text:
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# Aggregate demand in Keynesian analysis

## Key points

• Aggregate demand is the sum of four components: consumption, investment, government spending, and net exports.
• Consumption can change for a number of reasons, including movements in income, taxes, expectations about future income, and changes in wealth levels.
• Investment can change in response to its expected profitability, which in turn is shaped by expectations about future economic growth, the creation of new technologies, the price of key inputs, and tax incentives for investment. Investment can also change when interest rates rise or fall.
• Government spending and taxes are determined by political considerations.
• Exports and imports change according to relative growth rates and prices between two economies.
• Disposable income is income after taxes.
• An inflationary gap exists when equilibrium is at a level of output above potential GDP.
• A recessionary gap exists when equilibrium is at a level of output below potential GDP.

## Aggregate demand in Keynesian analysis

The Keynesian perspective focuses on aggregate demand. The general idea being that firms produce output only if they expect it to sell.
Thus, while the availability of the factors of production determines a nation’s potential gross domestic product, or GDP, the amount of goods and services actually being sold—known as real GDP—depends on how much demand exists across the economy. You can see this concept represented graphically in the diagram below.
This Keynesian view of the AD/AS model shows that with a horizontal aggregate supply, a decrease in demand leads to a decrease in output but no decrease in prices.
Image credit: Figure 1 in "Aggregate Demand in Keynesian Analysis" by OpenStaxCollege, CC BY 4.0
Keynes argued that, for reasons we'll explain shortly, aggregate demand is not stable—it can change unexpectedly.
Suppose the economy starts where start text, A, D, 0, end text intersects start text, S, R, A, S, end text at start text, P, 0, end text and start text, Y, p, end text in the diagram above. Because start text, Y, p, end text is potential output, the economy is at full employment. But because aggregate demand is volatile, it can easily fall. Thus, even if we start at start text, Y, p, end text, if aggregate demand falls, we find ourselves in what Keynes termed a recessionary gap. The economy is in equilibrium but with less than full employment, as shown at start text, Y, 1, end text. Keynes believed that the economy would tend to stay in a recessionary gap, with its attendant unemployment, for a significant period of time.
Similarly—though not shown in the figure—if aggregate demand increases, the economy could experience an inflationary gap, where demand is attempting to push the economy past potential output. As a consequence, the economy would experience inflation.
The key policy implication for either situation is that government needs to step in and close the gap, increasing spending during recessions and decreasing spending during booms, to return aggregate demand to match potential output.
Since aggregate demand is total spending, economy-wide, on domestic goods and services, economists also refer to it as total planned expenditure. We can calculate aggregate demand by adding up its four components: consumption expenditure, investment expenditure, government spending, and spending on net exports—exports minus imports.

## What determines consumption expenditure?

Consumption expenditure is spending by households and individuals on durable goods, nondurable goods, and services. Durable goods are things that last and provide value over time, such as automobiles. Nondurable goods are things like groceries—once you consume them, they are gone. Services are intangible things consumers buy, like healthcare or entertainment.
Keynes identified three factors that affect consumption:
• Disposable income: For most people, the single most powerful determinant of how much they consume is how much income they have in their take-home pay. This left-over income is also also known as disposable income, which is income after taxes.
• Expected future income: Consumer expectations about future income also are important in determining consumption. If consumers feel optimistic about the future, they are more likely to spend and increase overall aggregate demand. News of recession and troubles in the economy will make them pull back on consumption.
• Wealth or credit: When households experience a rise in wealth, they may be willing to consume a higher share of their income and to save less. When the US stock market rose dramatically in the late 1990s, for example, US rates of saving declined, probably in part because people felt that their wealth had increased and there was less need to save. How do people spend beyond their income when they perceive their wealth increasing? The answer is borrowing. On the other side, when the US stock market declined about 40% from March 2008 to March 2009, people felt far greater uncertainty about their economic future, so rates of saving increased while consumption declined.
Finally, Keynes noted that a variety of other factors combine to determine how much people save and spend. If household preferences about saving shift in a way that encourages consumption rather than saving, then aggregate demand will shift out to the right.

## What determines investment expenditure?

Spending on new capital goods is called investment expenditure. Investment falls into four categories: producer’s durable equipment and software, new nonresidential structures, changes in inventories, and residential structures. The first three types of investment are conducted by businesses, while the last is conducted by households.
Keynes’s treatment of investment focuses on the key role of expectations about the future in influencing business decisions.
When a business decides to make an investment in physical assets—like plants or equipment—or in intangible assets—like skills or a research and development project—that firm considers both the expected benefits of the investment, like future profits, and the costs of the investment, such as interest rates.
The clearest driver of the benefits of an investment is expectations for future profits. When an economy is expected to grow, businesses perceive a growing market for their products. Their higher degree of business confidence will encourage new investment. For example, in the second half of the 1990s, US investment levels surged from 18% of GDP in 1994 to 21% in 2000. However, when a recession started in 2001, US investment levels quickly sank back to 18% of GDP by 2002.
Interest rates also play a significant role in determining how much investment a firm will make. Just as individuals need to borrow money to purchase homes, businesses need financing when they purchase big ticket items. The cost of investment thus includes the interest rate. Even if the firm has the funds, the interest rate measures the opportunity cost of purchasing business capital. Lower interest rates stimulate investment spending and higher interest rates reduce it.
Many factors can affect the expected profitability on investment. For example, if the price of energy declines, then investments that use energy as an input will yield higher profits. If the government offers special incentives for investment—for example, through the tax code—then investment will look more attractive; conversely, if government removes special investment incentives from the tax code or increases other business taxes, then investment will look less attractive.
Keynes believed that business investment is the most variable of all the components of aggregate demand.

## What determines government spending?

The third component of aggregate demand is spending by federal, state, and local governments. Although the United States is usually thought of as a market economy, government still plays a significant role in the economy. Government provides important public services such as national defense, transportation infrastructure, and education.
Keynes recognized that the government budget offered a powerful tool for influencing aggregate demand. Not only could aggregate demand be stimulated by more government spending—or reduced by less government spending—but consumption and investment spending could be influenced by lowering or raising tax rates.
Keynes concluded that during extreme times like deep recessions, only the government had the power and resources to move aggregate demand.

## What determines net exports?

Exports are products produced domestically and sold abroad, and imports are products produced abroad but purchased domestically. Since aggregate demand is defined as spending on domestic goods and services, export expenditures add to aggregate demand, while import expenditures subtract from aggregate demand.
Two sets of factors can cause shifts in export and import demand: changes in relative growth rates between countries and changes in relative prices between countries.
The level of demand for a nation’s exports tends to be most heavily affected by what is happening in the economies of the countries that would be purchasing those exports. For example, if major importers of US-made products like Canada, Japan, and Germany have recessions, exports of US products to those countries are likely to decline since quantity of a nation’s imports is directly affected by the amount of income in the domestic economy. More income will bring a higher level of imports.
Exports and imports can also be affected by relative prices of goods in domestic and international markets. If US goods are relatively cheaper compared with goods made in other places—perhaps because a group of US producers has mastered certain productivity breakthroughs—then US exports are likely to rise. If US goods become relatively more expensive—perhaps because a change in the exchange rate between the US dollar and other currencies has pushed up the price of inputs to production in the United States—then exports from US producers are likely to decline.

## A quick overview of what drives aggregate demand

We've gone over all the components of aggregate demand and what drives them to change above, but you may find it useful to refer to the table below that summarizes this information.
Determinants of aggregate demand
Reasons for a decrease in aggregate demandReasons for an increase in aggregate demand
ConsumptionRise in taxes, fall in income, rise in interest, desire to save more, decrease in wealth, fall in future expected incomeConsumptionDecrease in taxes, increase in income, fall in interest rates, desire to save less, rise in wealth, rise in future expected income
InvestmentFall in expected rate of return, rise in interest rates, drop in business confidenceInvestmentRise in expected rate of return, drop in interest rates, rise in business confidence
GovernmentReduction in government spending, increase in taxesGovernmentIncrease in government spending, decrease in taxes
Net exportsDecrease in foreign demand, relative price increase of US goodsNet exportsIncrease in foreign demand, relative price drop of S. goods

## Summary

• Aggregate demand is the sum of four components: consumption, investment, government spending, and net exports.
• Consumption can change for a number of reasons, including movements in income, taxes, expectations about future income, and changes in wealth levels.
• Investment can change in response to its expected profitability, which in turn is shaped by expectations about future economic growth, the creation of new technologies, the price of key inputs, and tax incentives for investment. Investment can also change when interest rates rise or fall.
• Government spending and taxes are determined by political considerations.
• Exports and imports change according to relative growth rates and prices between two economies.
• Disposable income is income after taxes.
• An inflationary gap exists when equilibrium is at a level of output above potential GDP.
• A recessionary gap exists when equilibrium is at a level of output below potential GDP.

