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# Is the empty set a subset of itself?

Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty set, have infitinite emptiness e.g. do all sets including the empty set contain infinitely many empty sets?

• Yes, but not a proper subset. – Berci Mar 19 '13 at 11:34
• You refer to the empty set, suggesting that you already know that there is a unique set that is empty, so what do you mean by "contain infinitely many empty sets"? – Trevor Wilson Mar 19 '13 at 18:55
• possible duplicate of Empty set does not belong to empty set – Marc van Leeuwen Mar 20 '13 at 10:26
• Oops, I thought this was a duplicate, but I did not look closely enough. Unfortunately it seems I cannot un-vote to close :-( – Marc van Leeuwen Mar 20 '13 at 10:29
• @NickRosencrantz Regarding your last comment, $\emptyset \notin \emptyset$, because for all $x$ we have $x \notin \emptyset$ by the definition of $\emptyset$. – Trevor Wilson Mar 20 '13 at 19:05

There is only one empty set. It is a subset of every set, including itself. Each set only includes it once as a subset, not an infinite number of times.

• I guess it is a stupid question, but what is a set which includes any set more than 1 time ? – Dominic Michaelis Mar 20 '13 at 10:27
• @DominicMichaelis: A multiset can include multiple copies of elements. Not a stupid question at all. – Ross Millikan Mar 20 '13 at 13:11
• @DominicMichaelis That is a good question. Thank you for asking. Never heard of that before. – BCLC Jul 29 '14 at 23:35

Let $A$ and $B$ be sets. If every element $a\in A$ is also an element of $B$, then $A\subseteq B$.

Flip that around and you get

If $A\not\subseteq B$, then there exists some element $x\in A$ such that $x\notin B$.

If $A$ is the empty set, there are no $x$s in $A$, so in particular there are no $x$s in $A$ that are not in $B$. Thus $A\not\subseteq B$ can't be true. Furthermore, note that we haven't used any property of $B$ in the previous line, so this applies to every set $B$, including $B=\emptyset$.

(From a wider standpoint, you can think of the empty set as the set for which $x\in \emptyset\implies P$ is true for every statement $P$. For example, every $x$ in the empty set is orange; also, every $x$ in the emptyset is not orange. There is no contradiction in either of these statements because there are no $x$'s which could provide counterexamples.)

The empty set is subset of the empty set, as every element of the empty set is an element of the empty set. But $0$ is not in the empty set.

$A \subseteq B$ when $x\in A \implies x\in B$. As $x\in A \iff x\in A$ we see that $A \subseteq A$ is always true, when $A$ is a set.

A value is a value not a set, sometimes $0$ is defined as the empty set but then $0$ is the empty set and not the number.

This happens for example in category theory, as you are only interested in abstract sets, and all sets of the same cardinality are in a sense the same, you just title finite sets by their cardinality.

Any set is a subset of itself. Let be $$A=\{x\,\vert\,\varphi(x)\}.$$ As $$\varphi(x)\implies \varphi(x),$$ we have $$A\subseteq A.$$

Because the reflexivity of $\subseteq$, for all $A$ set $A \subseteq A$ is true. For $A = \emptyset$ we have that $\emptyset \subseteq \emptyset$, so the empty set is a subset of itself.

The intersection of two sets is a subset of each of the original sets. So if $\phi$ is empty set and A is any set then $\phi\cap A$ is $\phi$ which means $\phi$ is subset of A(which is any set) and $\phi$ is a subset of $\phi$ which implies empty set is subset of itself.

By the definition of subset this question means: Is every element of the empty set a member of the empty set? The answer must be yes, since the empty set doesn't contain any elements.

An empty subset can be selected from any set, including the empty set itself, using selection criteria that cannot be satisfied, e.g. selecting those elements $x$ such that $x\neq x$.

See my formal proof (in DC Proof format) at: http://dcproof.com/ExistenseOfNullSet.htm

There is only one empty set. If $\phi_1$ and $\phi_2$ are empty sets, then $\ = \phi_2$.

See my formal proof (in DC Proof format) at: http://dcproof.com/UniquenessOfNullSet.htm

Edit: For any set $X$ we can select an empty subset $S$ such that:

$\forall a:[a\in S\iff a\in X\land a\neq a]$

or

$\forall a:[a\in S\iff a\in X\land a\notin X]$


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Mean Reversion Strategies: Introduction, Trading, Strategies and More

Almost every trading strategy in the market tries to exploit two prevailing phenomena: momentum and mean reversion. In this article, we will talk about the mean reversion and mean reversion strategies for trading.

What is Mean Reversion?

In finance, mean reversion theory states that security prices and economic indicators such as interest rates will tend to revert to the historical mean prices.

For example, many investors had bought Infosys stock when the price crashed after the resignation of CEO, Mr. Vishal Sikka because they expect Infosys stock to trade higher in the future when the management issues are solved.

The mean reversion principles can be applied to fundamental factors such as buying a stock with low PE and expecting the PE to rise to historical average PE or industry PE.

A value investor commonly follows this approach to buy the stock for long-term investments. Similarly, these principles can be applied using technical indicators to create a short-term mean reversion trading strategies.

Mean Reversion Strategies

Statistical arbitrage strategies use mean reversion principle to take advantage of the price inefficiencies between a group of securities. It is one of the popular types of trading strategies in quantitative trading. There are various types of statistical arbitrage trading strategies:

In directional trading, each instrument’s signal is independent of the other’s instruments.

For example, over the past one month, the price of crude oil has increased by $2. One day, without any significant news or factor, crude oil price changes by$20. By the mean reversion principle, you can expect the price of crude oil to fall in the coming days such that the average change in the price of crude oil remains the same and you can exploit this opportunity.

In this type of trading strategies trading signals depend on the two or more cointegrated instruments. Pairs trading is one of the most famous examples of such a type of trading style.

Since we are using two cointegrated instruments to make a trade, therefore the name pairs trading. But it’s not always a pair, it could also be triplets or could be more than that. If you found five cointegrated stocks, you can make a portfolio of that and do the trading.

Let’s say you have a pair of instruments with similar fundamentals, belonging to the same sectors and similar economic links. For example, stocks like Google and Microsoft or Facebook and Twitter.

Since they have similar fundamentals, you expect both the stocks to behave similarly. You also expect the ratio or spread of such stocks to remain constant with time. However, due to a temporary change in the demand and supply and other factors, there might be a divergence in the spread between the pairs.

In such scenarios, one security outperforms the other. According to the mean reversion principle, you expect this divergence to revert to normal with time. In such scenarios, when there is a temporal divergence, you can perform the pairs trade. That’s buying the underperforming security and selling the outperforming security.

Correlation vs Cointegration

Most of the people are confused between correlation and cointegration, and they often think the same. But that’s not the case.

When two price series move in the same or opposite direction, then there is some correlation between the price series. If one price series moves in either up or down direction and other price series also move in the same direction, there is a positive correlation between them.

While one price series moves in the upward or downward direction, the other moves opposite to that, then both series are negatively correlated.

Cointegration is a statistical property of two or more price series that indicates if a linear combination of the series is stationary, then both series are cointegrated with each other.

For example, if the linear combination of two stocks is stationary, both stocks are cointegrated with each other.  A price series is said to be stationary if it’s mean and variance is constant over time.

Statistical test for cointegration: Augmented dickey fuller or ADF test is one of the statistical tests for cointegration. In Python, this can be easily done through the statsmodels library.

As explained in the principle of pairs trading, the ratio or spread between stocks must converge to the mean over time for pairs trading to work. That is, both stocks must be cointegrated with each other.

Just looking at the correlation between the stocks might give you spurious results because the prices of the two stocks may keep on increasing without ever mean-reverting.  There is a myth that goes around that the two correlated instruments must be cointegrated and vice versa.

Let’s define two price series such that they are correlated but not cointegrated.

By looking at both price series, they seem correlated with each other but what about cointegration?

For two price series to be cointegrated, the spread/ratio between two price series should be stationary. Let’s take the difference between the above series.

The spread is trending. It is clear from the results that both price series are correlated but not cointegrated. So we can conclude that the prices can be correlated but not cointegrated.

Now consider another scenario where prices are cointegrated but not correlated.

It seems evident that both price series are not correlated, as there seems no relation between them.

The spread hovers between -10 to +10. So, we can say that the two price series can be cointegrated without being correlated.

Before implementing pairs trading, we need to check whether the price series are cointegrated or not so that their spread/ratio reverts to the mean.

Selection of Pairs

How to select stock pairs?

Suppose you have a large universe of stocks. The first step is to segregate stocks on the basis of market capitalization, sector, daily traded volume etc. After segregating, you can check for a correlation between the securities in each group.

The correlation helps to filter the number of pairs to a more manageable set. Once you get the securities in a small set of groups, you can check for cointegrated pairs within the group and select the cointegrated pairs.

How to select forex pairs?

The basic idea behind selecting pairs in forex is similar to that of stocks. We need to find countries that have similar economic fundamentals.

Some of the suitable candidates are

• EUR/USD and CHF/USD,
• USD/KRW (US dollar/South Korean won) and  USD/HKD (US dollar/Hong Kong dollar).

These pairs belong to the same economic zone as Euro and Swiss Franc belong to the Eurozone. Doing pairs trading in the currency market has its advantage, such as liquidity in currencies is higher, thus lowering transaction costs.

How to select pairs in the futures market?

In the futures market, there are not many good pairs, even with similar economic exposure. This may be due to differences in demand and supply. Therefore, in futures, you can’t merely rely on economic exposure for choosing pairs.

Steps to implement pairs trading in Python

1. Select forex pairs
2. Calculate ratio and check for cointegration
3. Create feature to generate trading signals
4. Define entry and exit point
5. Calculate pnl

Step 1 - Select forex pairs

As discussed above, we select pairs having similar economic fundamentals. Euro and Swiss Franc both belong to a similar economic zone, so we start with this pair, but we can’t just only rely on the fundamental factors.

Step 2 - Calculate ratio and check for cointegration

We calculate the ratio between the currency pairs. If the ratio is stationary, then we can say that the currency pairs are cointegrated. We are using an ADF test to check whether the ratio is stationary or not. One thing to remember while using the ADF test is that the test result changes by changing the order of ratio.

The p-value from the ADF test for the EURUSD/CHFUSD ratio is less than the 0.05 significance level. Therefore we can say that the ratio is stationary. But the p-value for the ratio of CHFUSD/EURUSD is greater than 0.05 significance level. So, this ratio is not stationary.

We will use the EURUSD/CHFUSD ratio for pairs trading. Let’s take a look at the cointegrated ratio to make sure this makes sense.

Step 3 - Create a feature to generate trading signals

The absolute ratio isn’t very useful in statistical terms. It can be observed through the above ratio graph that it does look like it moves around a stable mean. We need to normalize the ratio. This is done using z-score.

Z score is defined as:

Z Score = (Value — Mean) / Standard

Now it’s easier to observe that the ratio moves around the mean, but sometimes it divergences from the mean, which we can take advantage of.

Step 4 - Define entry and exit points

If the z score crosses below the lower threshold, then we buy and exit the position when it reaches the mean. If the z score crosses above the upper threshold, then we sell and exit the position when it reaches the mean.

Further Improvements

The strategy can be further optimised using different values of the lookback period of moving average and standard deviation.

The feature that we used to define the entry and exit position is z-score. You can use a different variant of z-score such as:

z score: (15-day moving average — 50-day moving average) / 50-day standard deviation

Another approach is to use the Bollinger Band for signal generation.

Stop loss

You can set the stop loss above and below your threshold level.

