\theta \left( \frac{2 - 2\sin^2 i + (n^2 -1)}{\sin^2 i}\right)^{1/2}$$$$\frac{\Delta n}{n}\simeq \Delta\theta\left(2\cot^2 i +\frac{n^2-1}{\sin^2 i}\right)^{1/2}$$

• Hi, may I know how do you get to the last expression? – WeiShan Ng Mar 31 '18 at 10:43


text:
# Calculating the $\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}$

Question: If $$s \in \mathbb{N}$$ is it true: $$\sum_{x=1}^\infty\left[\prod_{k=0}^s{s\choose k}x^k\right]^{-1}={\zeta\left[{s+1 \choose 2}\right] \above 1.5pt \prod_{k=1}^s{s \choose k}};$$ where $$\zeta\left[{s+1 \choose 2}\right]$$ is the Riemann zeta function$$^1$$ evaluated at the triangular numbers$$^2$$ ?

I believe I can show the answer to the question is yes by way of inspection. I cannot get an actual explicit calculation that answers the question affirmatively - that is what I am asking for !

Solution to question by inspection: For short handedness I write $$\zeta_{\times}(s)=\sum_{x=1}^\infty\Bigg[\prod_{k=0}^s{s\choose k}x^k\Bigg]^{-1}$$ I computed a small list of values of $$\zeta_{\times}(s)$$ for small $$s$$:

$$\begin{array}{ l | l } s & 2 & 3 & 4 & 5 & 6 & 7\\ \hline \zeta_{\times}(s) & \frac{\zeta(3)}{2} & \frac{\zeta(6)}{9} & \frac{\zeta(10)}{96} & \frac{\zeta(15)}{2500} & \frac{\zeta(21)}{162000}& \frac{\zeta(28)}{26471025}\\ \end{array}$$

Inspection suggests that $$\zeta_{\times}(s)$$ has numerator equal to the Riemann zeta function at $${s+1 \choose 2}.$$ The denominators of $$\zeta_{\times}(s)$$ can be written explicitly as $$2,9,96,2500,162000,\ldots$$; which appear to be the integer sequence A001142$$^3$$. Listed in the reference section is the following paper by Lagarias: Products of Binomial Coefficients and Unreduced Farey Fractions.$$^4$$. Lagarias defines the unreduced Farey fractions$$^5$$ to be the ordered sequence of all reduced and unreduced fractions between $$0$$ and $$1$$ with denominator of size at most $$s.$$ Following Lagarias' notation: I write $$G_s$$ for the set of unreduced Farey fractions and let $$|G_s|$$ and $$G_s^*$$ denote the cardinality and the product of all elements of $$G_s$$ respectively. Lagarias shows the following results: $$|G_s|={s+1 \choose 2}$$ and $$G_s^*=\prod_{k=1}^s{s \choose k}.$$ I put everything together and get $$\zeta_{\times}(s)={\zeta\big(|G_s|\big) \above 1.5pt G_s^*}$$