## Self-check questions

In the Keynesian framework, which of the following events might cause a recession? Which might cause inflation? Sketch AD/AS diagrams to illustrate your answers.
• A large increase in the price of the homes people own
• Rapid growth in the economy of a major trading partner
• The development of a major new technology offering profitable opportunities for business
• An increase in interest rates
• A decrease in the price of a good imported from a major trading partner
In a Keynesian framework, using an AD/AS diagram, which of the following government policy choices offer a possible solution to recession? Which offer a possible solution to inflation?
• A tax increase on consumer income
• A surge in military spending
• A reduction in taxes for businesses that increase investment
• A major increase in what the US government spends on healthcare

## Review questions

• Name some economic events not related to government policy that could cause aggregate demand to shift.
• Name some government policies that could cause aggregate demand to shift.

## Critical-thinking questions

• The Conference Board’s Global Economic Outlook 2015 projects China’s growth between 2015 and 2019 to be about 5.5%. TheInternational Business Times reports that China is the United States’ third largest export market, with exports to China growing 294% over the last 10 years. Explain what impact China has on the US economy.
• What may happen if growth in China continues or contracts?

## Want to join the conversation?

• Answers to the critical thinking questions?
• According to the text on this article: "Rapid growth by a major trading partner will increase demand for exports. Aggregate demand will shift to the right and may cause inflation if it goes beyond potential GDP."

Here are key points to keep in mind:
1. America is one of China's largest exporting partners.
2. As China's economy grows, it's demand for American exports also grow. If China's economy contracts, it's demand for American exports will also contract.
3. Exports are a component of GDP. An increase in exports will shift the aggregate demand curve to the right. A decrease in exports will shift aggregate demand to the left.

(Answer to question 1) Change in China's economy impacts the American economy by having some power to shift the US aggregate supply to the left or right.

(Answer to question 2) Growth in China's economy will result in American aggregate demand shifting to the right. This could lead to the US growing its real GDP if its current real GDP is below potential GDP, but if real GDP is above potential GDP, then inflation will result with little increase in real GDP. Contraction in China's economy will result in American aggregate demand shifting to the left. This could help America's economy "cool down" if the American economy has already surpassed its potential GDP, or cause the American economy to go into a recession if the American economy is already below its potential GDP.
• Why is investment is considered as an unstable component of aggregate demand?
• According to Keynes, levels of investment are determined in large part by expectations. Since human expectations are volatile, it follows that investment is volatile.
• What is the different between micro and macro economy
(1 vote)
• Micro and Macro are two main branches of the economics. Micro helps to understand the economy in the light of consumers, firms and production while Macros' main objects are the overall economy, like GDP (output), Inflation (overall price changes) and Unemployment.


text:
# Testing¶

When we write code, we want to be sure that it is doing exactly what we thinking it is doing. Unfortunately, no one is perfect and this property is transferred to the code that we write. So we, as conscious, programmers, should write tests for all of your code ensuring that is it doing the right thing in all circumstances.

Typically we apply tests for our code that the function level, where there is, at least a test for each of the functions in our program. However, in order to achieve full test coverage, it may be necessary to have more than one test for a given function. Consider the code below, which performs simple temperature conversions, using logic to change the arithmetic as appropriate.

def temperature_conversion(value, unit):
"""
Temperature conversion that is dependent on
the input unit.

Args:
value (float): Temperature to be converted.
unit (str): Unit of the original temperature.

Returns:
(float): Temperature converted to Kelvin.
"""
if unit == 'C':
return value + 273.15
elif unit == 'F':
return (value + 459.67) * 5 / 9
else:
raise ValueError("The supported units are C or F.")

In this example, our code will raise an error if the unit argument is neither 'C' or 'F'.

We can now write some tests for this function. We know that the melting point is water is 0 oC, 273.15 K and 32 oF. Therefore our tests should reflect this knowledge.

import numpy as np

c = temperature_conversion(0, 'C')
np.testing.assert_almost_equal(c, 273.15)
print('Test 1 Passed!')

Test 1 Passed!

f = temperature_conversion(32, 'F')
np.testing.assert_almost_equal(f, 273.15)
print('Test 2 Passed!')

Test 2 Passed!

These two tests will cover two of the branches of our function. The np.testing.assert_almost_equal function is an element of the NumPy testing library. This test function checks if the two arguments are approximately equal. This must be approximate due to the floating point arithmetic problems. For an integer, string, or Boolean where the value does not need to be approximate, the np.testing.assert_equal function may be used. If the values are not almost equal, and error will be thrown.

np.testing.assert_almost_equal(c, 270.15)
print('Test 3 Passed!')

---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
/tmp/ipykernel_1967/3447443753.py in <module>
----> 1 np.testing.assert_almost_equal(c, 270.15)
2 print('Test 3 Passed!')

/opt/hostedtoolcache/Python/3.7.12/x64/lib/python3.7/site-packages/numpy/testing/_private/utils.py in assert_almost_equal(actual, desired, decimal, err_msg, verbose)
597         pass
598     if abs(desired - actual) >= 1.5 * 10.0**(-decimal):
--> 599         raise AssertionError(_build_err_msg())
600
601

AssertionError:
Arrays are not almost equal to 7 decimals
ACTUAL: 273.15
DESIRED: 270.15

Notice that the Python kernel never reaches the print statement.

However, so far our tests do not cover every possible operation of the function. We have a test for the if and elif branches, but not the else branch. Therefore, we must introduce another type of test, that checks if an error is raised.

np.testing.assert_raises(ValueError, temperature_conversion, 0, 'c')
print('Test 4 Passed!')

Test 4 Passed!

This function takes the expected error type, the function and the input arguments and will pass if the error is thrown under these circumstances.

## Test-driven development¶

Test-driven development (TDD) is a methodology for computer programming, where the tests are written first for a particular use case and then the operational code is written such that it passes these tests. This methodology is different from that applied to this point in the course, however, it is very popular and powerful when applied properly. In the exercise below, you are asked to try your hand at some test-driven development.

format: plain


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# Math Help - Prove linearly independent

1. ## Prove linearly independent

Prove the statement:

If u, v, and w are linearly independent, then u+v, v, and w are also linearly independent.

2. Originally Posted by ssl000
Prove the statement:

If u, v, and w are linearly independent, then u+v, v, and w are also linearly independent.
Given: $\alpha \vec{u} + \beta \vec{v} + \gamma \vec{w} = \vec{0}$ only when $\alpha = \beta = \gamma = 0$ are equal to zero.

Now re-write $\alpha \vec{u} + \beta \vec{v} + \gamma \vec{w}$ as $\alpha (\vec{u} + \vec{v})+ \delta \vec{v} + \gamma \vec{w} = \vec{0}$ ....

3. Originally Posted by mr fantastic
Given: $\alpha \vec{u} + \beta \vec{v} + \gamma \vec{w} = \vec{0}$ where not all of the $\alpha, \beta, \gamma$ are equal to zero.

This can be re-written as $\alpha (\vec{u} + \vec{v})+ \delta \vec{v} + \gamma \vec{w} = \vec{0}$ where $\delta = \beta - \alpha$ ....
Wouldn't that prove that its linearly dependent, since $\delta = \beta - \alpha$?

4. Originally Posted by ssl000
Wouldn't that prove that its linearly dependent, since $\delta = \beta - \alpha$?
Sorry I misread the question. I've edited my reply accordingly.


text:
# Grid $2\times3$ colored with four colours

I have a $2\times3$ grid which is colored with four colours. How many colourings do there exists such that every two adjacent squares have different colours? I don't know how to start.

• Start by trying to solve similar problems for smaller grids and fewer colors. – MJD Jan 14 '17 at 18:39
• @MJD What is the basic idea? – Problemsolving Jan 14 '17 at 18:45
• This idea should work but not in general: Top middle cell has 4 choices, neighbors have 3 choices, then take cases for the last two cells. – greenturtle3141 Jan 14 '17 at 18:47

You have $4\cdot 3=12$ ways to color the $1$st column. Now given that you've made your choice of colors for the $1$st column, the number of ways to color the $2$nd column will drop down from $12$ to $7$. The same happens for the 3rd column, once you've colored the $2$nd one . So the total number is $$12\cdot 7 \cdot 7=588.$$
• Let A, B,C, D be the four colors. Suppose the $1$st column is (A,B) (A on the top, and B on the bottom). Can you list the possibilities for the $2$nd column? – MathChat Jan 14 '17 at 20:23


text:
# Inequality with restriction

I have been trying to prove without success the following inequality. If $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$ then prove that $a+b+c+d+\frac{1}{abcd}\geq 18$.