For example, in the above strategy, the set threshold was plus/minus 2 standard deviations. You can set the stop loss at plus/minus 3 standard deviations. When the ratio/spread crosses that threshold, you can exit the position. Another approach is to exit the position when a prefixed loss is hit.

Holding Period

You can keep the position for a day, week or month and exit after that. How long you can keep the position can be found using a concept known as the half-life. It tells how long it would take the time series to revert to the mean. It gives an idea of the expected holding period for a particular trade.

When you exit the position based on time, you will wait for the price to revert to mean to initiate new positions.

Check out this webinar to learn about mean reversion strategies.

Conclusion

Mean reversion strategy will only work on a range-bound or stationary stock price series. The strategy will not work on a stock price series that trends higher or trends lower for a long time.

To check if a price series is stationary or not a statistical technique called an augmented Dickey-Fuller (ADF) test is the most commonly used.

Unfortunately, most stock price series are not stationary. But fortunately, we can create a portfolio of stocks such that the market value of that portfolio is stationary and implement a profitable mean reversion trading strategy on that portfolio. This forms the basis on which pairs trading, triplet trading.

Learn Mean Reversion Strategies in detail in the Quantra course.

Disclaimer: The views, opinions, and information provided within this guest post are those of the author alone and do not represent those of QuantInsti®. The accuracy, completeness, and validity of any statements made or the links shared within this article are not guaranteed. We accept no liability for any errors, omissions or representations. Any liability with regards to infringement of intellectual property rights remains with them.

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## how to find center of gravity of a vibrating screen

how to find center of gravity of a vibrating screen SCREEN CAPACITY CALCULATION VIBFEM Formula: A = B * S * D * V * H * T * K * Y * P * O * W * F “A”, the calculated capacity per square foot of screen area in tons per hour.how to find center of gravity of a vibrating screen,how to find center of gravity of a vibrating screen /images/crusher/233.jpg SCREEN CAPACITY CALCULATION VIBFEM Formula: A = B * S * D * V * H * T * K * Y * P * O * W * F “A”, the calculated capacity per square foot of screen area in tons per hour. get price /images/crusher/115.jpg Screening Theory and PracHow To Find Center Of Gravity Of A Vibrating Screen,how to find center of gravity of a vibrating screen; . Isolate a vibrating screen which weighs a total of 16400 pounds. Get Price. SCREENING ON A SCREEN WITH A VIBRATING SIEVE. used a risk of torsional vibrations of the frame around the center of gravity may appear. . calculate such sieve length that guarantees the desired efficiencyDynamics and screening characteristics of a vibrating,,01/07/2009· The equations of motion for any point on the vibrating screen are xD = 0.00035Dy sin t - (0.00368 + 0.00038Dy ) cos t yD = (-0.00382 - 0.00035 Dx )sin t + 0.00038Dx cos t Eq.(8) shows that the center of gravity traces an approximate circle and that the amplitude in the horizontal and vertical directions is between 3.5 mm and 5 mm. Fig. 3 shows how the center of gravity moves in threeSCREEN CAPACITY CALCULATION - VIBFEM,screen under the force of gravity. When viewing a screen opening from above, the more horizontal the screen deck lays, the larger the opening appears. This difference in effective screen opening between flat and incline gives flat screens greater capacity for the same wire opening size. Figure 2 is a demonstration of the effect.Vibrating screen | SKF,distance between the centres of gravity of the counterweights and shaft axis: R = 80 mm; distance between the bearings: 3 m; lubrication method: grease; operating temperature of the bearings: T = 75 °C (165 °F) environment: the screen may be located outdoors, in harsh, dusty and humid conditions; required SKF rating life: 20 000 hCalculating the G-force of a Vibrating Screen,How do you calculate the accelerating / g-force of a screen deck? To do this, you will need three pieces of information: Number of poles of the motor Stroke of the screen (measured) Frequency of the current to the motor From the number of poles of the motor, the moto's shaft rotational speed can be calculated, using the following formula: w = (2 X 60 X f) / n where: w = shaft rotational speed,how to find center of gravity of a vibrating screen,how to find center of gravity of a vibrating screen /images/crusher/233.jpg SCREEN CAPACITY CALCULATION VIBFEM Formula: A = B * S * D * V * H * T * K * Y * P * O * W * F “A”, the calculated capacity per square foot of screen area in tons per hour. get price /images/crusher/266.jpg Screening Theory and PracDynamics and screening characteristics of a vibrating,,01/07/2009· The equations of motion for any point on the vibrating screen are xD = 0.00035Dy sin t - (0.00368 + 0.00038Dy ) cos t yD = (-0.00382 - 0.00035 Dx )sin t + 0.00038Dx cos t Eq.(8) shows that the center of gravity traces an approximate circle and that the amplitude in the horizontal and vertical directions is between 3.5 mm and 5 mm. Fig. 3 shows how the center of gravity moves in threeVibrating Screen Capacity Calculations – MEKA,Unidirectional Method. The material velocity of linearly vibrated screens can be obtained from the diagrams contained in appendix A-1. i = angle of incidence of the line of force in relation to the horizontal plane. e = eccentricity (mm) App = peak-to-peak (mm) = 2 x e.

## 5 Ways to Calculate Center of Gravity - wikiHow

04/06/2020· This will give you the distance from the datum to the center of gravity of the object. To do this, simply divide 1180 ft. x lb. by 130 lbs. 1180 ft. x lb. ÷ 130 lbs = 9.08A dynamic optimization theoretical method for heavy,Horizontal coordinate of the center of gravity $$y$$ (m): Vertical coordinate of the center of gravity $$\theta$$ ($$^\circ$$): Pitching angle of the vibrating screen $$A$$ (m): Horizontal distance between the center of gravity and the unload side spring support $$B$$ (m): Horizontal distance between the center of gravity and the load side spring $$A'$$ (m):Characteristics and efficiency of a new vibrating screen,,01/10/2013· When the screen surface is vibrating, the following force balance equations hold: (5) m a sin β − m g cos α = N, m a cos β + m g sin α = F, where m is the mass of a particle (kg); a is the acceleration of the particle, given by a = l ψ ¨ = l (2 L (h 2 − h 1) ω 2 [(h 2 − h 1) 2 (1 − cos ω t) (cos ω t + (1 / 2)) 2 + 4 L 2 cos ω t]) / ([4 L 2 + (h 2 − h 1) 2 (1 − cos ω t) 2] 2) (m/s 2); in which ψ ¨ is the angularCenter of Gravity Definition, Equation and Calculation,,Calculation of Center of Gravity. Calculate the center of gravity of following diagram: Step 1: Draw Free Body Diagram of the System. Step 2: Find Weight Distance Moment with Reference to Datum. Datum is the arbitrary starting point on the end of the slab. Lets suppose, we choose point A as datum and find(PDF) A new method for gravity separation: Vibrating,In this study, the Central Composite Design (CCD) was applied by using Design-Expert (version 6.0.5) for modeling and optimizing the effect of operating variables on the performance of gravity,Center of gravity - A basic explanation of balancing weight,Artwork: How to find an object's center of gravity: 1) Hang the object from a point on its edge and it will rotate until its center of gravity is directly under that point. Hang a plumbline (a weight on a string) from the same point. Draw a line parallel to the stringFind Center of a Blob (Centroid) Using OpenCV,To find the center of the blob, we will perform the following steps:-1. Convert the Image to grayscale. 2. Perform Binarization on the Image. 3. Find the center of the image after calculating the moments.Vibrating Screen Capacity Calculations – MEKA,Unidirectional Method. The material velocity of linearly vibrated screens can be obtained from the diagrams contained in appendix A-1. i = angle of incidence of the line of force in relation to the horizontal plane. e = eccentricity (mm) App = peak-to-peak (mm) = 2 x e.BRECON smart vibration technology: Know How Center,Each weight at rotation creates a centrifugal force vector whose direction is determined by the centre of motion (centre of rotor) and the centre of gravity. Since the total centrifugal force is determined by the resulting centrifugal force vectors, counterbalancingA dynamic optimization theoretical method for heavy,Horizontal coordinate of the center of gravity $$y$$ (m): Vertical coordinate of the center of gravity $$\theta$$ ($$^\circ$$): Pitching angle of the vibrating screen $$A$$ (m): Horizontal distance between the center of gravity and the unload side spring support $$B$$ (m): Horizontal distance between the center of gravity and the load side spring $$A'$$ (m):

## Characteristics and efficiency of a new vibrating screen,

01/10/2013· When the screen surface is vibrating, the following force balance equations hold: (5) m a sin β − m g cos α = N, m a cos β + m g sin α = F, where m is the mass of a particle (kg); a is the acceleration of the particle, given by a = l ψ ¨ = l (2 L (h 2 − h 1) ω 2 [(h 2 − h 1) 2 (1 − cos ω t) (cos ω t + (1 / 2)) 2 + 4 L 2 cos ω t]) / ([4 L 2 + (h 2 − h 1) 2 (1 − cos ω t) 2] 2) (m/s 2); in which ψ ¨ is the angular accelerationProblem and Solution of Linear Vibrating Screen - Xinhai,In brief, the vibration source of linear vibrating screen is vibration motor. The materials on the screen can be upsprung and moved by an exciting force of a certain magnitude in a fixed direction. The tilt angle of linear vibrating screen is a certain value. Some vibrating screen can adjusted while some can’t be adjusted. Commonly adjusting the direction of exciting force is the solution under this circumstance.Center of Gravity Definition, Equation and Calculation,,Calculation of Center of Gravity. Calculate the center of gravity of following diagram: Step 1: Draw Free Body Diagram of the System. Step 2: Find Weight Distance Moment with Reference to Datum. Datum is the arbitrary starting point on the end of the slab. Lets suppose, we choose point A as datum and findCenter of gravity - A basic explanation of balancing weight,Artwork: How to find an object's center of gravity: 1) Hang the object from a point on its edge and it will rotate until its center of gravity is directly under that point. Hang a plumbline (a weight on a string) from the same point. Draw a line parallel to the stringDewatering screen - IFE Aufbereitungstechnik GmbH,These screens dewater material and sludge and produce bulk material which can be transported with standard means. They are also used to recover solid particles from liquids, muds or sludges. The robust screen frame is manufactured in either a welded or a bolted and glued design, depending on the application. The angle of vibratory agitation is optimized to reduce moisture of the solids to minimum values. The unit is supported by a screen deck inclination of +3° (rising). All screenFind Center of a Blob (Centroid) Using OpenCV,To find the center of the blob, we will perform the following steps:-1. Convert the Image to grayscale. 2. Perform Binarization on the Image. 3. Find the center of the image after calculating the moments.Make a Center Finder : 9 Steps (with Pictures) - Instructables,Use the center finder to mark the center from a couple of directions. You will need to turn the table tennis ball over to mark the extension of the axis. Do that by making a circle on a piece of paper, the diameter of which is the same as the PVC collar. Mark the center of the circle. Align the center,,,


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Tag Info

My answer to How does the Hubble parameter change with the age of the universe? (which is itself adapted from Equation for Hubble Value as a function of time) explains how to calculate the scale factor. In fact we calculate the time as a function of the scale factor rather than the other way around. The equation we use is:  t(a) = \frac{1}{H_0}\int_0^a ...


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10,837 Pages

## REPLACED

This blog has been replaced. A new J Function blog should be referred to instead. The blog can be found here.

## The (old) J Function

The J function is a reasonably fast growing function. This blog will explain the structure of the function and will give various calculated values. The J Function will then be used by a brand new version of the Alpha function. Click here for more information about the Alpha Function.