and I am done. $$\blacksquare$$

Source and motivation of the problem: Curiosity is the main driving source of the problem. Here it goes: If $$x$$ and $$s$$ are positive numbers I write $$\zeta(s)$$ for the Riemann zeta function evaluated at the number $$s.$$ For ease of typing I write Pascal's Triangle$$^{6}$$ like this: $$\text{ }\begin{matrix} 1&&&&\\ 1&1&&&\\ \color{red}{1}&\color{red}{2}&\color{red}{1}&&\\ \color{blue}{1}&\color{blue}{3}&\color{blue}{3}&\color{blue}{1}&\\ 1&4&6&4&1\\ \vdots \end{matrix}$$ and recall that the first row in Pascal's triangle is numbered at $$0$$. Now look at the second and third rows of the triangle, highlighted in red and blue respectively and observe: $$\sum_{x=0}^\infty{ 1 \above 1.5pt \color{red}{1}+\color{red}{2}x+\color{red}{1}x^2}=\zeta(2)$$ and $$\sum_{x=0}^\infty{ 1 \above 1.5pt \color{blue}{1}+\color{blue}{3}x+\color{blue}{3}x^2+\color{blue}{1}x^3}=\zeta(3).$$ This observation shows that $$\zeta(s)$$ can be recovered from the rows of Pascal's triangle - in particular I can write $$\sum_{x=0}^\infty\frac{1}{(1+x)^s}=\zeta(s).$$ On the other hand using the binomial theorem$$^{7}$$ allows me to compute $$(1+n)^s$$ explicitly with the formula: $$\sum_{k=0}^s{s\choose k}x^k$$ in which case after substitution: $$\zeta(s)=\sum_{x=0}^\infty\bigg[\sum_{k=0}^s{s\choose k}x^k\bigg]^{-1}$$ Now out of total curiosity I decided to swap the inner summation inside the big brackets and swap it out with a product, noting carefully that if I do that I need to shift the starting point in the outer summation to $$1$$ otherwise I would be dividing by zero. For short handedness I wrote $$\zeta_{\times}(s)=\sum_{x=1}^\infty\Bigg[\prod_{k=0}^s{s\choose k}x^k\Bigg]^{-1}$$ Numerical inspection suggested $$\zeta_{\times}(s) ={\zeta\left[{s+1 \choose 2}\right] \above 1.5pt \prod_{k=1}^s{s \choose k}}$$ I double checked my numerical hunch against Sloan's Database and encountered the paper by Lagarias. In Lieu of the fact that Lagarias has an explicit formulae for $$\log G_s^*$$ (see Lagarias paper or reference above for notation) I primarily became interested in computing $$\log \zeta_{\times}(s).$$ If I could answer the question affirmatively then I would know that $$\log \zeta_{\times}(s)=\log \zeta(|G_s|)-\log(G_s^*)$$

• What is the source of this problem? It's quite an interesting sum. – Carl Schildkraut Oct 12 '18 at 17:14

$$\zeta_{\times(s)} =\sum_{x=1}^\infty\Bigg[\prod_{k=0}^s{s\choose k}x^k\Bigg]^{-1}$$
Maybe I'm just being naive, but since $$\prod_{k=0}^s{s\choose k}x^k =\prod_{k=0}^s{s\choose k}\prod_{k=0}^sx^k =G_s^*x^{s(s+1)/2}$$, $$\zeta_{\times}(s) =\sum_{x=1}^\infty \dfrac1{G_s^*}x^{s(s+1)/2} =\dfrac1{G_s^*}\sum_{x=1}^\infty x^{s(s+1)/2} =\dfrac{\zeta(s(s+1)/2)}{G_s^*}$$.


text:
# Young's Double Slit Experiment lab

1. Jan 19, 2014

### braeden

In class we did a lab regarding the wave nature of light. We pretty much conducted Young's double slit experiment. For the lab we were supposed to establish the relationship between the distance from the slits to the viewing screen and the distance between successive maxima. Can anyone explain to me, physics wise, why as the distance between the viewing screen the distance between successive maxima increases proportionally as well. I can obviously figure it looking at the formula but need a explanation of what's going on for my lab. Thanks

2. Jan 19, 2014

### ZapperZ

Staff Emeritus
Er... this is straightforward trigonometry.

These maxima and minima are at particular angles. So if you look at the first maximum, it comes out of the slit at a fixed angle. So if you consider A as the distance to the screen and O is the distance to the first maximum, then if you keep the angle $\theta$ fixed, can't you see that O will get larger as A increases?

Your a trig. function (I'll let you figure this out yourself which one is relevant) to relate the angle, distance A, and distance O, you should be able to get the exact relationship between all three.

In the future, please write down exactly the mathematical description that you know, and start from there.

Zz.

3. Jan 19, 2014

### jostpuur

It almost appears as if you are trying to make the original question sound as dumb. I wouldn't approve of that, because it is not obvious at all why the maxima and minima would come at particular angels. They do so only as an approximation when the distance between the slits is signifigantly smaller than the other distances in the situation. In order to prove this nicely, one of the most obvious ways (IMO) is to apply the approximation

$$\sqrt{1 + x} \approx 1 + \frac{x}{2}$$

So that's the level of math that is required for this.

4. Jan 19, 2014

### ZapperZ

Staff Emeritus
Er... no. I was pointing out the nature of the problem. Indicating that this is probably a math question might provide the OP a place to look. Furthermore, the OP indicated that he/she has the mathematical "formula" to look at. Knowing exactly what it is is the starting point since that is something the OP knows.

My philosophy in teaching has ALWAYS been to start where the understanding stops!