-
You should describe the approaches you did try, even if unsuccessful. –  Casteels Feb 27 '14 at 9:57
This is just a guess and I have not worked it out yet... Put $a = \cos\phi\cos\theta$, $b = \sin\phi\cos\theta$, $c = \cos\phi\sin\theta$ and $d = \sin\phi\sin\theta$ with $\theta,\phi\in[0,\pi/2]$. Does that help? –  L__ Feb 27 '14 at 10:06
@L__: be careful, you only parametrized the intersection of the hypersphere with $ad=bc$... –  zozoens Feb 27 '14 at 10:12

First, when $a=b=c=d=\dfrac{1}{2}$, the equality holds.
Now try to use $$\sum a+32\times\frac{1}{32abcd}\ge 36 \sqrt[36]{\frac{abcd}{(32abcd)^{32}}}.$$
Notice also that $1=a^2+b^2+c^2+d^2\ge 4\sqrt{abcd}$, we complete the proof.
@user61810 $4+32=36$. –  Ma Ming Mar 2 '14 at 9:07


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# How can we prove that $F_p \in \mathcal {P} \lor Φ?$

I have created a conjecture which I do not know how to prove. The conjecture is written like:

$F_p \in \mathcal {P} \lor Φ$

$p$ = prime number in correspondence to the appearance of a number in the Fibonacci Sequence.

$\mathcal {P}$ = The set of prime numbers in the Fibonnaci Sequence.

$Φ$ = The set of Fibonacci numbers that have $1+$ divisors which cannot be divided into any earlier Fibonacci numbers.

My conjecture, is that if you have a prime-th Fibonacci number, then that number is either going to be prime or have factors of which are not factors of any earlier Fibonacci numbers. Here is a list of the first $10$ Fibonacci numbers.

1) $1$

2) $1$

3) $2$

4) $3$

5) $5$

6) $8$

7) $13$

8) $21$

9) $34$

10) $55$

As we can see, all prime-th Fibonacci numbers in the list $\in$ $\mathcal {P}$ except for the $2^{nd}$ Fibonacci number because $1$ is not considered a prime, but we will consider it a prime to all numbers. Now as we list the following $20$ Fibonacci numbers this time, we will start to notice that not all prime-th Fibonacci numbers are prime. The $19^{th}$ Fibonacci number is $4181$ which is not a prime number.

$4181 = 37 \times 113$ $\therefore$ $4181 \in Φ$ $\because$ it's factors $(37, 113)$ are not factors of any Fibonacci number before $19$.

All prime-th Fibonacci numbers after $4181$ are prime, but then when we come across the $31^{st}$ Fibonacci number being $1346269$, which is also not a prime number.

$1346269 = 557 \times 2417$ $\therefore$ $134629 \in Φ$ $\because$ it's factors $(557, 2417)$ are not factors of any Fibonacci number before $1346269$.

The most common factor I assume to be divided into fibonnaci numbers is $2^3$ where I assume that $2^3 | F_{6 + 12n} : n \in \mathbb{W}$.

Now at first you would believe that $α(F_p \in \mathcal {P}) > α(F_p \in Φ)$ : $α(χ) =$ amount of times $χ$ appears. But as $P_y \rightarrow \infty$ the gap between $(F_p \in \mathcal {P})_y \land (F_p \in \mathcal {P})_{y + 1} \rightarrow \infty$ at a much faster rate the greater the value of $y$ : $y \in \mathbb{Z^+}$ $\therefore$ $α(F_p \in \mathcal {P}) < α(F_p \in Φ)$ the greater the value of $y$, and when $y > 137$ then $α(F_p \in \mathcal {P}) \ll α(F_p \in Φ)$.

$φ$ can be expressed as a limit:

$$\lim_{k \rightarrow \infty} \frac{F_k + 1}{F_k} = φ : k \in \mathbb{Z^+}$$

$\because$ $F_k = F_{k + 1} - F_{k - 1}$ (you can carry out the mathematics from there). However even if I substitute $k$ as $p$, and then substitute $F_p$ with their unique prime factorisation $\iff$ $F_p \in Φ$, it does not get me anywhere.

Is there a way to prove that this conjecture works either via this approach or another? The largest Fibonnaci prime I know in the sequence so far is $F_{9311}, (9311 = prime)$ so if that is a key fact to finding a proof, please provide it in your answer if you have one. It would be much appreciated.

If you do not know what any of the various symbols in this question mean, I will post an edit to this question tomorrow to explain their definition which will be when I see your answers $-$ I have to go to sleep now. G'night.

Your conjecture is equivalent to stating that for all $n < p$, $\gcd(F_p, F_n) = 1$ if $p$ is prime.
That this is true directly follows from the identity $\gcd(F_m, F_n) = F_{\gcd(m, n)}$.
• So if $n < p, F_{gcd(p,n)} = 1 \therefore φ = \frac{F_{gcd(p,n)} + \sqrt 5}{2F_{gcd(p,n)}}$ – George N. Missailidis Jul 24 '17 at 22:22


text:
To find total cost, to the nearest cent, to cool the house for this 24-hour period

To find total cost, to the nearest cent, to cool the house for this 24-hour period
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

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Alannej
Given:
Rate at which the cost of cooing the house accumulates = \$0.05
Concept Used:
Applications of integral
Calculation:
Rate at which the cost of cooing the house accumulates
$=0.05{\int }_{5.231}^{18.770}\left(80-10\mathrm{cos}\left(\frac{\pi }{12}\right)-78\right)dt$
$=0.05{\int }_{5.231}^{18.770}\left(2-10\mathrm{cos}\left(\frac{\pi t}{12}\right)\right)dt$
$=0.05\left[2{\int }_{5.231}^{18.770}dt-10{\int }_{5.231}^{18.770}\mathrm{cos}\left(\frac{\pi t}{12}\right)dt\right]$
$=0.05\left[2\left(t\right){|{}_{\left(5.231\right)}^{18.770}-\frac{120}{\pi }\left(\mathrm{sin}\left(\frac{\pi t}{12}\right)\right)|}_{5.231}^{18.770}\right]$
putting the limits to get answer
$=0.05{\int }_{5.231}^{18.770}\left(80-10\mathrm{cos}\left(\frac{\pi }{12}\right)-78\right)dt=5.096$


text:
# Integral problem

1. Aug 11, 2014

### Wicketer

Is there a non-zero continuous function, f, R -> R such that ∫f(x) dx over (c, c^2+1) = 0, for all c in R? Hope this makes sense. I've been trying to find a more formal way of approaching this rather than just visualizing it with graphs. Any thoughts?

Many thanks.

2. Aug 11, 2014

### maajdl

Try to see if you can built a primitive F(c) such that F(c²+1)=F(c) .
If you can find one, then f is its derivative.

I guess it might be useful to observe that c²+1 > c .
Defining F in the interval [0,1] might be all that is necessary.
Will F be continuous, will its derivative be continuous, ???

Did you give the full statement of the question?

Try to graph such a function.
Give it a try.

Last edited: Aug 11, 2014
3. Aug 11, 2014

### HallsofIvy

Generally, the "zero-function" is f(x)= 0 for all x so that "non- zero" means "not equal to 0 for at least one x". But we can have f(x)= 0 for x= 1, non-zero for x not equal to 0, and its Riemann integral, $\int_a^b f(x)dx$, from any a to any b, is non-zero. If you use the Lebesque function, then we can have f(x) equal to zero on any set of measure 0 and have $\int_a^b f(x)d\mu$ non-zero.

4. Aug 11, 2014

### disregardthat

following up on maajdl's comment, the existence of such a function f is equivalent to the existence of a strictly increasing or decreasing differentiable function F : R --> R such that F(c^2+1) = F(c).

But, it is easily seen that no such function can exist, because the derivative of F(x^2+1) is equal to the derivative of F(x), which means that 2xf(x^2+1) = f(x). Plugging in x = 0 shows that f(0) = 0. Assuming you mean non-zero as in f can not take the value 0.


text:
# How to make grooves without editing geometry

I'm trying to cut grooves onto a curved surface without directly editing the geometry. This is my desired end result that I achieved by Ctrl+B to bevel and E to extrude down (Exactly what I don't want to do).
Is there a way I can do this with modifiers or something like that? I will occasionally need to edit the fine details of the mesh and I can't do it whilst these grooves are physically preset, and I need them there for reference. I also can't bake a normal map for it because the materials are geometry defined and my change too. Thanks in advance.

