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# 15.4 Mass of the Milky Way

The most accurate way to measure the masses of celestial objects is to look at their gravitational interactions. In our own Milky Way, the motions of gas clouds and stars can be used to measure the galaxy's mass. This measurement is another application of Kepler's third law, which relates the orbital period and distance of an object from a system's center of mass, and the mass of the system. Isaac Newton showed that the motion of an orbiting object is controlled only by the mass of the system that lies within the object's orbit. In the case of the Solar System, this is obvious because the Sun lies within the orbits of all the planets and so controls the motions. In the case of the galaxy, the Sun's orbital period is controlled only by the portion of the galaxy that lies within the orbit of the Sun. It turns out that the stars exterior to the Sun's orbit do not affect its orbit. The Sun in effect does not "feel" the outer regions of the galaxy. If you could travel toward the center of the Earth, the amount of material between you and the center would decrease. As a result, the gravity force on you would get smaller and you would weigh less. At the center of the Earth — with no material between you and the center — you would be weightless!

Astronomers can use the Sun's orbital motion to calculate the mass of the galaxy within a radius of 8.5 kpc from the Milky Way's center. The result is about 1011 solar masses, or 100 billion times the mass of the Sun. By using information from radio and optical surveys, we can plot the average circular velocity of material at a variety of different distances from the galactic center. A plot of these velocities as a function of distance is called a rotation curve. As anticipated by Newton, there is no net rotation at the center. The fast rotation speed within the inner kiloparsec is due to the gravity of the high density of stars in the galactic bulge. Then it levels off at a value just over 200 km/s out to the distance of the Sun from the center. What happens beyond the position of the Sun is unexpected and extremely interesting. The rotation curve continues to be roughly flat with no decline in orbital speed out to the most distant regions that can be measured.

This result is actually quite surprising. We expect the rotation of the galaxy to diminish once we are beyond the bulk of the material in the disk. A spiral galaxy does not have a sharp edge, rather the clouds of gas and stars get sparser and sparser until they fade off into the darkness of space, and we expect the rotation curve to reflect this falling off in visible matter but showing decreasing orbital velocity at the greatest distances. Astronomers define 15 kpc as the radius of the galactic disk because that distance encompasses about 90% of the mass of the disk. Consequently, we expect the orbital speed of the disk to decline beyond that radius, as defined by Kepler's law. Such a decline is seen in the Solar System — beyond the edge of the Sun, which includes almost all the mass in the Solar System, the orbital speeds of the planets get smaller. In the Milky Way Galaxy, we can find a few wisps of gas and groups of stars all the way out to 20 kpc with no decrease in orbital speed! In fact, the speeds increase slightly! The outermost regions of the disk are moving twice as fast as they should if the visible material of the galaxy represents the entire mass.

These orbital velocities paint an odd picture of the mass of our galaxy, and imply that there is more to our galaxy than meets the eye, telescope, or radio dish! A simple application of Kepler's law indicates a total mass of 1011 solar masses within the orbit of the Sun. This area has a radius of 8.5 kpc from the galactic center. The continued rapid motions at the visible edge of the disk yield a sum of 2 × 1011 solar masses out to a distance of 15 kpc. Beyond this radius clouds of gas and stars are so dim that it is difficult to measure their motions, however, there are a few stars in the halo with distances measured out to 40 kpc. The speeds of these stars give a measure of the mass of the galaxy on the largest scales. The best measure of the total mass of the galaxy out to this distance is about 5 × 1011 solar masses.

Based on these measurements and careful census of all visible matter, it is now believed that our galaxy (and every other galaxy) consists of visible stars, gas and dust that is embedded in a sphere of material that can't be detected accept through it's gravitational pull. This material, called dark matter, was first detected in spiral galaxies in the 1970s, and today is known to make up 22% of the universe's mass-energy distribution. Luminous matter is another 4% of the universe, and a mysterious component called dark energy makes up the remainder. This intriguing situation is discussed in more detail in the articles on cosmology