Do you really think that, at this level, this is not a far-field situation? Near-field 2-slit experiment is highly unusual. Based on the description given in the OP, I see this as being a standard, straightforward 2-slit experiment in an intro lab. Do you see otherwise?

Zz.

5. Jan 20, 2014

### jostpuur

I believe the original problem was a far-field situation. I also believe that braeden has had difficulty understanding what it means.


text:
# Colorado Slips To 10th Place In National Solar Energy Rankings

September 3, 2015

Colorado has slipped in a national ranking of solar energy for the third year in a row, according to a study by Environment Colorado. The group analyzed the number of solar energy kilowatts in each state relative to its population size.

Colorado ranked 10th in 2014, down one slot from 2013. That’s despite the state requirement that utilities get more of their power from renewable energy sources by 2020.

In order to remain a solar energy leader, Colorado needs to think of renewable energy standards as a floor, not as a ceiling, says Anna McDevitt with Environment Colorado.

"So let’s blow past them," she said.

Critics of the state’s renewable energy standard say it's costing ratepayers too much for too little benefit.


text:
# Table with overlaping colors

I'm interested in making a table with overlaping colors. What I mean is that I would like to add an additional color to the first column of my table that would have opacity set to 20% and if it were possible I'd like it to blend with the row colors. I had a real hard time to find an answer to this. I would really appreciate some help :)

What I have so far is:

\documentclass[12pt, a4paper, oneside, onecolumn]{memoir}

\usepackage[table]{xcolor}
\definecolor{ltable}{HTML}{EDE7FF}
\definecolor{dtable}{HTML}{D9CEFF}

\begin{document}

\rowcolors{3}{ltable}{dtable}
\begin{tabular}{ |c|c| }
\hline
\rowcolor{dtable} \multicolumn{2}{|c|}{some text} \\ \hline
1 & text \\ \hline
2 & text \\ \hline
3 & text \\ \hline
4 & text \\ \hline
5 & text \\ \hline
\end{tabular}

\end{document}

The overlaping color would start on 1 and end on 5. For excercise purposes it could just be black with opacity set on 20%.

You can use tikz to do it. I made a table using matrix and then applied the colors on a background layer. The result that I obtained is given below. I am think you can improve and get the desired result.

MWE

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{matrix,backgrounds,calc}
\definecolor{oddrow}{HTML}{00FF00}
\definecolor{evenrow}{HTML}{0000FF}
\definecolor{column1}{HTML}{FF0000}

\begin{document}
\begin{tikzpicture}
\matrix(tab)[matrix of nodes,nodes={inner sep=5pt,minimum width=1cm},column sep=2mm]
1 & text \\
2 & text \\
3 & text \\
4 & text \\
5 & text \\
};

\begin{scope}[on background layer]
\fill[column1,opacity=0.5] (tab-1-1.north west) rectangle (tab-5-1.south east); column1 color
\foreach \rw in {1,3,...,5} %odd rows
\draw[draw=black,fill=oddrow,opacity=0.5] (tab-\rw-1.north west) rectangle (tab-\rw-2.south east);
\foreach \rw in {2,4,...,4} %even rows
\draw[draw=black,fill=evenrow,opacity=0.5] (tab-\rw-1.north west) rectangle (tab-\rw-2.south east);
\draw (tab-1-1.north west) rectangle ($(tab-1-2.north east)+(0,0.5)$)node[midway]{Some text};
\draw ($(tab-1-1.north west)+(0,0.5)$) rectangle ($(tab-1-2.north east)+(0,1)$)node[midway]{Header};
\end{scope}
\end{tikzpicture}
\end{document}

• Honestly I was afraid Tikz was the fastest solution I find it confusing. Now, I understand the first part: making the two columns of squares. Thanks to another question on this platform I was able to do this: \begin{tikzpicture} \matrix(tab)[matrix of nodes, every node/.style={text depth=.5ex,text height=2ex, minimum width=10mm, draw}, draw=red] {%table without header 1 & text \\ 2 & text \\ 3 & text \\ 4 & text \\ 5 & text \\ }; \end{tikzpicture} But could you point me to a guide so I can figure out how to use/modify the rest of your code? – shantais Oct 29 '18 at 21:47


text:
# Symbolic FindInstance

I'm wondering if there is any way of doing something like the following: FindInstance[x < y, {x}, Reals] and getting, as a possible return value, {x -> y - 1}.