## Introduction

For an introduction, it will be useful to refer to my other blogs on the Strong D function. The following notation rules will also help to understand the behaviour of Strong D Functions and the basic structure of the J Function.

Notation:

$$D(m_{[x]}) = D(m_1,m_2,...,m_x)$$

$$D(m_{[x]},n_{[y]}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)$$

$$D(1,0_{[y]}) = D(D(1_{[y]})_{[y]})$$

$$D(m,n_{[y]}) = D(m-1,D(m,n_{[y-1]},n_y-1)_{[y]})$$

Basic J Function Structure:

The J Function consists of a series of 3-tuple parameters. This simple example uses one 3-tuple only:

$$J(k,m,n) = D(k,0_{[m]},n)$$

Note that the equivalent Strong D Function will have m+2 parameters in total, all parameters except for the first and last will be zero.

Basic J Function Examples:

$$J(2,0,3) = D(2,3)$$

$$J(3,1,1) = D(3,0,1)$$

$$J(4,2,0) = D(4,0,0,0)$$

$$J(n,n-1,n) = D(n,0_{[n-1]},n) >> f_{\omega^2}(n)$$

## Definition

The J Function consists of a series of 3-tuple parameters. Each 3-tuple is simply 3 grouped input parameters. 3-tuples are delimited from each other with a semi-colon, instead of a comma. Here is a general example:

$$J(1,2,3;4,5,6;7,8,9;...;...;...)$$

J Function behaviour follows similar rules to the Strong D Function. The rules are as follows:

$$J(k,m,n) = D(k,0_{[m]},n)$$

$$J(k,m,n) = J(k-1,m,J(k,m,n-1))$$

$$J(k,m,0) >> J(k-1,m,J(k-1,m,k))$$    Explanation will be provided

$$J(1,m,0) >> J(J(1,m-1,2^m-1),m-1,J(1,m-1,2^m-1))$$    Explanation will be provided

Note that:

$$J(0,m,n) = J(0,0,n) = J(n) = D(n)$$ Because of the Leading Zero rule: L1

and

$$J(0,0,0) = J(0) = D(0) = 1$$

When more than one 3-tuple is used we get more complex behaviour. The general rule is:

$$J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))$$

But it will be easier to follow the examples when f = g = 0 < h:

$$J(h;k,m,n) = J(h;k-1,m,J(h;k,m,n-1))$$

and then these rules follow from the definitions above.

$$J(h;k,m,0) >> J(h;k-1,m,J(h;k-1,m,k))$$

and

$$J(h;1,m,0) >> J(h;J(h;1,m-1,2^m-1),m-1,J(h;1,m-1,2^m-1))$$

Some simpler evaluations are:

$$J(h;0,0,n) = J(h;n) = J(h;J(h;n-1))$$

and

$$J(h;0,0,0) = J(h-1;J(h-1;h,h,h),J(h-1;h,h,h),J(h-1;h,h,h))$$

and

$$J(1;0,0,0) = J(0;J(0;1,1,1),J(0;1,1,1),J(0;1,1,1))$$

$$>> f_{\omega^2}(J(1,1,1)) = f_{\omega^2}(D(1,0,1)) >> f_{\omega^2}(f_3(6))$$

## Calculated Examples up to J(1;0,0,n)

$$J(1;0,0,1) = J(1;1) = J(1;J(1;0)) >> f_{\omega^2}(f_{\omega^2}(f_3(6))) >> f_{\omega^2}^2(f_3(6))$$

and

$$J(1;0,0,n-1) >> f_{\omega^2}^{n}(f_3(6)) >> f_{\omega^2+1}(n)$$ when $$n < f_3(6)$$

or

$$J(1;0,0,n) >> f_{\omega^2+1}(n)$$ when $$n < f_{\omega^2}(f_3(6))$$

## Calculated Examples up to J(1;n,0,n)

$$J(1;1,0,0) = J(1;0,0,J(1;0,0,1)) >> f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))$$

and

$$J(1;1,0,1) = J(1;0,0,J(1;1,0,0)) >> f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6)))$$

and

$$J(1;1,0,n-1) >> f_{\omega^2+1}^n(n) = f_{\omega^2+2}(n)$$ when $$n < f_{\omega^2}^2(f_3(6))$$

or

$$J(1;1,0,n) >> f_{\omega^2+2}(n)$$ when $$n < f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))$$

then

$$J(1;2,0,0) = J(1;1,0,J(1;1,0,2)) >> f_{\omega^2+2}(f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6))))$$

and

$$J(1;2,0,n) >> f_{\omega^2+3}(n)$$

and

$$J(1;3,0,n) >> f_{\omega^2+4}(n)$$

then

$$J(1;n,0,n) >> f_{\omega^2+\omega}(n)$$

## Calculated Examples up to J(1;n,n,n)

$$J(1;1,1,0) = J(1;J(1;1,0,1),0,J(1;1,0,1)) >> f_{\omega^2+\omega}(J(1;1,0,1))$$

and

$$J(1;1,1,1) = J(1;1,1,J(1;1,1,0)) >> f_{\omega^2+\omega}^2(J(1;1,0,1))$$

then

$$J(1;1,1,n) >> f_{\omega^2+\omega}^n(n) = f_{\omega^2+\omega+1}(n)$$

and

$$J(1;n,1,n) >> f_{\omega^2+\omega+n}(n)$$ = f_{\omega^2+\omega.2}(n)\)

then

$$J(1;1,2,n) >> f_{\omega^2+\omega.2}^n(n) = f_{\omega^2+\omega.2+1}(n)$$

and

$$J(1;n,2,n) >> f_{\omega^2+\omega.2+n}(n)$$ = f_{\omega^2+\omega.3}(n)\)

then

$$J(1;n,3,n) >> f_{\omega^2+\omega.4}(n)$$

and

$$J(1;n,n,n) >> f_{\omega^2+\omega.n}(n) >> f_{\omega^2.2}(n)$$

## Calculated Examples up to J(n;0,0,0)

$$J(2;0,0,0) = J(1;J(1;2,2,2),J(1;2,2,2),J(1;2,2,2)) >> f_{\omega^2.2}(J(1;2,2,2))$$

and

$$J(2;0,0,1) = J(2;0,0,J(2;0,0,0)) >> f_{\omega^2.2}^2(J(1;2,2,2))$$

then

$$J(2;0,0,n) >> f_{\omega^2.2+2}(n)$$

and

$$J(2;n,0,n) >> f_{\omega^2.2+\omega}(n)$$

and

$$J(2;n,n,n) >> f_{\omega^2.3}(n)$$

or

$$J(3;0,0,0) >> f_{\omega^2.3}(n)$$

then

$$J(n-1;n,n,n) >> f_{\omega^2.n}(n)$$

or

$$J(n;0,0,0) >> f_{\omega^3}(n)$$

## Calculated Examples up to J(1,0,n;n,n,n)

$$J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))$$

then

$$J(1,0,0;k,m,n) = J(1,0,0;J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1))$$

and

$$J(1,0,0;0,0,0) = J(0,0,1;Z,Z,Z) = J(1;Z,Z,Z)$$

where $$Z = J(0,0,1;D(1,0),D(1,0),D(1,0))$$ Explanation will be provided

or $$Z = J(1;D(1,0),D(1,0),D(1,0)) = J(1;3,3,3)$$

then

$$J(1,0,0;0,0,0) = J(1;J(1;3,3,3),J(1;3,3,3),J(1;3,3,3))$$

$$>> f_{\omega^2.2}(J(1;3,3,3))$$

and

$$J(1,0,0;0,0,1) >> f_{\omega^2.2}^2(J(1;3,3,3))$$

$$J(1,0,0;0,0,n) >> f_{\omega^2.2+1}(n)$$

$$J(1,0,0;n,0,n) >> f_{\omega^2+\omega}(n)$$

$$J(1,0,0;n,n,n) >> f_{\omega^3}(n)$$

then

$$J(1,0,1;0,0,0) >> J(1,0,0;J(1,0,0;4,4,4),J(1,0,0;4,4,4),J(1,0,0;4,4,4))$$ Explanation will be provided

$$>> f_{\omega^3}(f_{\omega^3}(4))$$

and

$$J(1,0,1;0,0,n) >> f_{\omega^3+1}(n)$$

$$J(1,0,1;n,0,n) >> f_{\omega^3+\omega}(n)$$

$$J(1,0,1;n,n,n) >> f_{\omega^3+\omega^2}(n)$$

then

$$J(1,0,2;0,0,n) >> f_{\omega^3+\omega^2+1}(n)$$

$$J(1,0,2;n,0,n) >> f_{\omega^3+\omega^2+\omega}(n)$$

$$J(1,0,2;n,n,n) >> f_{\omega^3+\omega^2.2}(n)$$

then

$$J(1,0,3;n,n,n) >> f_{\omega^3+\omega^2.3}(n)$$

and

$$J(1,0,n;n,n,n) >> f_{\omega^3.2}(n)$$

## Calculated Examples up to J(2,0,n;n,n,n)

Without proof, calculations should continue with:

$$J(2,0,0;0,0,n) >> f_{\omega^3.2+1}(n)$$

$$J(2,0,0;n,0,n) >> f_{\omega^3.2+\omega}(n)$$

$$J(2,0,0;n,n,n) >> f_{\omega^3.2+\omega^2}(n)$$

and

$$J(2,0,1;n,n,n) >> f_{\omega^3.2+\omega^2.2}(n)$$

$$J(2,0,2;n,n,n) >> f_{\omega^3.2+\omega^2.3}(n)$$

$$J(2,0,h-1;n,n,n) >> f_{\omega^3.2+\omega^2.h}(n)$$

and

$$J(2,0,n;n,n,n) >> f_{\omega^3.3}(n)$$

## Calculated Examples up to J(n,n,n;n,n,n)

Without proof, calculations should continue with:

$$J(3,0,n;n,n,n) >> f_{\omega^3.4}(n)$$

$$J(f,0,n;n,n,n) >> f_{\omega^3.f}(n)$$

and

$$J(n,0,n;n,n,n) >> f_{\omega^4}(n)$$

then

$$J(1,1,0;0,0,n) >> f_{\omega^4+1}(n)$$

$$J(1,1,0;n,0,n) >> f_{\omega^4+\omega}(n)$$

$$J(1,1,0;n,n,n) >> f_{\omega^4+\omega^2}(n)$$

$$J(1,1,n;n,n,n) >> f_{\omega^4+\omega^3}(n)$$

$$J(n,1,n;n,n,n) >> f_{\omega^4.2}(n)$$

$$J(n,g,n;n,n,n) >> f_{\omega^4.g}(n)$$

and

$$J(n,n,n;n,n,n) >> f_{\omega^5}(n)$$

## Growth Rate of the J Function

$$J(3;0,0,0) >> f_{\omega^3}(3) = f_{\omega^{\omega}}(3)$$

$$J(4,0,4;4,4,4) >> f_{\omega^4}(n) = f_{\omega^{\omega}}(4)$$

$$J(5,5,5;5,5,5) >> f_{\omega^5}(n) = f_{\omega^{\omega}}(5)$$

The growth rate of the J Function therefore appears to be:

$$J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega}}(n)$$ with n+1 parameters in the J function

Each group of 3 parameters of value n represents one 3-tuple in the J Function. If we use the notation N to represent the 3-tuple n,n,n, then the growth rate of the J function may reach:

$$J(N_{[(n+1)/3]}) >> f_{\omega^{\omega}}(n)$$ with n+1 parameters in the J function

$$J(N_{[n]}) = J(N;N;...;N) = J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega^2}}(n)$$

## Some calculations for n=3

$$J(4,0,1) >> f_{\omega}(3)$$

$$J(3,1,1) >> f_{\omega.2}(3)$$

$$J(3,2,1) >> f_{\omega^2}(3)$$

$$J(3;0,0,0) >> f_{\omega^{\omega}}(3)$$

$$J(3;3,3,3;3,3,3) >> f_{\omega^{\omega.2}}(3)$$

If we use the notation T to represent the 3-tuple 3,3,3, then we can continue:

$$J(3;T;T;T) = J(3;3,3,3;3,3,3;3,3,3) >> f_{\omega^{\omega^2}}(3)$$

Look forward to any comments and questions. If anybody is interested, the J Function was named by my wife. The full name is the Juki Function.