You can add a groove using a boolean modifier.

My process would be as follows:

Duplicate some faces from your original, then separate them into another object.

Scale them down, and extrude out some thickness to get the inverse of the groove you want:

Add a boolean modifier on the first object set to difference, and select the new object as the target:

Then, just hide the second object to see the groove:

You could make your extrusion mechanically, as you have, but to 0 height, make a vertex group from the extruded vertices, and aim a Displace modifier at it, followed by a Weld to disappear it when you set the displacement to 0.

• Advantages over the Boolean given by @person ? Not many.. unless the Boolean is playing up because of necessarily coincident faces
• Disadvantages .. you can only edit the unmodified mesh, which contains doubled vertices.

So probably, the Boolean is better.


text:
How to add an empty line between two formulars

I have the following code

\listfiles
\documentclass[oneside,12pt]{scrartcl}
\usepackage[ngerman]{babel}
%\usepackage{amsmath}
\usepackage[fixamsmath,disallowspaces]{mathtools}
\begin{document}

\underline{Verschiebare Ger{\"a}te}

\begin{math}
P_j(t,s_j) = \begin{cases}
0 & \text{for } t < s_j \\
Q_j(t-s_j) & \text{for  }s_j \leq t\leq s_j+p_j  \\
0 & \text{for  } t>s_j+p_j
\end{cases}
\end{math}

\newline

\begin{math}
\text{with }s_j = r_j+ \Delta t\\
\Delta t\leq tDoF
\end{math}

\end{document}

And I would like to add an empty line between the two math-parts. Unfortunately the command \newline or \\ does not work in this context. Can you help me?

math is the same as \( or \$ and designed for inline math not displays. You want a display environment such as equation or align that will automatically offset the expression with vertical space and not require forced adjustments, for example:

\documentclass[oneside,12pt]{scrartcl}
\usepackage[ngerman]{babel}
%\usepackage{amsmath}
\usepackage[fixamsmath,disallowspaces]{mathtools}
\begin{document}

\section*{Verschiebare Ger{\"a}te}

\begin{equation*}
P_j(t,s_j) = \begin{cases}
0 & \text{for } t < s_j \\
Q_j(t-s_j) & \text{for  }s_j \leq t\leq s_j+p_j  \\
0 & \text{for  } t>s_j+p_j
\end{cases}
\end{equation*}
with
\begin{align*}
s_j = r_j+ \Delta t\\
\Delta t\leq tDoF
\end{align*}

\end{document}


text:
# What is meant by “constant” in Liouville's Theorem?

Liouville's Theorem states that:

Every holomorphic* function for which there exists a positive number M such that $|f(z)| \le M$ for all $z \in \mathbb C$ is constant.

I'm using this to prove the Fundamental Theorem of Algebra and I understand that for the proof to be complete:

It is sufficient to show any $p(z)$ has one root, for by division we can then write $p(z)=(z-z_0)g(z)$, with $g$ of lower degree. Note that if $$p(z)=a_n z^n + a_{n-1} z^{n-1} + ... + a_0,$$ then as $|z| \to \infty$, $|p(z)| \to \infty$. \cite{FTANY} This follows as: $$p(z)= z^n \left| a_n + \frac{a_{n-1}}{z} + ... + \frac{a_0}{z^n} \right|.$$ Assume $p(z) \ne 0,$ so $1/p(z)$ is bounded for $|z| \le R$ by continuity. Thus, $1/p(z)$ is bounded, entire function, which must be constant. Thus, $p(z)$ is constant, a contradiction which implies $p(z)$ must have a zero (our assumption.)

What I want to understand is why showing that $f$ is constant shows that $f$ has a root?

• It shows that if $f$ has no roots, then it is constant. This is eqivalent to stating that if $f$ is not constant, then it has at least one root. – user228113 Apr 29 '15 at 17:49

1) Let $p(z)$ be a nonconstant polynomial.
2) Assume $p(z)$ has no root.
3) Conclude $p$ is constant.


text:
# How do you solve logx-log2=1?

Aug 10, 2016

x = 20

#### Explanation:

Using the following $\textcolor{b l u e}{\text{laws of logarithms}}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\log x - \log y = \log \left(\frac{x}{y}\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(A\right)$
This applies to logarithms to any base.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\log}_{b} a = n \Leftrightarrow a = {b}^{n}} \textcolor{w h i t e}{\frac{a}{a}} |}}} \ldots \ldots . . \left(B\right)$

$\text{Using (A) } \log x - \log 2 = \log \left(\frac{x}{2}\right)$

A logarithm expressed as log x , usually indicates that the base is 10.

$\text{Using (B) } {\log}_{10} \left(\frac{x}{2}\right) = 1 \Rightarrow \frac{x}{2} = {10}^{1} = 10$

Thus $\frac{x}{2} = 10 \Rightarrow x = 20$


text:
NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area Ex 11.4

# NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area Ex 11.4

## NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area Ex 11.4

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NCERT Solutions for Class 7 Maths Chapter 11 Perimeter and Area Exercise 11.4
Ex 11.4 Class 7 Maths Question 1.
A garden is 90 m long and 75 m broad. A path 5 m wide is to be built outside and around it. Find the area of the path. Also find the area of the garden in hectare.
Solution:

Given: Length = 90 m
Area of the garden = l × b
= 90 m × 75 m = 6750 m2
Length of the garden including path
= 90m + 5m + 5m = 100 m
Breadth of the garden including path
= 75m + 5m + 5m = 85m
Area of the garden including path
= l × b
= 100 m × 85 m = 8500 m2
Area of the path = 8500 m2 – 6750 m2 = 1750 m2
Hence, required area of path = 1750 m2 and area of the garden = 6750 m2 = 0.675 ha

Ex 11.4 Class 7 Maths Question 2.
A 3 m wide path runs outside and around a rectangular park of length 125 m and breadth 65 m. Find the area of the path.
Solution:
Length of the park = 125 m
Breadth of the park = 65 m
Area of the park = l × b
= 125 m × 65 m = 8125 m2

Length of the park including path
= 125 m + 3m + 3m = 131 m
Breadth of the park including path
= 65m + 3m + 3m = 71m
Area of the park including path
= 131 m × 71 m = 9301 m2
∴ Area of the path
= 9301 m2 – 8125 m2 = 1176 m2
Hence, the required area = 1176 m2.

Ex 11.4 Class 7 Maths Question 3.
A picture is painted on a cardboard 8 cm long and 5 cm wide such that there is a margin of 1. 5 cm along each of its sides. Find the total area of the margin.
Solution:

Length = 8 cm, breadth = 5 cm
Area of the cardboard = l × b
= 8 cm × 5 cm = 40 cm2
Width of the margin = 1.5 cm
Length of the inner cardboard
= 8 cm – 1.5 × 2 cm
= 8 cm – 3 cm = 5 cm
= 5 cm – 1.5 × 2 cm
= 5 cm – 3 cm = 2 cm
Area of the inner rectangle = l × b
= 5 cm × 2 cm = 10 cm2 Area of the margin
= 40 cm2 – 10 cm2 = 30 cm2
Hence, the required area = 30 cm2.