Currently FindInstance will complain that y is a non-constant expression..

-
I am assuming you would not be satisfied with FindInstance[x - y < 0, {x, y}, Reals] (* {x->0,y->1} *) –  chris Oct 26 '12 at 9:59
At this point, I think that I will just implement a function which uses CylindricalDecomposition. If I do successfully do that, I'll post it as an answer here. –  jnhnum1 Oct 27 '12 at 16:33


text:
# Writting an integral of a transmission/interface problem

Hi everyone,

I have a advection-diffusion problem with an interface, let’s say \Omega = \Omega_1 \cup \Omega_2, \Gamma = \partial \Omega_1 \cap \partial \Omega_2, where [T] = 0, [k\frac{\partial T}{\partial n}] = 0 on \Gamma.
I need to write the integral of something like this
\int_\Gamma \frac{\partial T_1}{\partial n} -\frac{\partial T_2}{\partial n} = \int_\Gamma (\frac{1}{k_1} - \frac{1}{k_2}) k_1 \frac{\partial T_1}{\partial n} = \int_\Gamma (\frac{1}{k_1} - \frac{1}{k_2}) k_2 \frac{\partial T_2}{\partial n}

How can i write the expression \frac{\partial T_1}{\partial n}, \frac{\partial T_2}{\partial n} or in general f(T_1, T_2) on \Gamma for a continuos problem?
I have done T_1 = T*(region==1) and T_2 = T*(region==2) , and after that compute the normal derivative, but i’m not getting what i want to get, and i think if i do that T_1 is different to T_2 in \Gamma.

Any idea?

Best regards,
Rodrigo

First you must clearly define the Gamma edge and number it for example by gam. Then you do:
real K1=1./k1;
real K2=1./k2;
real K=K1-K2;
int1d(Th2,gam)(K*(N.x*(dx(T2))+N.y*(dy(T2)));


text:
# Showing that $a_n = 1-(\frac{-2}{3})^n$ is not bounded

I have the sequence $a_n = 1-(\frac{-2}{3})^n$, and I need to show if it is bounded.

I was first under the assumption that, if a sequence is monotonically increasing/decreasing and bounded that the sequence converges, it also meant that if a sequence converges, there had to be an upper or lower bound. But looking at the graph of the sequence in Wolfram, it turns out to be neither; it is a midpoint, around which the function oscillates, and it oscillates around $\displaystyle{\lim_{n \to \infty} a_n = 1-(\frac{-2}{3})^n} = 1$

So to show that a sequence converges, show that it monotonically increases or decreases and that has a bound. Alright. But how to show that a sequence, like $a_n = 1-(\frac{-2}{3})^n$, is not bounded.

I'm new to all this, so please take it easy ;-)

• $a_n$ is bounded. Its magnitude is always less than 2 e.g. There are more ways to show sequences converge then the way you mentioned. That's a sufficient condition for one type of sequence. – muaddib Jun 25 '15 at 13:04
• a converging sequence is bounded – user190080 Jun 25 '15 at 13:07
• It is? Then I must be doing something wrong on Wolfram - the graph is showing that it is oscillating farther and farther if traveling left on the x-axis. I typed in the sequence as $a_n = 1-(\frac{-2}{3})^n$ in Wolfram. – Garth Marenghi Jun 25 '15 at 13:08
• Consider $a_{2n}$ and $a_{2n+1}$ – Claude Leibovici Jun 25 '15 at 13:08
• Show what did you typed. May be, a typo ? – Claude Leibovici Jun 25 '15 at 13:09

You are right that monotonic, bounded sequences converge, but other sequences converge too. This is an example of a sequence which is not monotonic, but it does converge. You have indicated this yourself in saying that $$\lim_{n\to\infty}a_n=1.$$ Bounded means there exists $M\in\mathbb R$ such that $|a_n|<M$ for all $n$. Since only non-convergent sequences are unbounded, this sequence is bounded.

To prove that $\{a_n\}$ converges to 1, we have to show that we can draw as small an interval around 1 as we would like, and that after a while, the sequence doesn't leave that interval. So we pick the size of the interval. Call this size $\varepsilon$. That means we need to find a number $N$ such that if $n\geq N$, $|a_n-1|<\varepsilon$. Since the sequence $(-2/3)^n$ gets very small as $n\to\infty$, we can pick $N$ which is large enough so that $|(-2/3)^N|<\varepsilon$. Then if $n\geq N$, $|(-2/3)^n|\leq|(-2/3)^N|<\varepsilon$, so $$|a_n-1|=|1-(-2/3)^n-1|=|(-2/3)^n|\leq|(-2/3)^N|<\varepsilon.$$ Thus, for all $n\geq N$, $a_n$ is close to $1$, and the sequence converges.