Cheers B1mb0w.

The J Function


text:
# 2.6 Problem-solving basics for one-dimensional kinematics  (Page 2/2)

 Page 2 / 2

## Step 5

Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units . This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made. However, be warned that correct units do not guarantee that the numerical part of the answer is also correct.

## Step 6

Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important—the goal of physics is to accurately describe nature. To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units. Your judgment will improve as you solve more and more physics problems, and it will become possible for you to make finer and finer judgments regarding whether nature is adequately described by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to mechanically solve a problem.

When solving problems, we often perform these steps in different order, and we also tend to do several steps simultaneously. There is no rigid procedure that will work every time. Creativity and insight grow with experience, and the basics of problem solving become almost automatic. One way to get practice is to work out the text’s examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and progressing to the more difficult. Once you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text.

## Unreasonable results

Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at $0\text{.}{\text{40 m/s}}^{2}$ for 100 s, his final speed will be 40 m/s (about 150 km/h)—clearly unreasonable because the time of 100 s is an unreasonable premise. The physics is correct in a sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving—it also builds intuition in judging whether nature is being accurately described.

Use the following strategies to determine whether an answer is reasonable and, if it is not, to determine what is the cause.

## Step 1

Solve the problem using strategies as outlined and in the format followed in the worked examples in the text . In the example given in the preceding paragraph, you would identify the givens as the acceleration and time and use the equation below to find the unknown final velocity. That is,

$v={v}_{0}+\text{at}=0+\left(0\text{.}\text{40}\phantom{\rule{0.25em}{0ex}}{\text{m/s}}^{2}\right)\left(\text{100}\phantom{\rule{0.25em}{0ex}}\text{s}\right)=\text{40}\phantom{\rule{0.25em}{0ex}}\text{m/s}.$

## Step 2

Check to see if the answer is reasonable . Is it too large or too small, or does it have the wrong sign, improper units, …? In this case, you may need to convert meters per second into a more familiar unit, such as miles per hour.

$\left(\frac{\text{40 m}}{s}\right)\left(\frac{\text{3.28 ft}}{m}\right)\left(\frac{\text{1 mi}}{\text{5280 ft}}\right)\left(\frac{\text{60 s}}{\text{min}}\right)\left(\frac{\text{60 min}}{\text{1 h}}\right)=89 mph$

This velocity is about four times greater than a person can run—so it is too large.

## Step 3

If the answer is unreasonable, look for what specifically could cause the identified difficulty . In the example of the runner, there are only two assumptions that are suspect. The acceleration could be too great or the time too long. First look at the acceleration and think about what the number means. If someone accelerates at $0\text{.}{\text{40 m/s}}^{2}$ , their velocity is increasing by 0.4 m/s each second. Does this seem reasonable? If so, the time must be too long. It is not possible for someone to accelerate at a constant rate of $0\text{.}{\text{40 m/s}}^{2}$ for 100 s (almost two minutes).

## Section summary

• The six basic problem solving steps for physics are:

Step 1 . Examine the situation to determine which physical principles are involved.

Step 2 . Make a list of what is given or can be inferred from the problem as stated (identify the knowns).

Step 3 . Identify exactly what needs to be determined in the problem (identify the unknowns).

Step 4 . Find an equation or set of equations that can help you solve the problem.

Step 5 . Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units.

Step 6 . Check the answer to see if it is reasonable: Does it make sense?

## Conceptual questions

What information do you need in order to choose which equation or equations to use to solve a problem? Explain.

What is the last thing you should do when solving a problem? Explain.

what is ohm's law
states that electric current in a given metallic conductor is directly proportional to the potential difference applied between its end, provided that the temperature of the conductor and other physical factors such as length and cross-sectional area remains constant. mathematically V=IR
ANIEFIOK
A body travelling at a velocity of 30ms^-1 in a straight line is brought to rest by application of brakes. if it covers a distance of 100m during this period, find the retardation.
what's acceleration
The change in position of an object with respect to time
Mfizi
Acceleration is velocity all over time
Pamilerin
hi
Stephen
It's not It's the change of velocity relative to time
Laura
Velocity is the change of position relative to time
Laura
acceleration it is the rate of change in velocity with time
Stephen
acceleration is change in velocity per rate of time
Noara
what is ohm's law
Stephen
Ohm's law is related to resistance by which volatge is the multiplication of current and resistance ( U=RI)
Laura
how i don understand
how do I access the Multiple Choice Questions? the button never works and the essay one doesn't either
How do you determine the magnitude of force
mass × acceleration OR Work done ÷ distance
Seema
Which eye defect is corrected by a lens having different curvatures in two perpendicular directions?
acute astigmatism?
the difference between virtual work and virtual displacement
How do you calculate uncertainties
What is Elasticity
using a micro-screw gauge,the thickness of a piece of a A4 white paper is measured to be 0.5+or-0.05 mm. If the length of the A4 paper is 26+or-0.2 cm, determine the volume of the A4 paper in: a). Cubic centimeters b). Cubic meters
what is module
why it is possible for an object(man) to stay on air without falling down?
its impossible, what do you mean exactly?
Ryan
Exactly
Emmanuella
it's impossible
Your
Why is it not possible to stand in air?
bikko
the air molecules are very light enough to oppose the gravitational pull of the earth on the man..... hence, freefall occurs
Arzail
because of gravitational forces
Pamilerin
this mostly occur in space
Stephen
what is physics
no life without physics ....that should tell you something
Exactly
Emmanuella
😎👍
E=MC^2
study of matter and energy and an inter-relation between them.
Minahil
that's how the mass and energy are related in stationery frame
Arzail
Ketucky tepung 10m
firdaus
Treeskin, 6m Cloud gam water 2m Cloud gam white 2m And buur
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Like dont have but have
firdaus
Two in one
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DNA card
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War right? My impesilyty again. Don't have INSURAN for me
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firdaus
PUSH FIRST. HAHAHAAHA
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firdaus
Kinetic energy is the energy due to montion of waves,electrons,atoms, molecule,substances an object s.
Emmanuella
Opjective 0
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firdaus
SOME N
firdaus
10.000m permonth. U use momentom with me
firdaus
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Hilal
plz anyone can tell what is meteor and why meteor fall in night? can meteor fall in the day
Hilal
meteor are the glowy (i.e. heated when the enter into our atmosphere) parts of meteoroids. now, meteoroids are the debris resulting from the collision of asteroids or comets. yes, it occurs in daytime too, but due to the daylight, we cant observe it as clearly as in night
Arzail
thank's
Hilal
hello guys
Waka
wich method we use to find the potential on a grounded sphere
Noman
hello
Pamilerin
Physics is the science that studies everything around us from the smallest things like quarks to the biggest things like galaxies. It's simply everything.
Laura
Good day everyone
Divine
It talks mainly about matter with related topics such as forces energy gravity and time. It's amazing
Laura
Hi
Alpha
Physics generally is the study of everything around us.
Steven
physics is the branch of sceince
shafiu
physics is the branch of sceince that deal with motion
shafiu
physics is the branch of sceince that deal with motion &energy
shafiu


text:
# nvidia.dali.fn.bbox_paste¶

nvidia.dali.fn.bbox_paste(*inputs, **kwargs)

Transforms bounding boxes so that the boxes remain in the same place in the image after the image is pasted on a larger canvas.

Corner coordinates are transformed according to the following formula:

(x', y') = (x/ratio + paste_x', y/ratio + paste_y')

Box sizes (if xywh is used) are transformed according to the following formula:

(w', h') = (w/ratio, h/ratio)

Where:

paste_x' = paste_x * (ratio - 1)/ratio
paste_y' = paste_y * (ratio - 1)/ratio

The paste coordinates are normalized so that (0,0) aligns the image to top-left of the canvas and (1,1) aligns it to bottom-right.

Supported backends
• ‘cpu’

Parameters

input (TensorList) – Input to the operator.

Keyword Arguments
• ratio (float or TensorList of float) – Ratio of the canvas size to the input size; the value must be at least 1.

• bytes_per_sample_hint (int or list of int, optional, default = [0]) –

Output size hint, in bytes per sample.

If specified, the operator’s outputs residing in GPU or page-locked host memory will be preallocated to accommodate a batch of samples of this size.

• ltrb (bool, optional, default = False) – True for ltrb or False for xywh.

• paste_x (float or TensorList of float, optional, default = 0.5) – Horizontal position of the paste in image coordinates (0.0 - 1.0).

• paste_y (float or TensorList of float, optional, default = 0.5) – Vertical position of the paste in image coordinates (0.0 - 1.0).

• preserve (bool, optional, default = False) – Prevents the operator from being removed from the graph even if its outputs are not used.

• seed (int, optional, default = -1) –

Random seed.

If not provided, it will be populated based on the global seed of the pipeline.


text:
# Is ETC2 Texture Compression available in OpenGL ES 2.0?

I have an ES 2.0 app and would like to use ETC2 texture compression if it is available (i.e. I am running on a device that supports ES 3.0 standard). Is it possible to use ETC2 in ES 1.1/2.0, and if so how can I query if a device supports it? For most forms of texture compression, there is an extension I can query for, but there is no extension for ETC2 since it is built in.

• No, you can't use etc2 in ES2, not as far as i know. Probably on Android, but i still don't think so. – Raxvan Apr 9 '14 at 15:13

There are a bunch of GL extensions related to ETC2 support:

OES_compressed_ETC2_RGB8_texture
OES_compressed_ETC2_sRGB8_texture
OES_compressed_ETC2_punchthroughA_RGBA8_texture
OES_compressed_ETC2_punchthroughA_sRGB8_alpha_texture
OES_compressed_ETC2_RGBA8_texture
OES_compressed_ETC2_sRGB8_alpha8_texture

The ES3 compatibility extension aggregates those (and a few others). You can query for these extensions to check support, presumably.

• Ok I think I misunderstood the extensions. When I open the "OpenGL Extensions Viewer" on android (on the newer Nexus 7), I see "Texture GL_COMPRESSED_RGBA8_ETC2_EAC". So i guess i should look for that when querying the extensions. – default Apr 9 '14 at 17:21
• Well, except that string is not in the extensions, nor are any of the strings you mentioned (on this qualcomm device). Perhaps if those strings aren't in the extensions, then i can only use ETC2 in an ES 3.0 context? Not sure. – default Apr 9 '14 at 17:51
• If the string isn't in the set of extension strings, it's not supported. – Josh Apr 9 '14 at 18:13


text:
# The Ideal Rocket Equation

## June 19, 2019

SpaceX's Falcon Heavy exhaust plumes [Arabsat-6A Mission].

The first gunpowder-powered rockets were invented centuries ago, and since then rocket propulsion technology has evolved to the point where we now use electric propulsion to drive our space probes to the edge of our solar system. The ideal rocket equation and Newton’s Laws of Motion, among other things (aerodynamics, etc.), dictate the performance of a rocket. The ideal rocket equation or the Tsiolkovsky rocket equation is fundamentally derived from Newton’s laws of motion and the conservation of momentum. Before we derive the ideal rocket equation ourselves, let’s revisit Newton’s Laws of Motion and the law of conservation of momentum.