Ex 11.4 Class 7 Maths Question 4.
A verandah of width 2.25 m is constructed all along outside a room which is 5.5 m long and 4 m wide. Find:
(i) the area of the verandah.
(ii) the cost of cementing the floor of the verandah at the rate of ₹ 200 per m2.
Solution:

Length of the room = 5.5 m
Breadth of the room = 4 m
∴ Area of the room = l × b = 5.5 m × 4 m = 22 m2
Width of the verandah = 2.25 m
Length of the room including verandah
= 5.5 m + 2 × 2.25 m = 10 m
Breadth of the room including verandah
= 4 m + 2 × 2.25 m = 8.50 m2
Area of the room including verandah = l × b
= 10 m × 8.50 m = 85 m2
(i) Area of the verandah = 85 m2 – 22 m2
= 63 m2
(ii) Cost of cementing the floor of the verandah = ₹ 63 × 200 = ₹12600

Ex 11.4 Class 7 Maths Question 5.
A path 1 m wide is built along the border and inside a square garden of side 30 m. Find:
(i) the area of the path.
(ii) the cost of planting grass in the remaining portion of the garden at the rate of ₹ 40 per m2.
Solution:

Area of the square garden = (Side)2
= 30 m × 30 m = 900 m2
Length of the garden excluding the path = 30 m – 2 × 1 m = 28 m
∴ Area of the garden excluding the path = 28 m × 28 m = 784 m2
(i) Area of the path = 900 m2 – 784 m2
= 116 m2
(ii) Cost of the planting the remaining portion at the rate of ₹ 40 per m2
= ₹ 40 × 784 = ₹ 31,360

Ex 11.4 Class 7 Maths Question 6.
Two cross roads, each of width 10 m, cut a right angles through the centre of a rectangular park of length 700 m and breadth 300 m and parallel to its sides. Find the area of the roads. Also find the area of the park excluding cross roads. Give the answer in hectares.
Solution:

Length of the road parallel to the length of the park = 700 m
Width of the road = 10 m
∴ Area of the road = l × b = 700 m × 10 m = 7000 m2
Length of the road parallel to the breadth of the park = 300 m
Width of the road = 10 m Area of this road = l × b = 300 m × 10 = 3000 m2
= 7000 m2 + 3000 m2 – Area of the common portion
= 10,000 m2 – 10 m × 10 m
= 10,000 m2 – 100 m2
= 9900 m2 = 0.99 ha
Area of the park = l × b
= 700 m × 300 m = 210000 m2
Area of the park excluding the roads
= 210000 m2 – 9900 m2
= 200100 m2 = 20.01 ha

Ex 11.4 Class 7 Maths Question 7.
Through a rectangular field of length 90 m and breadth 60 m, two roads are constructed which are parallel to the sides and cut each other at right angles through the centre of the fields. If the width of each road is 3 m, find
(i) the area covered by the roads.
(ii) the cost of constructing the roads at of the rate of ₹ 110 per m2.
Solution:

Length of the road along the length of the field = 90 m
∴ Area of this road = l × b
= 90 m × 3 m = 270 m2
Similarly, the area of the road parallel to the breadth of the field = l × b
= 60 m × 3 m = 180 m2 Area of the common portion
= 3m × 3m = 9m2
(i) Area of the two roads
= 270 m2 + 180 m2 – 9 m2
= 450 m2 – 9 m2 = 441 m2
(ii) Cost of constructing the roads
= ₹ 110 × 441 = ₹ 48,510

Ex 11.4 Class 7 Maths Question 8.
Pragya wrapped a card around a circular pipe of radius 4 cm and cut off the length required of the cord. Then she wrapped it around a square box of side 4 cm (also shown). Did she have any cord left? (π = 3.14)

Solution:
Length of the cord = Circumference of the circular pipe
= 2πr = 2 × 3.14 × 4 = 25.12 cm
Perimeter of the square box
= 4 × side = 4 × 4 cm = 16 cm
Length of the cord left
= 25.12 cm – 16 cm = 9.12 cm
Yes, 9.12 cm cord is left.

Ex 11.4 Class 7 Maths Question 9.
The given figure represents a rectangular lawn with a circular flower bed in the middle. Find:
(i) the area of the whole land.
(ii) the area of the flower bed.
(iii) the area of the lawn excluding the area of the flower bed.
(iv) the circumference of the flower bed.

Solution:
(i) Length of the lawn = 10 m
Breadth of the lawn = 5 m
Area of the lawn = l × b
= 10 m × 5 m = 50 m2
(ii) Area of the circular flower bed = πr2
$$=\frac{22}{7} \times 2 \times 2=\frac{88}{7} \mathrm{m}^{2}=12.57 \mathrm{m}^{2}$$
(iii) Area of the lawn excluding the area of the flower bed

Ex 11.4 Class 7 Maths Question 10.
In the following figures, find the area of the shaded portion.

Solution:
(i) Area of the rectangle = l × b
= 18 cm × (6 cm + 4 cm)
= 18 cm × 10 cm = 180 cm2
Area of right triangle
$$=\frac{1}{2} \times b \times h=\frac{1}{2} \times 6 \times 10=30 \mathrm{cm}^{2}$$
Area of right ∆BCE = $$\frac{1}{2}$$ × b × h
= $$\frac{1}{2}$$ × 8 × 10 =40 cm2
Area of the two right triangles
= 30 cm2 + 40 cm2 = 70 cm2
= 180 cm2 – 70 cm2 = 110 cm2
(ii) Area of the square PQRS = (Side)2
= (20)2 = 400 cm2

Area of the three triangles
= 50 cm2 + 100 cm2 + 100 cm2 = 250 cm2
= 400 cm2 – 250 cm2 = 150 cm2

Ex 11.4 Class 7 Maths Question 11.
Find the area of the quadrilateral ABCD. Here, AC = 22 cm, BM = 3 cm, DN = 3 cm, and BM ⊥ AC, DN ⊥ AC.

Solution:


text:
# Math Help - Find Delta

1. ## Find Delta

Hi guys, I really need help.
I'm stucking at this question, any suggestion? Can you explain how to solve it step by step?

f(x)= 1/(x-1)
Find the largest δ such that if 0 < |x – 2| < δ then |f(x) – 1| < 0.01

2. ## Re: Find Delta

Originally Posted by dangbeau
Hi guys, I really need help.
I'm stucking at this question, any suggestion? Can you explain how to solve it step by step?
f(x)= 1/(x-1)
Find the largest δ such that if 0 < |x – 2| < δ then |f(x) – 1| < 0.01
This is not a tutorial service, therefore do not ask for a step-by-step solution.
You can pay a tutor for that. This is free.

Note that $|f(x)-1|=\left|\frac{x-2}{x-1}\right|$.

If $0<\delta<1~\&~|x-2|<\delta$ then $-\delta or $1-\delta or $\left|\frac{x-2}{x-1}\right|=\frac{|x-2|}{|x-1|}< \frac{\delta}{1-\delta}$

3. ## Re: Find Delta

Yay I got it now
you are my lifesaver, thank you so muchhhh


text:
# Working Seminar on Complex Geometry – Mori’s existence theorem on rational curves

Dr. Fei Ye (HKU)


text:
# 尊龙官网app

•    |
•    |
•    |
• English

The Brunn-Minkowski theory for C-coconvex sets

In this talk, we will discuss the $L_p$ Brunn-Minkowski theory associated with the $C$-coconvex sets for $p\in [0,1)$ and the $L_p$ Minkowski problem associated with the $C$-coconvex sets for $p\in R$. In particular, we will talk about our solution to an open problem raised by Schneider in [Schneider, Adv. Math., 332 (2018), pp. 199-219]. This talk is based on a joint work with Jing Yang and Deping Ye.


text:
# American Institute of Mathematical Sciences

2016, 13(1): 101-118. doi: 10.3934/mbe.2016.13.101

## The global stability of coexisting equilibria for three models of mutualism

 1 Department of Mathematics, Technical University of Iaşi, Bd. Copou 11, 700506 Iaşi, Romania 2 Department of Financial Mathematics, Jiangsu University, ZhenJiang, Jiangsu, 212013, China, China