To answer your other question, to prove that a sequence is unbounded, we have to prove that for all $M\in\mathbb R$, there exists $n$ such that $a_n>M$. This is usually not too difficult (when, indeed, the sequence is unbounded).

Note that $a_n$ is bounded, it is contained in the interval $[1/3,1+2/3]$ because $\left|\left(\frac{-2}{3}\right)^n\right| \leq \frac{2}{3}$ for every $n$

Hints for showing convergence:

1) Show that $\lim_{n\to \infty}\left(\frac{-2}{3}\right)^n = 0$

2) Use the fact that if $\lim_{n\to \infty} x_n = x$ and $\lim_{n\to \infty} y_n = y$ then $\lim_{n\to \infty} x_n -y_n= x-y$

just for the boundedness:

we assume $a_n$ converges to $a$, so therefore we know that $\forall \epsilon >0$ there exists $n_0(\epsilon)$ st

$$\forall n \ge n_0(\epsilon):|a_n-a|<\epsilon$$ so you have your bound for those $a_{n}$, since $a_n\in[a-\epsilon,a+\epsilon]$, then just take the $\max\{|a_n|, n<n_0(\epsilon)\}$ and you're pretty much done.


text:
## Molecularity

$K = \frac{k_{forward}}{k_{reverse}}$

ian_haliburton_1f
Posts: 61
Joined: Fri Sep 28, 2018 12:24 am

### Molecularity

Does anyone know the term for when 4 molecules are required to collide? In other words, what comes after termolecular?

Kate Chow 4H
Posts: 61
Joined: Fri Sep 28, 2018 12:28 am

### Re: Molecularity

A guess, but I would assume tetramolecular.

Joon Chang 2F
Posts: 59
Joined: Fri Sep 28, 2018 12:25 am

### Re: Molecularity

Yes I think a reaction that involves four molecules colliding simultaneously would be called a tetramolecular reaction. Given that termolecular reactions are uncommon due to three molecules having to collide simultaneously, tetramolecular reactions would be very rare.


text:
# How do I get the last 100 lines of a file in Unix?

## How do you find the last line of a file in Unix?

To look at the last few lines of a file, use the tail command. tail works the same way as head: type tail and the filename to see the last 10 lines of that file, or type tail -number filename to see the last number lines of the file. Try using tail to look at the last five lines of your .

## How do I get the last 100 commands in Linux?

In Linux, there is a very useful command to show you all of the last commands that have been recently used. The command is simply called history, but can also be accessed by looking at your . bash_history in your home folder. By default, the history command will show you the last five hundred commands you have entered.

## How do I print the last line of a file in Linux?

Linux tail command syntax

Tail is a command which prints the last few number of lines (10 lines by default) of a certain file, then terminates. Example 1: By default “tail” prints the last 10 lines of a file, then exits. as you can see, this prints the last 10 lines of /var/log/messages.


text:
Math Help - Solve the folowing system of equations

1. Solve the folowing system of equations

Solve the folowing system of equations
$\begin{cases} & \text{ if } {x}^{3}(1-x)+{y}^{3}(1-y) =12xy+18\\ & \text{ if } |3x-2y+10|+|2x-3y|=10 \end{cases}$ (Không nói nên lời)

2. Re: Solve the folowing system of equations

Your post is not displaying correctly (at least for me).
What kind of code is that?
Is that supposed to be LaTeX?

3. Re: Solve the folowing system of equations

Is it supposed to be like this?

Solve the folowing system of equations

$\begin{cases} \text{ if } {x}^{3}(1-x)+{y}^{3}(1-y) =12xy+18\\ \text{ if } |3x-2y+10|+|2x-3y|=10 \end{cases}$

5. Re: Solve the folowing system of equations

Help you how? We have no idea what you know or can do with this, and so can not give any hints, since you have shown no work of your own.