• Newton’s 1st Law: An object at rest will stay at rest, or an object in motion will stay in motion unless acted upon by an external force.

• Newton’s 2nd Law: The vector sum of the forces $$\vec{F}$$ acting on an object is equal to the mass $$m$$ of the object multiplied by its acceleration vector $$\vec{a}$$.

• Newton’s 3rd Law: For every action there is an equal and opposite reaction.

Momentum is the quantity of motion of a moving body, measured as a product of its mass and velocity. The Conservation of momentum is a general law of physics according to which the quantity called momentum that characterizes motion never changes in an isolated collection of objects; that is, the total momentum of a system remains constant. Momentum is a vector, involving both the direction and the magnitude of motion so the momenta of objects going in opposite directions can cancel to yield an overall sum of zero.

Before launch, the total momentum of a rocket and its fuel is zero. During launch, the downward momentum of the expanding exhaust gases just equals in magnitude the upward momentum of the rising rocket, so that the total momentum of the system remains constant—in this case, at zero value.

The essence of engines that we use, whether flying through the atmosphere or flying through space, is we carry a certain amount of material inside the spacecraft and we push that material out at a certain speed, and that makes us go forward. The fluid that we push to move forward is called the working fluid of the engine, and the speed at which we push out material is known as exit velocity or exhaust velocity. Rockets have a limited size. We can only load a certain amount of propellant on board the rocket. So there’s a real question of how much performance can we get out of a given amount of propellant.

# Deriving The Ideal Rocket Equation

For simplicity, assuming we’re in free space (where we don’t have to worry about atmospheric drag and gravity), there’s no force acting on the rocket except the force of the rocket engine itself. The initial mass of a rocket ($$\mathrm{m}_{\mathrm{i}}$$) is equal to the sum of the mass of the payload ($$\mathrm{m}_{\mathrm{PL}}$$), the mass of the structure ($$\mathrm{m}_{\mathrm{struct}}$$) and the mass of the propellant ($$\mathrm{m}_{\mathrm{prop}}$$).

$\mathrm{m}_{\mathrm{i}} \ = \ \mathrm{m}_{\mathrm{PL}} \ + \ \mathrm{m}_{\mathrm{struct}} \ + \ \mathrm{m}_{\mathrm{ prop }}$

To calculate the thrust of the engine, we multiply the mass flow rate at the exhaust ($$\mathrm{\dot{m}}$$), and the velocity at the exit or the exhaust velocity ($$\mathrm{V}_{\mathrm{e}}$$). By conservation of momentum, we’ll get the forward motion of the rocket.

$\mathrm{F} = \mathrm{\dot{m}} \mathrm{V}_{\mathrm{e}}$

Where the mass flow rate ($$\mathrm{\dot{m}}$$), is a time derivative of mass:

$\mathrm{\dot{m}} = \frac{\mathrm{dm}}{\mathrm{dt}}$

Once the rocket has expended it’s propellant, the final mass of the rocket is equal to the initial mass of the rocket minus the mass of the propellant.

$\mathrm{m}_{\mathrm{f}} \ = \ \mathrm{m}_{\mathrm{i}} \ - \ \mathrm{m}_{\mathrm{prop}}$ $\mathrm{m}_{\mathrm{f}} \ = \ \mathrm{m}_{\mathrm{PL}} \ + \ \mathrm{m}_{\mathrm{struct}}$

Now, we can derive the ideal rocket equation.

$\mathrm{F} = \mathrm{-V}_{\mathrm{e}} \frac{\mathrm{dm}}{\mathrm{dt}}$

Note: $$\mathrm{V}_{\mathrm{e}}$$ is negative relative to the the rocket’s motion. By conservation of momentum:

$\mathrm{F \ = \ ma \ = \ m \frac{dV}{d t} \ = \ -V_{e} \frac{dm}{dt}}$ $\mathrm{m \frac{dV}{dt} \ = \ -V_{e} \frac{dm}{dt}}$ $\mathrm{m \ dV = -V_{e} \ dm}$ $\mathrm{dV=-V_{e} \frac{d m}{m}}$ $\int_{\mathrm{V}_{\mathrm{i}}}^{\mathrm{V}_{\mathrm{f}}} \mathrm{dV}=-\mathrm{V}_{\mathrm{e}} \int_{\mathrm{m}_{\mathrm{i}}}^{\mathrm{m}_{\mathrm{f}}} \frac{\mathrm{dm}}{\mathrm{m}}$ $\mathrm{\underbrace{V_{\mathrm{f}}-\mathrm{V}_{\mathrm{i}}}_{\Delta \mathrm{V}} \ = \ -\mathrm{V}_{\mathrm{e}} \ln\frac{\mathrm{m}_{\mathrm{f}}}{\mathrm{m}_{\mathrm{i}}} \ = \ \mathrm{V}_{\mathrm{e}} \ln \frac{\mathrm{m}_{\mathrm{i}}}{\mathrm{m}_{\mathrm{f}}}}$

Final equation:

$\mathrm{\Delta V=V_{e} \ln \frac{m_{i}}{m_{f}}}$

Exponential form:

$\mathrm{\frac{\Delta V}{V_{e}}=\ln \frac{m_{i}}{m_{f}}}$ $\mathrm{\frac{m_{i}}{m_{f}}=e^{\frac{\Delta V}{V_{e}}}}$

# Implications of the Ideal Rocket Equation

When we change the rocket equation from the logarithmic form to an exponential form, we see that the ratio of the initial to the final mass of the rocket is the exponential of the change in velocity, which we get from burning the propellant, divided by the exhaust velocity.

$\mathrm{\frac{m_{i}}{m_{f}}=e^{\frac{\Delta V}{V_{e}}}}$

The performance of our rocket engine depends on the mass flow rate at the exhaust and the exhaust velocity. But what fraction of the rocket’s total mass (propellant fraction) has to be propellant in order to get us the velocity ($$\mathrm{\Delta V}$$) we need?

$\mathrm{m}_{\mathrm{prop}} \ = \ \mathrm{m}_{\mathrm{i}} \ - \ \mathrm{m}_{\mathrm{f}}$ $\mathrm{\frac{m_{i}}{m_{f}}=e^{\frac{\Delta V}{V_{e}}}}$ $\mathrm{\frac{m_{f}}{m_{i}}=e^{-\frac{\Delta V}{V_{e}}}}$ $\mathrm{m_{f}=m_{i} e^{-\frac{\Delta V}{V_{e}}}}$ $\begin{split} \therefore \mathrm{m_{prop}} \ & \mathrm{=} \ \mathrm{m_{i}} \mathrm{-} \mathrm{m_{f}} \\ & \mathrm{=} \ \mathrm{m_{i}-m_{i} e^{-\frac{\Delta V}{V_{e}}}} \\ & \mathrm{=} \ \mathrm{m_{i}}\left[\mathrm{1-e^{-\frac{\Delta V}{V_{e}}}}\right] \end{split}$

Propellant Fraction:

$\mathrm{\frac{m_{prop}}{m_{i}}=1-e^{-\frac{\Delta V}{V_{e}}}}$

Inert Fraction:

$\mathrm{\frac{m_{f}}{m_{i}}=e^{-\frac{ \Delta V}{V_{e}}}}$

Exhaust velocity is critical for performance, and chemical rockets (solid and liquid) have an exhaust velocity of about 2000 - 4500 m/s. Propulsion engineers like to talk about specific impulse ($$\mathrm{I_{sp}}$$); it’s a measure of how effectively a rocket uses propellant, or a jet engine uses fuel.

By definition, specific impulse is the total impulse (or change in momentum) delivered per unit of propellant consumed and is dimensionally equivalent to the generated thrust divided by the propellant mass flow rate or weight flow rate. If mass (kilogram, pound-mass, or slug) is used as the unit of propellant, then specific impulse has units of velocity. If weight (Newton or pound-force) is used instead, then specific impulse has units of time (seconds).

Multiplying flow rate by the standard gravity ($$\mathrm{g_{0}}$$) converts specific impulse from the mass basis to the weight basis. We can calculate the specific impulse of a rocket engine given the exhaust velocity and the acceleration of gravity at Earth.

$\mathrm{I_{sp}=\frac{V_{e}}{g_{0}}}$

Chemical rockets have a specific impulse of about 200 - 450 seconds.

# Effect of exhaust velocity on achieving Earth orbit

To get into Low Earth Orbit (LEO): $$\mathrm{\Delta V}$$ is about 9000 m/s (includes drag and gravity losses; ignores effect of Earth’s rotation).

Using a solid rocket motor, $$\mathrm{V}_{\mathrm{e}}$$ is about 2000 m/s; $$\mathrm{\frac{\Delta V}{V_{e}} \approx 4.5}$$. Thus, the rocket’s inert fraction will be:

$\mathrm{\frac{m_{f}}{m_{i}}=e^{-4.5}=0.011}$

The rocket’s propellant fraction will be:

$\mathrm{\frac{m_{prop}}{m_{i}}=1-e^{-4.5}=0.99}$

Now, if we were to use a high-performance liquid rocket engine with an exhaust velocity ($$\mathrm{V}_{\mathrm{e}}$$) of about 4000 m/s; $$\mathrm{\frac{\Delta V}{V_{e}} \approx 2.25}$$. How does this affect the amount propellant and payload we can carry?

The rocket’s inert fraction will be:

$\mathrm{\frac{m_{f}}{m_{i}}=e^{-2.25}=0.105}$

The rocket’s propellant fraction will be:

$\mathrm{\frac{m_{prop}}{m_{i}}=1-e^{-2.25}=0.89}$

Therefore, the higher the exhaust velocity, the lower the propellant fraction and higher the inert fraction. With a higher inert fraction, we are able to carry a heavier payload into orbit and keep the rocket structurally sound. This is why companies like SpaceX are constantly improving the performance of their rocket engines in order to carry heavier payloads to Low-Earth Orbit, the Moon and eventually to Mars.

SpaceX’s subscale Raptor engine has completed more than 1200 seconds of testing in less than two years.

In conclusion, the ideal rocket equation puts a limit on how much of the fraction of the mass of the rocket can be devoted to the actual structure and payload. In other words, we want to make rockets as light as possible. Splitting a rocket into sections known as stages and discarding them once the propellant has been expended is the best way to overcome the limitations put forth by the ideal rocket equation.


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# Tag Archives: Kunal Nayyar

## The Big Bang Theory of Trojan Asteroids

Season 06, Episode 03: “The Higgs Boson Observation”

In “The Higgs Boson Observation”, we learn that the subject of graduate student and Sheldon’s (Jim Parsons) new assistant, Alex Jensen (Margo Harshman), is on Trojan asteroids at the Earth-Sun L5 Lagrange point. This also happens to be Raj’s (Kunal Nayyar) area of expertise. But what exactly are Trojan asteroids and why are they of interest to astronomers and astrophysicists?

Trojans are minor planets or satellites that share an orbit with a planet but does not collide with it because it orbits around one of the two stable Lagrangian points (trojan points). The L4 and L5 lie approximately 60° ahead of and behind the larger body, respectively .

# Area of Research

These are the five Lagrangian points in a two-body system with one body far more massive than the other (e.g. the Sun and the Earth)

There are five points on or near Earth’s orbit known as Lagrange points where an asteroid will remain stationary with respect to Earth as it orbits around the Sun. These points mark the positions where the combined gravitational pull of two large bodies, such as the Earth and the Sun, provides the centripetal force needed to orbit with them.