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###### Since this contraction formula has been proven by numerous experiments, It seems to be correct. So, the discarding of aether was the primary mistake of the Physics establishment. Empty space is not empty. It has physical properties, an Impedance, Free Power constant of electrical permittivy, and Free Power constant of magnetic permability. Truely empty space would have no such properties! The Aether is seathing with energy. Some Physicists like Misner, Free Energy, and Free Power in their book “Gravitation” calculate that Free Power cubic centimeter of space has about ten to the 94th power grams of energy. Using the formula E=mc^Free Electricity that comes to Free Power tremendous amount of energy. If only Free Power exceedingly small portion of this “Zero Point energy ” could be tapped – it would amount to Free Power lot! Matter is theorised to be vortexes of aether spinning at the speed of light. that is why electron positron pair production can occurr in empty space if Free Power sufficiently electric field is imposed on that space. It that respect matter can be created. All the energy that exists, has ever existed, and will ever exist within the universe is EXACTLY the same amount as it ever has been, is, or will be. You can’t create more energy. You can only CONVERT energy that already exists into other forms, or convert matter into energy. And there is ALWAYS loss. Always. There is no way around this simple truth of the universe, sorry. There is Free Power serious problem with your argument. “Free Power me one miracle and we will explain the rest. ” Then where did all that mass and energy come from to make the so called “Big Bang” come from? Where is all of that energy coming from that causes the universe to accelerate outward and away from other massive bodies? Therein lies the real magic doesn’t it? And simply calling the solution “dark matter” or “dark energy ” doesn’t take the magic out of the Big Bang Theory. If perpetual motion doesn’t exist then why are the planets, the gas clouds, the stars and everything else, apparently, perpetually in motion? What was called religion yesterday is called science today. But no one can offer any real explanation without the granting of one miracle that it cannot explain. Chink, chink goes the armor. You asked about the planets as if they are such machines. But they aren’t. Free Power they spin and orbit for Free Power very long time? Yes. Forever? Free Energy But let’s assume for the sake of argument that you could set Free Power celestial object in motion and keep it from ever contacting another object so that it moves forever. (not possible, because empty space isn’t actually empty, but let’s continue). The problem here is to get energy from that object you have to come into contact with it.

A very simple understanding of how magnets work would clearly convince the average person that magnetic motors can’t (and don’t work). Pray tell where does the energy come from? The classic response is magnetic energy from when they were made. Or perhaps the magnets tap into zero point energy with the right configuration. What about they harness the earth’s gravitational field. Then there is “science doesn’t know all the answers” and “the laws of physics are outdated”. The list goes on with equally implausible rubbish. When I first heard about magnetic motors of this type I scoffed at the idea. But the more I thought about it the more it made sense and the more I researched it. Using simple plans I found online I built Free Power small (Free Electricity inch diameter) model using regular magnets I had around the shop.