Received  January 2015 Revised  June 2015 Published  October 2015

We analyze the dynamics of three models of mutualism, establishing the global stability of coexisting equilibria by means of Lyapunov's second method. This further establishes the usefulness of certain Lyapunov functionals of an abstract nature introduced in an earlier paper. As a consequence, it is seen that the use of higher order self-limiting terms cures the shortcomings of Lotka-Volterra mutualisms, preventing unbounded growth and promoting global stability.
Citation: Paul Georgescu, Hong Zhang, Daniel Maxin. The global stability of coexisting equilibria for three models of mutualism. Mathematical Biosciences & Engineering, 2016, 13 (1) : 101-118. doi: 10.3934/mbe.2016.13.101
##### References:
 [1] N. Apreutesei, G. Dimitriu and R. Strugariu, An optimal control problem for a two-prey and one-predator model with diffusion,, Comput. Math. Appl., 67 (2014), 2127. doi: 10.1016/j.camwa.2014.02.020. [2] J. L. Bronstein, U. Dieckmann and R. Ferrière, Coevolutionary dynamics and the conservation of mutualisms,, in Evolutionary Conservation Biology (eds. R. Ferrière, (2004), 305. doi: 10.1017/CBO9780511542022.022. [3] A. E. Douglas, The Symbiotic Habit,, Princeton University Press, (2010). [4] P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (2006), 337. doi: 10.1137/060654876. [5] P. Georgescu, Y.-H. Hsieh and H. Zhang, A Lyapunov functional for a stage-structured predator-prey model with nonlinear predation rate,, Nonlinear Anal.: Real World Appl., 11 (2010), 3653. doi: 10.1016/j.nonrwa.2010.01.012. [6] P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse,, Appl. Math. Comput., 219 (2013), 8496. doi: 10.1016/j.amc.2013.02.044. [7] P. Georgescu and H. Zhang, Lyapunov functionals for two-species mutualisms,, Appl. Math. Comput., 226 (2014), 754. doi: 10.1016/j.amc.2013.10.061. [8] B. S. Goh, Stability in models of mutualism,, Am. Nat., 113 (1979), 261. doi: 10.1086/283384. [9] W. G. Graves, B. Peckham and J. Pastor, A bifurcation analysis of a differential equations model for mutualism,, Bull. Math. Biol., 68 (2006), 1851. doi: 10.1007/s11538-006-9070-3. [10] G. W. Harrison, Global stability of predator-prey interactions,, J. Math. Biol., 8 (1979), 159. doi: 10.1007/BF00279719. [11] Y.-H. Hsieh, Richards model: A simple procedure for real-time prediction of outbreak severity,, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, 11 (2009), 216. [12] J. N. Holland and J.L. Bronstein, Mutualism,, in Population Dynamics, (2008), 2485. [13] J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology, 91 (2010), 1286. doi: 10.1890/09-1163.1. [14] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75. [15] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615. doi: 10.1007/s11538-005-9037-9. [16] A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Math. Med. Biol., 26 (2009), 309. doi: 10.1093/imammb/dqp009. [17] R. M. May, Models of two interacting populations,, in Theoretical Ecology: Principles and Application (ed. R. M. May), (1976), 78. [18] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. [19] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility,, Math. Biosci. Eng., 10 (2013), 369. doi: 10.3934/mbe.2013.10.369. [20] T. M. Palmer and A. K. Brody, Mutualism as reciprocal exploitation: African plant-ants defend foliar but not reproductive structures,, Ecology, 88 (2007), 3004. doi: 10.1890/07-0133.1. [21] L. V. Pienaar and K. J. Turnbull, The Chapman-Richards generalization of von Bertalanffy's growth model for basal area growth and yield in even-aged stands,, Forest Science, 19 (1973), 2. [22] F. J. Richards, A flexible growth function for empirical use,, J. Exp. Bot., 10 (1959), 290. doi: 10.1093/jxb/10.2.290. [23] J. Vandermeer and D. Boucher, Varieties of mutualistic interaction in population models,, J. Theor. Biol., 74 (1978), 549. [24] C. Vargas-De-León, Lyapunov functions for two-species cooperative systems,, Appl. Math. Comput., 219 (2012), 2493. doi: 10.1016/j.amc.2012.08.084. [25] C. Vargas-De-León, On the global stability of infectious diseases models with relapse,, Abstraction & Application, 9 (2013), 50. [26] C. Vargas-De-León and G. Gómez-Alcaraz, Global stability in some ecological models of commensalism between two species,, Biomatemática, 23 (2013), 139. [27] C. Wolin and L. Lawlor, Models of facultative mutualism: Density effects,, Am. Nat., 124 (1984), 843. doi: 10.1086/284320.

show all references

##### References:
 [1] N. Apreutesei, G. Dimitriu and R. Strugariu, An optimal control problem for a two-prey and one-predator model with diffusion,, Comput. Math. Appl., 67 (2014), 2127. doi: 10.1016/j.camwa.2014.02.020. [2] J. L. Bronstein, U. Dieckmann and R. Ferrière, Coevolutionary dynamics and the conservation of mutualisms,, in Evolutionary Conservation Biology (eds. R. Ferrière, (2004), 305. doi: 10.1017/CBO9780511542022.022. [3] A. E. Douglas, The Symbiotic Habit,, Princeton University Press, (2010). [4] P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (2006), 337. doi: 10.1137/060654876. [5] P. Georgescu, Y.-H. Hsieh and H. Zhang, A Lyapunov functional for a stage-structured predator-prey model with nonlinear predation rate,, Nonlinear Anal.: Real World Appl., 11 (2010), 3653. doi: 10.1016/j.nonrwa.2010.01.012. [6] P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse,, Appl. Math. Comput., 219 (2013), 8496. doi: 10.1016/j.amc.2013.02.044. [7] P. Georgescu and H. Zhang, Lyapunov functionals for two-species mutualisms,, Appl. Math. Comput., 226 (2014), 754. doi: 10.1016/j.amc.2013.10.061. [8] B. S. Goh, Stability in models of mutualism,, Am. Nat., 113 (1979), 261. doi: 10.1086/283384. [9] W. G. Graves, B. Peckham and J. Pastor, A bifurcation analysis of a differential equations model for mutualism,, Bull. Math. Biol., 68 (2006), 1851. doi: 10.1007/s11538-006-9070-3. [10] G. W. Harrison, Global stability of predator-prey interactions,, J. Math. Biol., 8 (1979), 159. doi: 10.1007/BF00279719. [11] Y.-H. Hsieh, Richards model: A simple procedure for real-time prediction of outbreak severity,, in Modeling and Dynamics of Infectious Diseases (eds. Z. Ma, 11 (2009), 216. [12] J. N. Holland and J.L. Bronstein, Mutualism,, in Population Dynamics, (2008), 2485. [13] J. N. Holland and D. L. DeAngelis, A consumer-resource approach to the density-dependent population dynamics of mutualism,, Ecology, 91 (2010), 1286. doi: 10.1890/09-1163.1. [14] A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75. doi: 10.1093/imammb/21.2.75. [15] A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615. doi: 10.1007/s11538-005-9037-9. [16] A. Korobeinikov, Stability of ecosystem: Global properties of a general predator-prey model,, Math. Med. Biol., 26 (2009), 309. doi: 10.1093/imammb/dqp009. [17] R. M. May, Models of two interacting populations,, in Theoretical Ecology: Principles and Application (ed. R. M. May), (1976), 78. [18] C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Eng., 6 (2009), 603. doi: 10.3934/mbe.2009.6.603. [19] A. V. Melnik and A. Korobeinikov, Lyapunov functions and global stability for SIR and SEIR models with age-dependent susceptibility,, Math. Biosci. Eng., 10 (2013), 369. doi: 10.3934/mbe.2013.10.369. [20] T. M. Palmer and A. K. Brody, Mutualism as reciprocal exploitation: African plant-ants defend foliar but not reproductive structures,, Ecology, 88 (2007), 3004. doi: 10.1890/07-0133.1. [21] L. V. Pienaar and K. J. Turnbull, The Chapman-Richards generalization of von Bertalanffy's growth model for basal area growth and yield in even-aged stands,, Forest Science, 19 (1973), 2. [22] F. J. Richards, A flexible growth function for empirical use,, J. Exp. Bot., 10 (1959), 290. doi: 10.1093/jxb/10.2.290. [23] J. Vandermeer and D. Boucher, Varieties of mutualistic interaction in population models,, J. Theor. Biol., 74 (1978), 549. [24] C. Vargas-De-León, Lyapunov functions for two-species cooperative systems,, Appl. Math. Comput., 219 (2012), 2493. doi: 10.1016/j.amc.2012.08.084. [25] C. Vargas-De-León, On the global stability of infectious diseases models with relapse,, Abstraction & Application, 9 (2013), 50. [26] C. Vargas-De-León and G. Gómez-Alcaraz, Global stability in some ecological models of commensalism between two species,, Biomatemática, 23 (2013), 139. [27] C. Wolin and L. Lawlor, Models of facultative mutualism: Density effects,, Am. Nat., 124 (1984), 843. doi: 10.1086/284320.
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# Geometry

posted by Mona

Points F, E, and D are on the sides line AB, line AC, and line BC, respectively, of right triangle ABC such that AFDE is a square. If AB = 12 and AC = 8, then what is AF?

1. Anonymous

Sss

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text:
Question

# In the hydrogen atom, the kinetic energy of the electron is $$3.4eV$$. The distance of that electron from the nucleus:

A
2.116˚A
B
0.529˚A
C
1.587˚A
D
21.16˚A

Solution

## The correct option is A $$2.116 \mathring A$$$$\dfrac{13.6}{n^{2}}=3.4 ev$$$$n= 2$$radius $$=0.529\times 2^{2}$$  $$A^{\circ}$$           $$=2.116A^{\circ}$$option A is correctChemistry

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Question 23670

Apr 6, 2015

This time, you add two solutions together and want the concentration of each ion present.

Because hydrochloric acid is a strong acid, it dissociates completely in aqueous solution to give

$H C {l}_{\left(a q\right)} \to {H}_{\left(a q\right)}^{+} + C {l}_{\left(a q\right)}^{-}$

As you can see, 1 mole of $H C l$ produces 1 mole of ${H}^{+}$ and 1 mole of $C {l}^{-}$ ions.