The Lagrangian points are approximate solutions to what is known as the three-body problem, a famous problem that attempts to model the motion of three bodies as they gravitationally affect each other. In essence, this problem dates back to 1687 when Sir Isaac Newton first published his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), often referred to as simply the “Principia”. Several scientists and mathematicians have all attempted to find an analytical solution to the problem or a simple equation that would describe the motions of all three bodies.

The three-body problem is difficult to solve because the three bodies tug against each other in chaotic and unpredictable ways. Only by simulating the problem on a computer can we see the objects paths. In comparison, the quantum three-body problem is relatively easy to solve and understand. However, approximate solutions can be useful. If the third mass is taken to be small enough that it does not affect the other two masses some interesting solutions crop up.

# Celestial Mechanics

To better understand how celestial bodies move in our solar system, we turn to Newton’s law of universal gravitation. This law states that every mass in the Universe attracts every other mass with a force that is proportional to the product of their masses. If we had two masses, $$M_{1}$$ and $$M_{2}$$, then the force between them would be proportional to the product of the two masses.
$F \propto M_{1}M_{2}$
This law also states that the force decreases with the square of the distance between them.
$F \propto \frac{1}{r^{2}}$
We can combine these into one equation that will help is analyze and solve the movement of celestial bodies anywhere in the Universe.
$F=G\frac{M_{1}M_{2}}{r^{2}}$
where $$G$$ is the gravitational constant. This law was deduced by Newton from observations and was formulated in the Principia.

If we were to look at the Sun and Earth system, we would have what is known as the two-body problem. While we may think this is a simple problem but in reality it isn’t. The Earth does not revolve around the Sun as if the more massive Sun was a stationary body but rather both the Earth and Sun revolve around a common point — the center of mass or barycenter. Though makes the problem a little more difficult, as this point is very close to the center of the Sun and we can make an approximation where the Sun is in the center and doesn’t move while the Earth moves around it.

If we use this technique, we can derive an equation that describes the Earth’s motion around the Sun. This is a classic “two-body” problem and its solutions describe the familiar elliptical orbits of the planets known since the time of Kepler. Unfortunately, when we add a third body to our equations of motion, such as a spacecraft lost in space between the Earth and the sun, we can no longer find an analytical solution; the equations become unsolvable.

While the three-body problem has no analytical solution, Joseph-Louis Lagrange in 1772 showed that if we restricted the mass of the third body to be so small that it couldn’t affect the other two, there were some solutions to be found. The solutions to this “restricted three-body problem” finds that the three bodies move in unison and always maintain the same position relative to each other.

These five points, where the third body stays in the same relative point in space, are known as Lagrange points and mark the positions where the combined gravitational pull of the two large masses precisely provides the force to orbit with them. They are labelled L1, L2, L3, L4 and L5 and also known as L-points or libration points.

## The Trojan Asteroids

The L4 and L5 libration points lie on the corners of two equilateral triangles. The points are balanced because the gravitational forces between the Earth and Sun keep any object in orbital equilibrium with the rest of the system. These points are sometimes referred to as “triangular Lagrange points” or “Trojan points” and come from the Trojan asteroids at the Sun-Jupiter L4 and L5 points. These asteroids were named after characters from Homer’s Iliad. Asteroids at the L4 point leads Jupiter and is referred to as the “Greek camp” while those are the L5 are referred to as the “Trojan camp”.

The Earth-Sun system, like the Jupiter-Sun system, also uses the same terms of reference. There is one known Greek asteroid in the Earth-Sun system, 2010 TK7, first detected in October 2010 by Dr. Martin Connors using the Wide-field Infrared Survey Explorer (WISE) . The asteroid’s designation come from the provisional naming system for minor planets. This asteroid which has a diameter of about 300 meters oscillates about the L4 Lagrangian point. The region around these points is known to contain interplanetary dust. As people tend to search for asteroids at much greater elongations, very few searches have been done at these locations. Currently, there are no confirmed or suspected L5 Trojans of Earth which makes both Raj’s and Alex’s work interesting and something to talk about.

## The Big Bang Theory of Trans-Neptunian objects

Season 02, Episode 04: “The Griffin Equivalency”

Rajesh Koothrappali (Kunal Nayyar) looks pleased as he talks about his inclusion into People magazine’s “30 Under 30 to watch” list for his discovery of the fictional trans-Neptunian object, 2008 NQ17 which exists beyond the Kuiper belt

In “The Griffin Equivalency” episode of The Big Bang Theory, Raj (Kunal Nayyar) announces that he has been chosen as one of People Magazine’s “30 visionaries under 30 years to watch” list for his discovery of the trans-Neptunian object 2008 NQ17. This fictional astronomical body exists just beyond the Kuiper Belt, a region of the Solar System that extends from the orbit of Neptune (about 30 Astronomical units(AU) away), to a distance of 50AU from the Sun. (1AU is roughly the distance of the Earth from the Sun).

The distribution and classification of the trans-Neptunian Objects in our solar system

The first trans-Neptunian object discovered was Pluto in 1930 and is the second most massive known dwarf planet in the Solar System. These objects are so small and distant that it wasn’t until 1992 that the second trans-Neptunian object orbiting the Sun, (15760) 1992 QB1, was discovered. As cryptic as these names sound, i.e 2008 NQ17 and 1992 QB1, the nomenclature comes from a provisional designation in astronomy that names objects immediately following their discovery. Once a proper orbit has been calculated, a permanent designation consisting of a number-name combination is given by the Minor Planet Center. Unfortunately for Raj, so many of these minor planets are discovered that most won’t be named for their discoverers.

# Provisional designation of Minor Planets

The provisional naming system for minor planets, of which dwarf planets, asteroids, trojans, centaurs, Kuiper belt objects, and other trans-Neptunian objects are a part of, have been in place since 1925. The first element in a minor planet’s provisional designation is the year of discovery, followed by two letters and sometimes a number.

The first letter indicates the “half-month” of the object’s discovery. “A” denotes a discovery in the first half of January (between January 1st and January 15th) while “B” denotes a discovery in the second half of January (between January 16th and January 31st). The letter “I” is not used so the letters go all the way up to “Y”. The first half is always the 1st through to the 15th of the month, regardless of the numbers of days in the second “half”.

First Letter (A-G)
A B C D E F G
Jan 1 Jan 16 Feb 1 Feb 16 Mar 1 Mar 16 Apr 1

First Letter (H-O)
H J K L M N O
Apr 16 May 1 May 16 Jun 1 Jun 16 Jul 1 Jul 16

First Letter (P-V)
P Q R S T U V
Aug 1 Aug 16 Sep 1 Sep 16 Oct 1 Oct 16 Nov 1

First Letter (W-Y)
W X Y
Nov 16 Dec 1 Dec 16

The second letter and the number indicate the order or discovery within that half-month.

Second Letter (A-K)
A B C D E F G H J K
1 2 3 4 5 6 7 8 9 10

Second Letter (L-U)
L M N O P Q R S T U
11 12 13 14 15 16 17 18 19 20

Second Letter (V-Z)
V W X Y Z
21 22 23 24 25

So the 14th minor planet discovered in the first half of June 2013 will get the provisional planetary designation 2013 LO. As modern techniques has made it (relatively) easier to detect these objects, there can be more than 25 discoveries in a given half-month. To solve this a subscript number is added to indicate the number of times that the letters have cycled through. So the 40th minor planet to be discovered in the first half of June 2013 will gain the designation 2013 LP1.
\begin{aligned} 40 & = 25 + 15 \\ \therefore P & = 15 \end{aligned}

# Planet Bollywood

Using this information we can calculate when Raj discovered “Planet Bollywood”. It was discovered some time between July 1-15 and it was the 441st minor planet to be discovered in that half month.
\begin{aligned} Q & = 16 \\ 25 \times 17 & = 425 \end{aligned}
And $$425 + 16 = 441$$.

It must have been a particularly busy month in the Big Bang Theory Universe. A quick check at the Minor Planet Center’s Provisional Designations webpage indicates that there were 137 astronomical bodies discovered in that half-month. That would make that last trans-Neptunian object to be discovered to receive the provisional designation 2008 NM4.

Long live Planet Bollywood!


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# Brill-Noether theory over the Hurwitz space

Isabel Vogt (UW)
Tuesday, October 6, 2020 - 2:30pm
via Zoom

Let C be a curve of genus g. A fundamental problem in the theory of algebraic curves is to understand maps of C to projective space of dimension r of degree d. When the curve C is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. Despite much study over the past three decades, a similarly complete picture has proved elusive in this case. In this talk, I will discuss recent joint work with Eric Larson and Hannah Larson that completes such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. In the course of our degenerative argument, we'll exploit a close relationship with the combinatorics of the affine symmetric group.

Abstract: In this talk we'll introduce the problem of understanding the space of degree d maps from an abstract curve C of genus g to P^r, and the key player: the Brill--Noether locus W^r_d(C). After computing some examples, I'll explain a little of what is expected" and known when the curve is general.


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# Riesz Representation Theorem (Hilbert Spaces)

## Theorem

Let $H$ be a Hilbert space.

Let $L$ be a bounded linear functional on $H$.

Then there is a unique $h_0 \in H$ such that

$\forall h \in H: L h = \left\langle{h, h_0}\right\rangle$

### Corollary

The norm of $L$ satisfies:

$\left\Vert{L}\right\Vert = \left\Vert{h_0}\right\Vert$

## Proof

If $L \equiv 0$ identically, then $Lh = 0 = \left\langle{h,0}\right \rangle$, and the theorem holds.

Otherwise, set:

$M = \ker{L} = L^{-1}\left(\{{0}\}\right)$.

Then $M$ is a subspace.

Because $L$ is bounded, it is continuous.

Because $\{0\}$ is closed,the continuity of $L$ implies that $M$ is closed.

Then we can decompose $H$ as a direct sum:

$H \cong M \oplus M^\perp$

As $L \not \equiv 0$, $M^\perp \ne \left\{{0}\right\}$.

Choose a $z \in M^\perp$ with norm $1$.

By linearity of $L$, for any $h \in H$,

 $\displaystyle L\left({zLh - hLz)}\right)$ $=$ $\displaystyle LzLh - LhLz$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle 0$ $\quad$ $\quad$

So $zLh - hLz \in \ker L = M$.

Then,

 $\displaystyle Lh$ $=$ $\displaystyle Lh \left \langle { z, z }\right \rangle$ $\quad$ $\Vert z \Vert = 1$ $\quad$ $\displaystyle$ $=$ $\displaystyle \left \langle {zLh,z}\right \rangle$ $\quad$ linearity in the first argument $\quad$ $\displaystyle$ $=$ $\displaystyle \left \langle {zLh - hLz + hLz,z}\right \rangle$ $\quad$ adding and subtracting $hLz$ in the first argument $\quad$ $\displaystyle$ $=$ $\displaystyle \left \langle {zLh - hLz,z}\right \rangle + \left \langle {hLz,z}\right \rangle$ $\quad$ linearity in the first argument $\quad$ $\displaystyle$ $=$ $\displaystyle \left \langle {hLz,z}\right \rangle$ $\quad$ $zLh - hLz \in M, z \in M^\perp$ $\quad$ $\displaystyle$ $=$ $\displaystyle \left \langle {h,z (Lz)^* }\right\rangle$ $\quad$ conjugate symmetry $\quad$

Thus $Lh = \left \langle{ h,h_0 }\right \rangle$ for $h_0 = z (Lz)^*$.