6. Re: Solve the folowing system of equations

Originally Posted by HallsofIvy
Help you how? We have no idea what you know or can do with this, and so can not give any hints, since you have shown no work of your own.
\begin{cases} \text{ if } {x}^{3}(1-x)+{y}^{3}(1-y) =12xy+18\\ \text{ if } |3x-2y+10|+|2x-3y|=10 \end{cases}

7. Re: Solve the folowing system of equations

We are not a homework service. We are here to help, not solve your problems for you. And multiple postings isn't going to solve your problem for you.

-Dan

8. Re: Solve the folowing system of equations

Originally Posted by topsquark

We are not a homework service. We are here to help, not solve your problems for you. And multiple postings isn't going to solve your problem for you.

-Dan
please do the problem for me

9. Re: Solve the folowing system of equations

thth776 is taking a vacation.

Still, if anyone knows how to start this I'd appreciate the hint. I'm kinda curious how to go about this one myself.

-Dan

Edit: Silly me. Just graph the thing. No worries then.

10. Re: Solve the folowing system of equations

There could be a different approach. Let us see if get a different idea. In the mean time have this
Hint: recollect the definition of modulus function, |x| = x for all x >= 0 and |x| = -x for all x <0.
For second case first go by the assumption that 3x-2y+10 >0 and 2x-3y>0, then proceed by solving with elimination by substitution.

11. Re: Solve the folowing system of equations

In other words, because 3x-2y+10 and 2x-3y are expressed as absolute-value functions, we have to assume both for when they are positive, and for when they are negative.

Because the two absolute value functions are being added together, they are commutative, and four possibilities will arise. Two of which will have real-valued solutions.

One of them is easy enough to solve.

The other requires solving a quartic function.

12. Re: Solve the folowing system of equations

Originally Posted by Lambin
In other words, because 3x-2y+10 and 2x-3y are expressed as absolute-value functions, we have to assume both for when they are positive, and for when they are negative.

Because the two absolute value functions are being added together, they are commutative, and four possibilities will arise. Two of which will have real-valued solutions.

One of them is easy enough to solve.

The other requires solving a quartic function.
Can you speak clearly?

13. Re: Solve the folowing system of equations

Sure. When you think about absolute value functions, by definition, we know that the outcome is always positive no matter whether we have a negative or positive value on the inside.

For example,

|3| = 3 and |-3| = 3.

It didn't matter whether it was a 3 or a -3, the outcome is a positive 3.

So, if we turned things around and wanted to solve for an unknown variable in an absolute value function:

|x| = ?

How would we know whether the x was a positive or negative value? Regardless of whether it is positive or negative, the outcome is going to be the same—it will be a positive outcome. So because they share the same outcome, we consider both the positive and negative value as solutions of the absolute-value function that are both valid. In this latter example, it would mean that we have to consider BOTH x and -x as solutions.

How does this connect with our problem in this post?

When we work with analytic expressions instead of numbers in absolute-value functions, we don't treat them differently.

So, with $|4x+3|=y$, we have to consider two things:

1. $4x+3=y$
2. $-(4x+3)=y$

See?

This would mean that for $|3x-2y+10|+|2x-3y|=10$, you would discover that there would be four equations. Think about what was shown earlier, and see how it relates with this problem. With these four equations, we have a system of equations with $x^3(1-y)+y^3(1-x)=12xy+18$.

You would have to solve from here—seeing where the four equations intersect with $x^3(1-y)+y^3(1-x)=12xy+18$.

14. Re: Solve the folowing system of equations

Originally Posted by Lambin
Sure. When you think about absolute value functions, by definition, we know that the outcome is always positive no matter whether we have a negative or positive value on the inside.

For example,

|3| = 3 and |-3| = 3.

It didn't matter whether it was a 3 or a -3, the outcome is a positive 3.

So, if we turned things around and wanted to solve for an unknown variable in an absolute value function:

|x| = ?

How would we know whether the x was a positive or negative value? Regardless of whether it is positive or negative, the outcome is going to be the same—it will be a positive outcome. So because they share the same outcome, we consider both the positive and negative value as solutions of the absolute-value function that are both valid. In this latter example, it would mean that we have to consider BOTH x and -x as solutions.

How does this connect with our problem in this post?

When we work with analytic expressions instead of numbers in absolute-value functions, we don't treat them differently.

So, with $|4x+3|=y$, we have to consider two things:

1. $4x+3=y$
2. $-(4x+3)=y$

See?