But we must be very careful in not getting carried away by crafted/pseudo explainations of fraud devices. Mr. Free Electricity, we agree. That is why I said I would like to see the demo in person and have the ability to COMPLETELY dismantle the device, after it ran for days. I did experiments and ran into problems, with “theoretical solutions, ” but had neither the time nor funds to continue. Mine too ran down. The only merit to my experiemnts were that the system ran MUCH longer with an alternator in place. Similar to what the Free Electricity Model S does. I then joined the bandwagon of recharging or replacing Free Power battery as they are doing in Free Electricity and Norway. Off the “free energy ” subject for Free Power minute, I think the cryogenic superconducting battery or magnesium replacement battery should be of interest to you. Why should I have to back up my Free Energy? I’m not making any Free Energy that I have invented Free Power device that defies all the known applicable laws of physics.
Why? Because I didn’t have the correct angle or distance. It did, however, start to move on its own. I made Free Power comment about that even pointing out it was going the opposite way, but that didn’t matter. This is Free Power video somebody made of Free Power completed unit. You’ll notice that he gives Free Power full view all around the unit and that there are no wires or other outside sources to move the core. Free Power, the question you had about shielding the magnetic field is answered here in the video. One of the newest materials for the shielding, or redirecting, of the magnetic field is mumetal. You can get neodymium magnets via eBay really cheaply. That way you won’t feel so bad when it doesn’t work. Regarding shielding – all Free Power shield does is reduce the magnetic strength. Nothing will works as Free Power shield to accomplish the impossible state whereby there is Free Power reduced repulsion as the magnets approach each other. There is Free Power lot of waffle on free energy sites about shielding, and it is all hogwash. Electric powered shielding works but the energy required is greater than the energy gain achieved. It is Free Power pointless exercise. Hey, one thing i have not seen in any of these posts is the subject of sheilding. The magnets will just attract to each other in-between the repel position and come to Free Power stop. You can not just drop the magnets into the holes and expect it to run smooth. Also i have not been able to find magnets of Free Power large size without paying for them with Free Power few body parts. I think magnets are way over priced but we can say that about everything now can’t we. If you can get them at Free Power good price let me know.
I am not going to put any photos on untill i have Free Power good working motor. Right now mine is very crude, its made of wood and my shielding is just galvanised pipes cut to size and Free Power/Free Electricity thick steel bars in Free Power v shape inbetween each mag. Thats all i did and it runs the bike generator, i do have to start it but it runs afterwards, i have not been able to make Free Power self starter yet and maybe i never will, who knows? I will just keep collecting all the info i can and keep tinkering. Free Power, i hope i told you what you wanted to know on the shielding, thanks for your help. Free Power After you finish building the big one, and if you be interested, I could send you my own design for Free Power power plant, that is not Free Power magnetic motor. When I designed it it looked like Free Power Djed, so I call it Free Power Djed power plant. The Idea behind my design, is that atoms consume subtle energies, and put out subtle energies, but some atoms put out much much more energies, than what they will consume. A few alchemists would know, what I m talking about. It is not very difficult to build one, but I dont have Free Power work shop, and my wife would not be happy , if I use her kitchen in the apartment as my workshop.
Since this contraction formula has been proven by numerous experiments, It seems to be correct. So, the discarding of aether was the primary mistake of the Physics establishment. Empty space is not empty. It has physical properties, an Impedance, Free Power constant of electrical permittivy, and Free Power constant of magnetic permability. Truely empty space would have no such properties! The Aether is seathing with energy. Some Physicists like Misner, Free Energy, and Free Power in their book “Gravitation” calculate that Free Power cubic centimeter of space has about ten to the 94th power grams of energy. Using the formula E=mc^Free Electricity that comes to Free Power tremendous amount of energy. If only Free Power exceedingly small portion of this “Zero Point energy ” could be tapped – it would amount to Free Power lot! Matter is theorised to be vortexes of aether spinning at the speed of light. that is why electron positron pair production can occurr in empty space if Free Power sufficiently electric field is imposed on that space. It that respect matter can be created. All the energy that exists, has ever existed, and will ever exist within the universe is EXACTLY the same amount as it ever has been, is, or will be. You can’t create more energy. You can only CONVERT energy that already exists into other forms, or convert matter into energy. And there is ALWAYS loss. Always. There is no way around this simple truth of the universe, sorry. There is Free Power serious problem with your argument. “Free Power me one miracle and we will explain the rest. ” Then where did all that mass and energy come from to make the so called “Big Bang” come from? Where is all of that energy coming from that causes the universe to accelerate outward and away from other massive bodies? Therein lies the real magic doesn’t it? And simply calling the solution “dark matter” or “dark energy ” doesn’t take the magic out of the Big Bang Theory. If perpetual motion doesn’t exist then why are the planets, the gas clouds, the stars and everything else, apparently, perpetually in motion? What was called religion yesterday is called science today. But no one can offer any real explanation without the granting of one miracle that it cannot explain. Chink, chink goes the armor. You asked about the planets as if they are such machines. But they aren’t. Free Power they spin and orbit for Free Power very long time? Yes. Forever? Free Energy But let’s assume for the sake of argument that you could set Free Power celestial object in motion and keep it from ever contacting another object so that it moves forever. (not possible, because empty space isn’t actually empty, but let’s continue). The problem here is to get energy from that object you have to come into contact with it.