Use the molarity of the two solutions to determine how many moles of each you mix

$C = \frac{n}{V} \implies n = C \cdot V$

n_("sol 1") = "0.140 M" * 16.0 * 10^(-3)"L" = "0.00224 moles HCl"

and

n_("sol 2") = "0.750 M" * 12.0 * 10^(-3)"L" = "0.009 moles HCl"

Automatically, each solutions brings that many ${H}^{+}$ and $C {l}^{-}$ ions to the mix.

The total volume of the new solution will be

${V}_{\text{total") = V_("sol 1") + V_("sol 2}}$

V_("total") = "16.0" + 12.0 = "28.0 cm"^(3)

The total number of ${H}^{+}$ and $C {l}^{-}$ moles will be

${n}_{{H}^{+}} = {n}_{\text{sol 1") + n_("sol 2}}$
${n}_{{H}^{+}} = 0.00224 + 0.009 = {\text{0.01124 moles H}}^{+}$

and

${n}_{C {l}^{-}} = {n}_{\text{sol 1") + n_("sol 2}}$
${n}_{C {l}^{-}} = 0.00224 + 0.009 = {\text{0.01124 moles Cl}}^{-}$

The new concentrations will be

C_(H^(+)) = "0.01124 moles"/(28.0 * 10^(-3)"L") = color(green)("0.401 M")

C_(Cl^(-)) = "0.01124 moles"/(28.0 * 10^(-3)"L") = color(green)("0.401 M")#


text:
# Definition of A posteriori. Meaning of A posteriori. Synonyms of A posteriori

Here you will find one or more explanations in English for the word A posteriori. Also in the bottom left of the page several parts of wikipedia pages related to the word A posteriori and, of course, A posteriori synonyms and on the right images related to the word A posteriori.

## Definition of A posteriori

A posteriori
A posteriori A pos*teri*o"ri [L. a (ab) + posterior latter.] 1. (Logic) Characterizing that kind of reasoning which derives propositions from the observation of facts, or by generalizations from facts arrives at principles and definitions, or infers causes from effects. This is the reverse of a priori reasoning. 2. (Philos.) Applied to knowledge which is based upon or derived from facts through induction or experiment; inductive or empirical.

## Meaning of A posteriori from wikipedia

- The Latin phrases a priori ('from the earlier') and a posteriori ('from the later') are philosophical terms po****rized by Immanuel Kant's Critique of...
- philosophy, it is common to call the knowledge gained a posteriori knowledge (in contrast to a priori knowledge). Empirical evidence is information that...
- A Posteriori is a studio album by German musical project Enigma. It is Enigma's 6th album. In December 2006, the album was nominated in the Best New Age...
- if they model real world languages. For example, if a naturalistic conlang is derived a posteriori from another language (real or constructed), it should...
- A posteriori (Latin, 'from the earlier') is a term used in philosophy and epistemology. A posteriori may refer to: A Posteriori, a 2006 music album by...
- maximum a posteriori estimation then estimates θ {\displaystyle \theta } as the mode of the posterior distribution of this random variable: θ ^ M A P ( x...
- In Bayesian statistics, the posterior probability of a random event or an uncertain proposition[clarification needed] is the conditional probability that...
- A posteriori necessity is a thesis in metaphysics and the philosophy of language, that some statements of which we must acquire knowledge a posteriori...
- according to them, it is knowable a posteriori that PTI and not Q is not metaphysically possible. These a posteriori physicalists endorse some version...
- phrase a-posteriori probability is also used as an alternative to empirical probability or relative frequency. The use of the phrase "a-posteriori" is reminiscent...


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text:
# \emph{} no longer works as expected in XeLaTeX / LuaLaTeX

Compiling with either XeLaTeX or LuaLaTeX, \emph{} no longer displays text as expected, i.e. in italics. \textit{} works fine. Here is a MWE demonstrating the issue.

\documentclass{article}

\usepackage{fontspec}

\begin{document}

\emph{lorem ipsum}

\textit{lorem ipsum}

\end{document}

Here is the output I get :

I have TeXLive 2016 (MacTeX) installed. I updated it an hour ago with sudo tlmgr update --self --all.

I guess there is something wrong but what ? I'd be happy to have some help. Thank you.

• This was already raised this morning (or afternoon or whatever time it is in Australia) on the fontspec issues list github.com/wspr/fontspec/issues/254 – David Carlisle Jan 23 '17 at 12:15
• it actually seems to be fontspec at fault here rather that lualatex/xelatex as if you recompile your document without the fontspec package it works as expected – ArTourter Jan 23 '17 at 12:15
• @DavidCarlisle thank you for the link, I didn't know... Yes @ArTourter you're right, but I need to be able to use fontspec anyway :( – pictho Jan 23 '17 at 12:17
• @pictho no there was no reason for you to know, I get mailed if an issue is posted there so easier for me to know:-) I just meant that (a) it's not your fault and (b) Will has already been told:-) – David Carlisle Jan 23 '17 at 12:20
• for now I would just put \let\emph\textit in your document so you can carry on. and expect a fontspec update to come soon with a fix. – David Carlisle Jan 23 '17 at 12:21


text:
naginterfaces.library.specfun.struve_h0¶

naginterfaces.library.specfun.struve_h0(x)[source]

struve_h0 returns the value of the Struve function of order , .

For full information please refer to the NAG Library document for s17ga

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/s/s17gaf.html

Parameters
xfloat

The argument of the function.

Returns
h0float

The value of the function at .

Raises
NagValueError
(errno )

is too large and the function returns zero.

Notes

struve_h0 evaluates an approximation to the Struve function of order zero, .

Please consult the NIST Digital Library of Mathematical Functions for a detailed discussion of the Struve function including special cases, transformations, relations and asymptotic approximations.

The approximation method used by this function is based on Chebyshev expansions.

References

NIST Digital Library of Mathematical Functions

MacLeod, A J, 1996, MISCFUN, a software package to compute uncommon special functions, ACM Trans. Math. Software (TOMS) (22(3)), 288–301


text:
# ALICE Theses

Posledne pridané:
2016-03-16
15:57
 Azimuthal Anisotropy of Strange and Charm Hadrons : Measured in Pb--Pb Collisions at 2.76 TeV / Perez Lara, Carlos Eugenio CERN-THESIS-2015-314 -

Fulltext
2016-03-11
11:06
 Diffraction in ALICE and trigger efficiencies / Navin, Sparsh ALICE is built to measure the properties of strongly interacting matter created in heavy-ion collisions [...] CERN-THESIS-2011-378. - 183 p.

Birmingham U. server - Fulltext - Full text
2016-03-11
11:03
 Proton-proton collisions at the Large Hadron Collider's ALICE Experiment: diffraction and high multiplicity / Matthews, Zoe Louise Diffraction in pp collisions contributes approximately 30 % of the inelastic cross section [...] CERN-THESIS-2011-379. - 220 p.

Birmingham U. server - Fulltext - Full text
2016-03-11
10:59
 $J/\psi$ production in proton-proton collisions at ALICE LHC / Kour, Ravjeet A Large Ion Collider Experiment (ALICE) studies the strong interaction part (Quantum Chromo Dynamics) of the Standard Model at the CERN Large Hadron Collider [...] CERN-THESIS-2011-380. - 169 p.

Birmingham U. server - Fulltext - Full text
2016-03-09
16:16
 Measurement of $\Xi^{-}$ yield at the NA57 experiment and a Monte Carlo study of the $\rho^{0}$ resonance at the ALICE experiment / Platt, Richard J CERN-THESIS-2008-245. - 181 p.

Birmingham U. server - Full text - Full text
2016-01-29
11:31
 Anisotropic flow and flow fluctuations at the large hadron collider / Zhou, You One of the fundamental questions in the phenomenology of Quantum Chromodynamics (QCD) is what the properties of matter are at the extreme densities and temperatures where quarks and gluons are in a new state of matter, the so-called Quark Gluon Plasma (QGP) [...] CERN-THESIS-2016-005. - 143 p.

1874/325588
Full text - Full text - CERN library copies
2016-01-15
15:09
 Multiplicity dependence of two-particle correlation in $\sqrt{s}$ = 7 TeV pp collisions at LHC-ALICE experiment / Bhom, Jihyun Early stage of universe or inside of neutron stars are supposed to be Quark Gluon Plasma (QGP) state [...] CERN-THESIS-2015-267 - 187 p.