This would mean that for $|3x-2y+10|+|2x-3y|=10$, you would discover that there would be four equations. Think about what was shown earlier, and see how it relates with this problem. With these four equations, we have a system of equations with $x^3(1-y)+y^3(1-x)=12xy+18$.

You would have to solve from here—seeing where the four equations intersect with $x^3(1-y)+y^3(1-x)=12xy+18$.
It was difficult . Can you solve clearly?

15. Re: Solve the folowing system of equations

Toàn là một lũ cùi mía, đ' đứa nào bít giải, dẹp mẹ cái trang này đi

Page 1 of 2 12 Last


text:
×

# Need help with this calculus problem

Given $$\displaystyle \pi = 2\int_{-1}^{1} \! \sqrt{1-x^2} \, \mathrm{d}x$$, use properties of integrals to compute the definite integral,

$$\displaystyle A = \int_{-2}^{2} \! (x-3) \sqrt{4-x^2} \, \mathrm{d}x$$

(Source: Tom Apostol - Calculus Volume 1)

Note by Danish Mohammed
3 years, 2 months ago

Sort by:

Got it finally! :D $$\displaystyle \int_ {-2}^2 (x - 3)\sqrt {4 - x^2}\, dx$$

$$=2 \displaystyle \int_{-2}^2 (x-3) \sqrt{1-\left(\frac{x}{2}\right)^2} \, dx$$

$$=4 \displaystyle \int_{-1}^1 (2 x-3) \sqrt{1-x^2} \, dx$$

$$=8 \displaystyle \int_{-1}^1 x \sqrt{1-x^2} \, dx-12 \int_{-1}^1 \sqrt{1-x^2} \, dx$$

$$=8(0)-6 \pi$$ (the first integral is zero as the integrand is odd)

$$=-6 \pi$$ · 3 years, 2 months ago


text:
# Excess Topologies in Metric Spaces

Revision as of 03:03, 17 September 2014; view current revision

### New Zealand Journal of Mathematics

Vol. 44, (2014), Pages 45-60

Gerald Beer

Department of Mathematics,

California State University Los Angeles,

5151 State University Drive, Los Angeles,

California 90032, USA}

Camillo Costantini

Dipartimento di Matematica,

Universit\{a} di Torino,

via Carlo Alberto 10,

10123 Torino, Italy} mailto:camillo.costantini@unito.it

Sandro Levi

Dipartimento di Matematica e Applicazioni,

Universit\{a} di Milano-Bicocca,

via Cozzi 53,

20125 Milano, Italy}

Abstract Given a family $\mathcal {S}$ of nonempty subsets of a metric space $\langle X,d \rangle$ containing the singletons, we consider topologies on the nonempty subsets of X generated by families of excess functionals of the from $\{e_d(S,\cdot) : S \in \mathcal{S}\}$. Such topologies can be broken into lower and upper halves: the lower (resp. upper) half is the weakest topology on the nonempty subsets such that each excess functional in the family is upper (resp. lower) semicontinuous. Remarkably, one such topology can be stronger than another while its lower half can fail to be. We also study excess topologies generated by families of the form $\{e_d(\cdot,S) : S \in \mathcal{S}\}$; the results we give exhibit a decided lack of symmetry with those for the former class of excess topologies. This paper can be viewed as a sequel to a recent paper by the authors on gap topologies [3]. While the methodology and point-of-view are similar, the subject matter here is considerably more subtle.

Keywords excess functional, excess topology, Wijsman topology, dual Wijsman topology, Hausdorff distance, gap functional, quasi-uniformity, strict inclusion, bornology

Classification (MSC2000) 54B20.


text:
# Bignums makes running sprite-local blocks fail

Here is a project to reproduce the bug.
Turn bignums off, then run . It works as expected, making 'Label' say "It works!" for 2 seconds. However, if you turn bignums on, it fails to work.

Good catch. I have no idea why. @djdolphin, any ideas?

I thought maybe it was messing up the "2" in "for 2 secs," but no, calling SAY directly isn't broken, just running another sprite's one.

Looks like "wishes" need to be added to the bignums version of InputSlotMorph.prototype.evaluate...

Sigh, any minute now Jens is going to yell at me about how this shows why libraries overwriting Snap! primitives is a bad idea, but when I say we should just include this library in the core Snap! he'll yell at me again.