To show uniqueness, assume $h_0$ and $h_1$ both satisfy the above equation for all $h \in H$:

 $\displaystyle \left \langle {h,h_0}\right \rangle$ $=$ $\displaystyle \left \langle {h,h_1}\right \rangle$ $\quad$ $\quad$ $\displaystyle \implies \ \$ $\displaystyle \left \langle {h,h_0}\right \rangle - \left \langle {h,h_1}\right \rangle$ $=$ $\displaystyle 0$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle \left \langle {h,h_0-h_1}\right \rangle$ $\quad$ additivity in the second argument $\quad$

The result follows from Setting $h = h_0 - h_1$ and invoking the positive definiteness of the inner product.

$\blacksquare$

## Source of Name

This entry was named for Frigyes Riesz.


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{PRDA} allows performing a prospective or retrospective design analysis to evaluate inferential risks (i.e., power, Type M error, and Type S error) in a study considering Pearson’s correlation between two variables or mean comparisons (one-sample, paired, two-sample, and Welch’s t-test).

For an introduction to design analysis and a general overview of the package see vignette("PRDA"). Examples for retrospective design analysis and prospective design analysis are provided in vignette("retrospective") and vignette("prospective") respectively.

## Installation

You can install the released version of PRDA from CRAN with:

install.packages("PRDA")

And the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("ClaudioZandonella/PRDA",
build_vignettes = TRUE)

## The Package

{PRDA} package can be used for Pearson’s correlation between two variables or mean comparisons (i.e., one-sample, paired, two-sample, and Welch’s t-test) considering an hypothetical value of ρ or Cohen’s d respectively. See vignette("retrospective") and vignette("prospective") to know how to set function arguments for the different effect types.

### Functions

In {PRDA} there are two main functions retrospective() and prospective().

#### • retrospective()

Given the hypothetical population effect size and the study sample size, the function retrospective() performs a retrospective design analysis. According to the defined alternative hypothesis and the significance level, the inferential risks (i.e., Power level, Type M error, and Type S error) are computed together with the critical effect value (i.e., the minimum absolute effect size value that would result significant).

Consider a study that evaluated the correlation between two variables with a sample of 30 subjects. Suppose that according to the literature the hypothesized effect is ρ = .25. To evaluate the inferential risks related to the study we use the function retrospective().

set.seed(2020) # set seed to make results reproducible

retrospective(effect_size = .25, sample_n1 = 30,
test_method = "pearson")
#>
#>  Design Analysis
#>
#> Hypothesized effect:  rho = 0.25
#>
#> Study characteristics:
#>    test_method   sample_n1   sample_n2   alternative   sig_level   df
#>    pearson       30          NULL        two_sided     0.05        28
#>
#> Inferential risks:
#>    power   typeM   typeS
#>    0.27    1.826   0.003
#>
#> Critical value(s): rho  =  ± 0.361

In this case, the statistical power is almost 30% and the associated Type M error and Type S error are respectively around 1.80 and 0.003. That means, statistical significant results are on average an overestimation of 80% of the hypothesized population effect and there is a .3% probability of obtaining a statistically significant result in the opposite direction.

To know more about function arguments and further examples see the function documentation ?retrospective and vignette("retrospective").

#### • prospective()

Given the hypothetical population effect size and the required power level, the function prospective() performs a prospective design analysis. According to the defined alternative hypothesis and the significance level, the required sample size is computed together with the associated Type M error, Type S error, and the critical effect value (i.e., the minimum absolute effect size value that would result significant).

Consider a study that will evaluate the correlation between two variables. Knowing from the literature that we expect an effect size of ρ = .25, the function prospective() can be used to compute the required sample size to obtain a power of 80%.

prospective(effect_size = .25, power = .80, test_method = "pearson",
display_message = FALSE)
#>
#>  Design Analysis
#>
#> Hypothesized effect:  rho = 0.25
#>
#> Study characteristics:
#>    test_method   sample_n1   sample_n2   alternative   sig_level   df
#>    pearson       122         NULL        two_sided     0.05        120
#>
#> Inferential risks:
#>    power   typeM   typeS
#>    0.797   1.119   0
#>
#> Critical value(s): rho  =  ± 0.178

The required sample size is n = 122, the associated Type M error is around 1.10 and the Type S error is approximately 0.

To know more about function arguments and further examples see the function documentation ?prospective and vignette("prospective").

### Hypothetical effect size

The hypothetical population effect size can be defined as a single value according to previous results in the literature or experts indications. Alternatively, {PRDA} allows users to specify a distribution of plausible values to account for their uncertainty about the hypothetical population effect size. To know how to specify the hypothetical effect size according to a distribution and an example of application see vignette("retrospective").

## Contributing to PRDA

The PRDA package is still in the early stages of its life. Thus, surely there are many bugs to fix and features to propose. Anyone is welcome to contribute to the PRDA package.

Please note that this project is released under a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.

#### Bugs and New Features

To propose a new feature or to report a bug, please open an issue on GitHub. See Community guidelines.

#### Future Plans

• Improve compute time by parallelizing the code
• Implement design analysis in the case of linear regression models

## Citation

To cite {PRDA} in publications use:

Zandonella Callegher, C., Pastore, M., Andreella, A., Vesely, A., Toffalini, E., Bertoldo, G., & Altoè G. (2020). PRDA: Prospective and Retrospective Design Analysis (Version 1.0.0). Zenodo. https://doi.org/10.5281/zenodo.4044214

A BibTeX entry for LaTeX users is

@Misc{,
author       = {Zandonella Callegher, Claudio and Pastore, Massimiliano and Andreella, Angela and
Vesely, Anna and Toffalini, Enrico and Bertoldo, Giulia and Altoè, Gianmarco},
title        = {PRDA: Prospective and Retrospective Design
Analysis},
year         = 2020,
publisher    = {Zenodo},
version      = {1.0.0},
doi          = {10.5281/zenodo.4044214},
url          = {https://doi.org/10.5281/zenodo.4044214}
}

## References

Altoè, Gianmarco, Giulia Bertoldo, Claudio Zandonella Callegher, Enrico Toffalini, Antonio Calcagnì, Livio Finos, and Massimiliano Pastore. 2020. “Enhancing Statistical Inference in Psychological Research via Prospective and Retrospective Design Analysis.” Frontiers in Psychology 10. https://doi.org/10.3389/fpsyg.2019.02893.

Bertoldo, Giulia, Claudio Zandonella Callegher, and Gianmarco Altoè. 2020. “Designing Studies and Evaluating Research Results: Type M and Type S Errors for Pearson Correlation Coefficient.” Preprint. PsyArXiv. https://doi.org/10.31234/osf.io/q9f86.

Gelman, Andrew, and John Carlin. 2014. “Beyond Power Calculations: Assessing Type S (Sign) and Type M (Magnitude) Errors.” Perspectives on Psychological Science 9 (6): 641–51. https://doi.org/10.1177/1745691614551642.


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# Intuition for Interchange Symmetry of the Riemann Tensor

Is there an intuitive/geometric picture for the interchange symmetry of the Riemann tensor? I have seen plenty of algebraic derivations, but would like to understand if the symmetry expresses something intuitive or obvious (in hindsight at least).

Here is what I have worked out so far (please let me know if I've gotten it wrong!)

I don't know if this notation is standard (I can't quite keep track of the variety of conventions) so I apologize if it's off. Work with a torsion free, metric connection, namely the Levi-Civita Connection.

Define the Reimann Curvature Tensor as the infinitesimal rotation matrix resulting from parallel transport of a vector $Z$ around a parallelogram-shaped loop defined by $X$ and $Y$. $$R(X,Y)Z = R_{\mu\nu\lambda}^\sigma X^\mu Y^\nu Z^\lambda$$ (1) Anti-symmetry in the first two indices. Reversing the direction of the loop makes the effect opposite. Therefore $(R_{\mu\nu})^\sigma_\lambda = -(R_{\nu\mu})^\sigma_\lambda$. (These parentheses makes the matrix explicit). In other words $R(X,Y) = - R(Y,X)$. $$R_{\mu\nu\lambda}^\sigma = -R_{\nu\mu\lambda}^\sigma$$

(2) Anti-symmetry in the second two indices. An infinitesimal rotation matrix (like a Lorentz transform) is anti-symmetric when its indices are lowered. Putting the upper index in the last position, $$g_{\sigma\tau} R^\tau_{\mu\nu\lambda} = R_{\mu\nu\lambda\sigma} = -R_{\nu\mu\sigma\lambda}$$

(3) First Bianchi Identity (via this question on Math Stack Exchange). The fact that the connection is Torsion-free forces the lateral faces of a cube to close. The Bianchi identity expresses that they form a triangle. $$R_{[\mu\nu\lambda]}^\sigma = 0$$

(4) Interchange symmetry. From these three properties, we can derive the interchange symmetry. For example in these notes. You can also get it, I have seen, by expanding out the Levi-Civita connection in terms of the metric.

I am looking for something other than pure index juggling! For example, some sort of picture similar to that in the top answer on that Math Stack Exchange post.

(5) Bonus: After this I'm going to get to the Second Bianchi identity, I imagine. Is there any good intuition for that? Wikipedia gives it as the following, though I'm willing to bet that the indices are in a different order than I put them. $$R_{ab[cd;e]}^{}= R_{abcd;e}^{}+R_{abde;c}^{}+R_{abec;d}^{}=0$$

I can elucidate a different perspective on the Riemann tensor. It is helpful to write it not as $$R^\sigma_{\mu\nu\lambda}$$ but rather as $$R^i_{j\mu\nu}$$ to start off with to distinguish the two types of indices, despite them running over the same range.

This is because if we consider the frame bundle which is in fact a $$GL(d,\mathbb R)$$ principal bundle, the connection one form (pulled back by a section) belongs to $$\Omega^1(M) \otimes T_eG$$, i.e. it is Lie algebra-valued form.

The Lie algebra is matrix valued, so when we take the curvature which is a Lie algebra valued two form, we have two indices $$ij$$ for this, and a further two $$\mu\nu$$ indices as it is a two form on the base space.

As such, clearly it must be anti-symmetric in the $$\mu\nu$$ indices, by what we know about forms. But why also anti-symmetric in $$ij$$? Recall with the existence of a metric we can restrict to orthogonal frames, and then $$ij$$ are anti-symmetric precisely because of the anti-symmetric matrices representing the Lie algebra of the orthogonal group.

Naturally the identities will not mean much unless either you are of Riemann sort, or, you start comprehending where it makes sense; put in contexts of say, space-time in GR, and further restricting to only acting upon vectors with physical meaning. For example start with Bel decomposition. Then trace it back to merely the index juggling.

Another obvious first move is the equation $R_{uv}=0$ holding in empty space.

Another one is the Einstein tensor equals Stress energy tensor being the one and only one way to put the principle of general covariance into an equation.

• For context: how about regular Riemannian geometry in 2, 3, or 4 dimensions (no time dimensions). The visualizations I linked to hold in this context. – Ravi Charan Mar 9 '18 at 17:17
• Beyond the basics of the connection-metric properties, I personally found it difficult to mentally visualize the various curvature tensor/scalar quantities in 4 dimensional case (which happen to all be procedurally given from the Riemann). Gravitation way and actual curves of types of curvatures are "$2/3$" of the physics and hence most essential. Not by spatial mental capabilities, but by physical phenomena. You typically need the tools afforded in the links when you pinpoint symmetries/isometries and the objective is to find the metric in the first place. – user76568 Mar 9 '18 at 18:53


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# Workshop on Homotopy Theory

## Date and Location

The workshop will take place on May 5th, 2018 at the University of Regensburg in the SFB Seminar room M311.

## Aims and Scope

The aim of this workshop is to bring together early career researchers working in topics related to homotopy theory to discuss recent advances, foster collaboration, and say farewell.