Fulltext - Full text
2016-01-06
14:54
 Ausrichtung des ALICE Übergangsstrahlungsdetektors sowie Zweiteilchenintensitätsinterferometrie identischer Pionen aus p+p Kollisionen bei LHC Energien von 900 GeV und 7 TeV / Huber, Sebastian This PhD thesis deals with results from the Large Hadron Collider (LHC), which provided first data in September 2009 [...] CERN-THESIS-2011-372. - 193 p.

Darmstadt, Tech. U. server - Fulltext - Full text
2016-01-06
14:45
 Production of light flavor hadrons and anti-nuclei at the LHC / Kalweit, Alexander With the recording of the first collisions of the Large Hadron Collider (LHC) in November 2009, a new era in the domain of high energy and relativistic heavy-ion physics has started [...] CERN-THESIS-2012-424. - 121 p.

Darmstadt, Tech. U. server - Fulltext - Full text
2016-01-06
14:40
 Dielectron production in proton-proton collisions with ALICE / Koehler, Markus K Ultrarelativistic hadron collisions, such as delivered since a couple of years at the Large Hadron Collider (LHC), provide new insights into the properties of strongly interacting matter at high temperatures and densities, which is expected to have existed a few of a millionth seconds after the big [...] CERN-THESIS-2015-259. - 109 p.

Darmstadt, Tech. U. server - Fulltext - Full text


text:
# Infected Mushroom

Whoops! Maybe you were looking for sell out?
For those without comedic tastes, the so-called experts at Wikipedia have an article about Infected Mushroom.
“I may have been born blind, but whenever I hear Amit Duvdevani sing, I wish that I had been born deaf instead.”
~ Stevie Wonder on Amit Duvdevani
“I heard mushrooms go well with cheese.”
~ Oscar Wilde on infected mushrooms
“You heard wrong.”
~ Mario on on the above
Erez Aisen

Infected Mushroom is an Israeli "psy"trance duo and an illegal drug in 26 countries. Formed by Erez Aisen and Amit Duvdevani in the the city of Haifa, located in the southern border of Lebanon. The duo has garnered a large international fanbase of 16 years old trance kiddies. Infected Mushroom are known for their consistent sonic reverse evolution exemplified by the subsequent albums, the stellar The Catering (1999), The classic Testicle Mushrooms (myth or menace?) (2000), The clubby B.N.P. Emo On Phire (2001), The mostly crappy Perverting Vegeta-Aryans (2003), the totally stupid The Stupid Vizier (2004), the boring-ass Bitches are Malicious (2007) and the extremely racist Legs of the Black Swimmer (2009) The decline of Infected Mushroom is blamed partly on copious consumption of cocaine from the buttocks of Duvdevani and partly on Aisen's child pornography addiction and the apparent need to get onto MTV.

## The Good Albums and the Nature of Seduction to Cheese grated items

1999's The Catering was the breakthrough debut album for Infected Mushroom. The album featured a dark, rhythmic atmosphere exemplified in the song "Psycho", which gathered much acclaim and popularity among nerds, hippies, psytrancers and other cult members. This album was one of the first albums to bring Israeli psytrance to more 'mainstream' audiences that included up to 30 people at its height, and contributed to the gradually growing popularity of the crappy genre of "psytrance" world-wide. It also had an excellent track which sampled the fascinating art-house intellectual thriller Independence Day. The following year (2000) the duo released the second best album, also being one of the most well known albums in the genre today, Testicle Mushroom (2000). The album showcased a track often cited as their most popular track, 'Bust A Nut'. This album is commonly referred to as "the last good album".

Amit Duvdevani

Professors Oskar von Beneckendorff III und Shneeddle von Hindenburg proved this theory of Inverse Inspiration in the summer of 2003 with quantum mathematics and a goat.

• $Psy + MTV = BIG-ego$
• $\Rightarrow a \sim \infty$ for finite xtc
• $\Rightarrow a \sim \infty$ for finite Porn
• $\Rightarrow L = \textrm{Cheese Factor 11...}$

It is the hope of psytrancers the world over that Infected Mushroom will kick their cocaine habit and spend less time on porn so that they can return to the glory days of time past, however Erez has stated in a recent radio interview that “we are past psytrance, we did that when we where kids.” And so the psytrance world will have to listen to talented producers instead.

It is believed that Duvdevani enjoys singing in Spanish so much is because he has nothing else to do while Aisen is on the keyboards.


text:
# For a frequency modulated signal, the modulation index is doubled. The average power of the modulated signal is

This question was previously asked in
TANGEDCO AE EC 2015 Official Paper
View all TANGEDCO Assistant Engineer Papers >
2. doubled
3. unaltered
4. none of the above

Option 3 : unaltered
Free
CT 1: Determinants and Matrices
2660
10 Questions 10 Marks 12 Mins

## Detailed Solution

Analysis:

The amplitude Ac is constant in a phase-modulated or a frequency modulated signal. RF power does not depend upon the modulation index.

A general expression for a phase or frequency modulated signal is:

$$\phi(t)=A_ccos[ω_ct+g(k_k,m(t))]$$

m(t) = the modulating signal

ωc = Carrier frequency

kk becomes kc for FM and kp for PM.

The average power (Pavg) is given by:

$$P_{avg}=\frac{A_c^2}{2}$$ (Always)

We observe that the transmitted power is independent of the modulation index in the of FM.

Transmitted power in AM is given by:

$${P_t} = {P_c}\left( {1 + \frac{{{\mu ^2}}}{2}} \right)$$

Pc = Carrier Power

∴ We conclude that If the modulation index increases, transmitted power increases in the of AM and is unchanged for FM.


text:
# Revisiting the positive DC corona discharge theory: Beyond Peek's and Townsend's law

Abstract : The classical positive Corona Discharge (CD) theory in cylindrical axisymmetric configuration is revisited in order to find analytically the influence of gas properties and thermodynamic conditions on the corona current. The matched asymptotic expansion of Durbin \& Turyn of a simplified but self-consitent problem is performed and explicit analytical solutions are derived. The mathematical derivation permits to express a new positive DC corona current-voltage charachteristic, either chosing dimensionless or dimensional formulation. In dimensional variables the current voltage law and the corona inception voltage explicitly depends on electrodes size and on physical gas properties such as ionization and photoionization parameters. The analytical predictions are successfully confronted with experiments and with Peek's and Townsend's laws. An analytical expression of the corona inception voltage $\varphi_{on}$ is proposed, which depends on known values of the physical parameters without adjustable parameters. As a proof of consistency, the classical Townsend current-voltage law $I=C\varphi(\varphi-\varphi_{on})$ is retrieved by linearizing the non-dimensional analytical solution. A brief parametric study showcases the interest of this analytical current model especially for exploring small corona wires or considering various thermodynamic conditions.
Keywords :
Document type :
Journal articles
Complete list of metadatas

Cited literature [29 references]

https://hal.archives-ouvertes.fr/hal-01806950
Contributor : Open Archive Toulouse Archive Ouverte (oatao) <>
Submitted on : Monday, June 4, 2018 - 11:51:56 AM
Last modification on : Friday, October 11, 2019 - 8:22:31 PM
Long-term archiving on : Wednesday, September 5, 2018 - 12:55:22 PM

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monrolin_20116.pdf
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### Citation

Nicolas Monrolin, Olivier Praud, Franck Plouraboué. Revisiting the positive DC corona discharge theory: Beyond Peek's and Townsend's law. Physics of Plasmas, American Institute of Physics, 2018, vol. 25 (n° 6), pp. 06350/13-06350/14. ⟨10.1063/1.5031780⟩. ⟨hal-01806950⟩

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text:
Lemma 68.16.8. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Assume that for every coherent $\mathcal{O}_ X$-module $\mathcal{F}$ we have $H^1(X, \mathcal{F}) = 0$. Then $X$ is an affine scheme.

Proof. The assumption implies that $H^1(X, \mathcal{F}) = 0$ for every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ by Lemmas 68.15.1 and 68.5.1. Then $X$ is affine by Proposition 68.16.7. $\square$

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text:
# Latin square

A Latin square of $n$ is an $n\times n$ array such that each column and each row are made with the same $n$ symbols, using every one exactly once .

Examples.

 $\left(\begin{array}[]{cccc}a&b&c&d\\ c&d&a&b\\ d&c&b&a\\ b&a&d&c\end{array}\right)\qquad\left(\begin{array}[]{cccc}1&2&3&4\\ 4&3&2&1\\ 2&1&4&3\\ 3&4&1&2\end{array}\right)$
Title Latin square LatinSquare 2013-03-22 12:14:34 2013-03-22 12:14:34 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 05B15