Political gridlock at it's finest.

Oh whoops I missed this one while fixing bignums bugs yesterday... I hope I turn out to be competent to fix this one...

V6.8.0 still needs patching... Something like

if (useBigNums) {
....
InputSlotMorph.prototype.evaluate = function () {
if (this.isNumeric) {return parseNumber(this.contents().text || '0')};
return window.bigNumbers.originalEvaluate.apply( this);
};
...

Project.

Okay, perhaps it's time to retire bignums altogether

Fixed. All I had to do was copy your own patch to evaluate into the bignums code. I'll make a PR for this but I also want to work on colors a little.


text:
# C(n,r)=C(r,r)⋅C(n−r,0)+C(r,r−1)⋅C(n−r,1)+C(r,r−2)⋅C(n÷r,2)+C(r,r−3):C(n−r,3)++C(r,1)⋅C(a−r,r−1)+C(r,0)⋅C(n−r,r)

Question
Probability
C(n,r)=C(r,r)⋅C(n−r,0)+C(r,r−1)⋅C(n−r,1)+C(r,r−2)⋅C(n÷r,2)+C(r,r−3):C(n−r,3)++C(r,1)⋅C(a−r,r−1)+C(r,0)⋅C(n−r,r)

2020-12-26
Suppose that you have n objects, and you line them up. You need to pick r objects. If you take r objects from the first r objects, then you need to take 0 objects from the rest of the objects, which you have n—r. Thus, you have O(r,r)- C(n—r,0) ways of doing this.
Similarly, if you pick k objects from the first r objects, you need to take r —k objects from the rest of the objects. You have O(r, k)-C(n—r,r—k) ways of doing this.
Since you can take Q, 1, ..., r objects from the first r objects, we have that 0
ways. Thus we have proven that PSKr C(n,r)=∑ C(r,k)*C(n-r,r-k) k=0ZSK
which we needed to prove.

### Relevant Questions

The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius $$\displaystyle{R}={7.4}\times{10}^{{-{15}}}$$ m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit $$\displaystyle={1.66}\times{10}^{{-{27}}}$$ kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
A manufacturer of digital phones has the following probability distribution for the number of defects per phone: xf(x) .89 1.07 2.03 3.01
(a) Determine the probability of 2 or more defects. (b) Is a randomly selected phone more likely to have 0 defects or 1 or more defects?
If you used a random number generator for the numbers from 1 through 20 to play a game, what is the theoretical probability of getting each of these outcomes? a. A multiple of 3 or a multiple of 7, P(multiple of 3 or multiple of 7) b. P( even or odd) c. P(prime or 1) d. How did you find the probabilities of these events?
The diagram shows a spinner. When the arrow is spun the probability of scoring a 2 is 0.3 . The arrow is spun twice and the scores are added. What is the probability that the total score is 6?6? Express the answer as a decimal. [ \sqrt { 2 } \text { . } ]


text:
# What is the Cartesian form of ( 3 , (-9pi)/4 ) ?

Nov 20, 2016

The cartesian coordinates are $\left(\frac{3 \sqrt{2}}{2} , \frac{- 3 \sqrt{2}}{2}\right)$

#### Explanation:

To convert from polar coordinates $\left(r , \theta\right)$ to cartesian coordinates $\left(x , y\right)$, we use the following equations

$x = r \cos \theta$

and $y = r \sin \theta$

Here $\left(r , \theta\right) = \left(3 , - 9 \frac{\pi}{4}\right)$

Therefore,

$x = 3 \cos \left(\frac{- 9 \pi}{4}\right) = 3 \cos \left(- 2 \pi - \frac{\pi}{4}\right) = 3 \cos \left(- \frac{\pi}{4}\right)$

$x = 3 \cos \left(\frac{\pi}{4}\right) = \frac{3 \sqrt{2}}{2}$

$y = 3 \sin \left(\frac{- 9 \pi}{4}\right) = 3 \sin \left(- 2 \pi - \frac{\pi}{4}\right) = 3 \sin \left(- \frac{\pi}{4}\right)$

$y = 3 \cdot - \frac{\sqrt{2}}{2} = \frac{- 3 \sqrt{2}}{2}$

The cartesian coordinates are $\left(\frac{3 \sqrt{2}}{2} , \frac{- 3 \sqrt{2}}{2}\right)$