## Lectures

• Hoang-Kim Nguyen (University of Regensburg)

## Financial Support

Limited financial support for the conference and younger participants has been provided by the SPP 1786: Homotopy theory and algebraic geometry and SFB 1085: Higher Invariants.

## Practical Information

MAP: Points of Interest.

PUBLIC TRANSIT: Local bus system. There are many useful buses typically, but the 6 will suffice for getting to and from the city center and the university (the nearest stop is the Universität stop). The work shop is on the third floor of the math building. The above maps may prove helpful.

ACCOMMODATION: We will reserve a block of hotel rooms at the [ www.hotel-central-regensburg.de Hotel Central].

One can also check the standard alternatives:

2. Airbnb

• TRAVEL: One can reach the university by following the instructions here.
• INTERNET: Access to eduroam is available throughout the university. If by some miracle, Eduroam works for you, then you can use that. Otherwise the Bavaria offers free wifi as well through BayernWLAN.

## Organizers

• Justin Noel (Regensburg)
• Georgios Raptis (Regensburg)

## Schedule

• All lectures will take place in the SFB Seminar Room in the math wing M311
• Registration and coffee breaks will be held nearby.
• On Saturday evening we will gather at Alte Linde.
Saturday
09:40 - 10:00 Registration and Coffee
10:00 - 10:55 Malkiewich
10:55 - 11:10 Coffee
11:10 - 12:05 Patchkoria
12:15 - 13:45 Lunch (Unikat)
13:45 - 14:40 Frankland
14:40 - 14:55 Coffee
14:55 - 15:50 Nguyen
15:50 - 16:00 Coffee
16:00 - 16:55 Land
16:55 - 17:05 Clean up
18:00 - 22:00 Dinner (Alte Linde)

## Registered Participants

• Peter Arndt (Düsseldorf)
• Uli Bunke (Regensburg)
• Denis-Charles Cisinski (Regensburg)
• Martin Frankland (Osnabrück)
• Moritz Groth (Bonn)
• Cary Malkiewich (Binghamton / MPIM)
• Niko Naumann (Regensburg)
• Hoang-Kim Nguyen (Regensburg)
• Justin Noel (Regensburg)
• Irakli Patchkoria (Bonn)
• Matan Prasma (Regensburg)
• Georgios Raptis (Regensburg)
• Daniel Schaeppi (Regensburg)
• Florian Strunk (Regensburg)
• Sean Tilson (Wuppertal)
• Koen van Woerden (Regensburg)

## Titles and Abstracts

• Cary Malkiewich (Periodic orbits and topological restriction homology):

I will talk about a project to import trace methods, usually reserved for algebraic K-theory computations, into the study of periodic orbits of continuous dynamical systems (and vice-versa). Our main result so far is that a certain fixed-point invariant built using equivariant spectra can be unwound into a more classical invariant that detects periodic orbits. As a simple consequence, periodic-point problems (i.e. finding a homotopy of a continuous map that removes its n-periodic orbits) can be reduced to equivariant fixed-point problems. This answers a conjecture of Klein and Williams, and allows us to interpret their invariant as a class in topological restriction homology (TR), coinciding with a class defined earlier in the thesis of Iwashita and separately by Luck. This is joint work with Kate Ponto.

• Irakli Patchkoria (Computations in real topological cyclic homology):

In this talk I will present recent computations in real topological cyclic homology (TCR). We will see that for a broad class of commutative rings with involution, the ring of components of TCR can be described in terms of Witt vectors over the components of the real topological Hochschild homology. We will also discuss connections to polynomial maps and extra functoriality of Witt vectors. Finally, we will mention the recent progress on TCR(F_p). This is joint work with E. Dotto and K. Moi.

• Martin Frankland (Towards the dual motivic Steenrod algebra in positive characteristic): Several tools from classical topology have useful analogues in motivic homotopy theory. Voevodsky computed the motivic Steenrod algebra and its dual over a base field of characteristic zero. Hoyois, Kelly, and Ostvaer generalized those results to a base field of characteristic $p$, as long as the coefficients are mod $\ell$ with $\ell \neq p$. The case $\ell = p$ remains conjectural.

In joint work with Markus Spitzweck, we show that over a base field of characteristic $p$, the conjectured form of the mod $p$ dual motivic Steenrod algebra is a retract of the actual answer. I will sketch the proof and possible applications. I will also explain how this problem is closely related to the Hopkins-Morel-Hoyois isomorphism, a statement about the algebraic cobordism spectrum MGL.

• Hoang-Kim Nguyen (Adjoint functor theorems in infinity-categories)
• Markus Land (Genuine L-theory):

In this talk I will give an overview of what this project is about and how it relates (hopefully) to higher Grothendieck-Witt theory and real algebraic K-theory.


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# My mobile number is a prime number

My personal mobile number 9565804301 is a prime number. What is a prime number? Any integer p greater than 1 is called a prime number if and only if its positive factors are 1 and the number p itself. In other words, the natural numbers which are completely divisible by 1 and themselves only and


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# Correlation and regression analysis

Series Development Geology Reference Manual Methods in Exploration Geological methods Correlation and regression analysis T. C. Coburn Web page PDF file (requires access)
Figure 1 Linear regression of x-on-y. Note the negative slope corresponding to a negative correlation. The regression line is determined so as to minimize the sum of squared deviations: $\sum _{i}{e_{i}^{2}}$

Correlation analysis, and its cousin, regression analysis, are well-known statistical approaches used in the study of relationships among multiple physical properties. The investigation of permeability-porosity relationships is a typical example of the use of correlation in geology.

The term correlation most often refers to the linear association between two quantities or variables, that is, the tendency for one variable, x, to increase or decrease as the other, y, increases or decreases, in a straight-line trend or relationship.[1] [2] The correlation coefficient (also called the Pearson correlation coefficient), r, is a dimensionless numerical index of the strength of that relationship. The sample value of r, which can range from -1 to +1, is computed using the following formula:

$r={\frac {\displaystyle \sum _{{i}}(x_{{i}}-{\bar {x}})(y_{{i}}-{\bar {y}})}{{\sqrt {\displaystyle \sum _{i}(x_{{i}}-{\bar {x}})^{2}\times \displaystyle \sum _{i}(y_{{i}}-{\bar {y}})^{2}}}}}$

where the summation is made over the n sample values available and where

• $x_{i}$ = ith observation of the first variable
• $y_{i}$ = corresponding ith observation of the second variable
• ${\bar {x}},{\bar {y}}$ = corresponding sample (arithmetic) means

Many useful types of correlation other than simple linear correlation exist. Correlation based on the ranks (numerical ordering) of measurements of two variables is an important alternative for simple linear correlation when the number of samples is small and/or the joint distribution of the two variables is not simple. Multiple correlation and partial correlation are useful when studying relationships involving more than two variables. Cross correlation and autocorrelation are important to the analysis of repeated patterns observed in time and space, such as depth-related data recorded from geological stratigraphic sequences. Autocorrelation can also be used to measure the degree of similarity among porosity values measured at different locations.

Whereas correlation describes the linear association among variables, regression involves the prediction of one quantity from the others. Regression analysis is that broad class of statistics and statistical methods that comprises line, curve, and surface fitting, as well as other kinds of prediction and modeling techniques.

The simplest type of regression analysis involves fitting a straight line between two variables (Figure 1). In this case, one of the quantities is called the independent or predictor variable (usually denoted x), while the other is called the dependent or predicted variable (usually denoted y). This approach is often referred to as simple linear regression, or y-on-x regression. It leads to the development of an empirical straight-line relationship between the two variables and has the following form:

$\widehat {y}=ax+b$

where

• a = slope
• b = y-intercept in an (x, y) coordinate system

One feature of this line is that it always passes through the centroid of the data (${\bar {x}},{\bar {y}}$) defined by the two arithmetic averages. The formulas for the parameters a and b are

$a={\frac {\displaystyle \sum _{{i}}(x_{{i}}-{\bar {x}})(y_{{i}}-{\bar {y}})}{\displaystyle \sum _{i}(x_{{i}}-{\bar {x}})^{2}}}$, and $b={\bar {y}}-a{\bar {x}}$

There are two primary statistical applications for the regression line. It can be used as a sample-based estimate of an underlying linear functional relationship among the quantities of interest, or it can be employed as a predictive tool. To use the regression line for prediction, which is the most common usage in geology, simply substitute various values of x into the equation y = ax + b and solve for the corresponding predicted values of y.

By substituting the original x-values into the equation, it is possible to derive a measure of "goodness of fit" for the line. For every original x-value, a predicted value of y can be computed that differs from the corresponding original y-value by a certain amount. This difference, y - y, is called the residual e (in the y-direction), or deviation from the regression line. Two consequences of the least squares approach for the determination of the parameters a and b are that the sum of these residuals is zero and the sum of their squared values is as small as possible.

The sum of squares of the residuals is the basis for two measures of goodness of fit: the error variance s2 and the multiple correlation coefficient (also called the coefficient of determination). These two measures are given by the

following formulas:

$s^{2}={\frac {\displaystyle \sum _{{i}}(y_{{i}}-\widehat {y_{{i}}})}{n-2}}$
$R^{2}=1-{\frac {\displaystyle \sum _{{i}}(y_{{i}}-\widehat {y_{{i}}})}{\displaystyle \sum _{{i}}(y_{{i}}-{\bar {y}})}}$

where n is the number of (x, y) pairs.

The error variance s2 measures the scatter in the data about the regression line. The square of the sample correlation coefficient, R2, takes values from 0 to 1 and can be interpreted as the fraction of the variation in the data explained by the regression line. The line is said to be a good fit of the data if s2 is small and R2 is simultaneously large.

Some important extensions of two-variable linear regression analysis of particular interest to geologists include the following:

• Fitting through the origin, or forcing the regression line through any fixed point
• Regression in reverse, which consists of rewriting the regression of y-on-x as
$x={\frac {(y-b)}{a}}$
• Modeling nonlinear relationships, such as
$\widehat {y}=a+bx+cx^{2}$

Various other functions of the x variable can be included in the previous relationship, such as polynomials and logarithms. Again, the regression parameters are determined so as to minimize the corresponding error variance.

## Multiple and multivariate regression

The most important extension of the two-variable case is to situations involving more than two variables. When there is still one dependent variable but many predictor variables, the fitting technique is called multiple linear regression. When there are also more than one dependent variable, the approach is called multivariate regression (see Multivariate data analysis). The methods of simple bivariate regression extend directly to these multivariate situations. A typical geological application of multiple regression is the prediction of fold thickness from various geometric attributes, as given by the following equation:

${\text{Thickness }}=a+b~({\text{attitude}})+c~({\text{tightness}})+d~({\text{asymmetry}})$

Polynomial regression, sometimes referred to as curve fitting, is a special type of multiple regression frequently used in the earth sciences to model nonlinear relationships. Spline fitting is another type of nonlinear curve fitting. A close relative to polynomial curve fitting is surface fitting, which has one or more spatial components as predictor variables.

The fitting of surfaces by least squares is an important component in most automated contouring software packages and is commonly used in computer generation of geological maps. Trend surface analysis, another mapping technique, is also based on the principles of least-squares fitting. Finally, some of the more specialized geostatistical techniques, such as kriging, are likewise rooted in the basic principles of least squares and multiple regression.

## References

1. Draper, N. R., and H. Smith, 1966, Applied regression analysis, 2nd ed.: New York, John Wiley, 709 p.
2. Snedecor, G. W., and W. G. Cochran, 1967, Statistical methods, 6th ed.: Ames, Iowa State Univ. Press, 593 p.