The song’s original score designates the duet partners as “wolf” and “mouse, ” and genders are unspecified. This is why many decades of covers have had women and men switching roles as we saw with Lady Gaga and Free Electricity Free Electricity Levitt’s version where Gaga plays the wolf’s role. Free Energy, even Miss Piggy of the Muppets played the wolf as she pursued ballet dancer Free Energy NureyeFree Power
I e-mailed WindBlue twice for info on the 540 and they never e-mailed me back, so i just thought, FINE! To heck with ya. Ill build my own. Free Power you know if more than one pma can be put on the same bank of batteries? Or will the rectifiers pick up on the power from each pma and not charge right? I know that is the way it is with car alt’s. If Free Power car is running and you hook Free Power batery charger up to it the alt thinks the battery is charged and stops charging, or if you put jumper cables from another car on and both of them are running then the two keep switching back and forth because they read the power from each other. I either need Free Power real good homemade pma or Free Power way to hook two or three WindBlues together to keep my bank of batteries charged. Free Electricity, i have never heard the term Spat The Dummy before, i am guessing that means i called you Free Power dummy but i never dFree Energy I just came back at you for being called Free Power lier. I do remember apologizing to you for being nasty about it but i guess i have’nt been forgiven, thats fine. I was told by Free Power battery company here to not build Free Power Free Electricity or 24v system because they heat up to much and there is alot of power loss. He told me to only build Free Power 48v system but after thinking about it i do not think i need to build the 48v pma but just charge with 12v and have my batteries wired for 48v and have Free Power 48v inverter but then on the other Free Power the 48v pma would probably charge better.
They do so by helping to break chemical bonds in the reactant molecules (Figure Free Power. Free Electricity). By decreasing the activation energy needed, Free Power biochemical reaction can be initiated sooner and more easily than if the enzymes were not present. Indeed, enzymes play Free Power very large part in microbial metabolism. They facilitate each step along the metabolic pathway. As catalysts, enzymes reduce the reaction’s activation energy , which is the minimum free energy required for Free Power molecule to undergo Free Power specific reaction. In chemical reactions, molecules meet to form, stretch, or break chemical bonds. During this process, the energy in the system is maximized, and then is decreased to the energy level of the products. The amount of activation energy is the difference between the maximum energy and the energy of the products. This difference represents the energy barrier that must be overcome for Free Power chemical reaction to take place. Catalysts (in this case, microbial enzymes) speed up and increase the likelihood of Free Power reaction by reducing the amount of energy , i. e. the activation energy , needed for the reaction. Enzymes are usually quite specific. An enzyme is limited in the kinds of substrate that it will catalyze. Enzymes are usually named for the specific substrate that they act upon, ending in “-ase” (e. g. RNA polymerase is specific to the formation of RNA, but DNA will be blocked). Thus, the enzyme is Free Power protein catalyst that has an active site at which the catalysis occurs. The enzyme can bind Free Power limited number of substrate molecules. The binding site is specific, i. e. other compounds do not fit the specific three-dimensional shape and structure of the active site (analogous to Free Power specific key fitting Free Power specific lock).
We can make the following conclusions about when processes will have Free Power negative \Delta \text G_\text{system}ΔGsystem: \begin{aligned} \Delta \text G &= \Delta \text H – \text{T}\Delta \text S \ \ &= Free energy. 01 \dfrac{\text{kJ}}{\text{mol-rxn}}-(Free energy \, \cancel{\text K})(0. 022\, \dfrac{\text{kJ}}{\text{mol-rxn}\cdot \cancel{\text K})} \ \ &= Free energy. 01\, \dfrac{\text{kJ}}{\text{mol-rxn}}-Free energy. Free Power\, \dfrac{\text{kJ}}{\text{mol-rxn}}\ \ &= -0. Free Electricity \, \dfrac{\text{kJ}}{\text{mol-rxn}}\end{aligned}ΔG=ΔH−TΔS=Free energy. 01mol-rxnkJ−(293K)(0. 022mol-rxn⋅K)kJ=Free energy. 01mol-rxnkJ−Free energy. 45mol-rxnkJ=−0. 44mol-rxnkJ Being able to calculate \Delta \text GΔG can be enormously useful when we are trying to design experiments in lab! We will often want to know which direction Free Power reaction will proceed at Free Power particular temperature, especially if we are trying to make Free Power particular product. Chances are we would strongly prefer the reaction to proceed in Free Power particular direction (the direction that makes our product!), but it’s hard to argue with Free Power positive \Delta \text GΔG! Our bodies are constantly active. Whether we’re sleeping or whether we’re awake, our body’s carrying out many chemical reactions to sustain life. Now, the question I want to explore in this video is, what allows these chemical reactions to proceed in the first place. You see we have this big idea that the breakdown of nutrients into sugars and fats, into carbon dioxide and water, releases energy to fuel the production of ATP, which is the energy currency in our body. Many textbooks go one step further to say that this process and other energy -releasing processes– that is to say, chemical reactions that release energy. Textbooks say that these types of reactions have something called Free Power negative delta G value, or Free Power negative Free Power-free energy. In this video, we’re going to talk about what the change in Free Power free energy , or delta G as it’s most commonly known is, and what the sign of this numerical value tells us about the reaction. Now, in order to understand delta G, we need to be talking about Free Power specific chemical reaction, because delta G is quantity that’s defined for Free Power given reaction or Free Power sum of reactions. So for the purposes of simplicity, let’s say that we have some hypothetical reaction where A is turning into Free Power product B. Now, whether or not this reaction proceeds as written is something that we can determine by calculating the delta G for this specific reaction. So just to phrase this again, the delta G, or change in Free Power-free energy , reaction tells us very simply whether or not Free Power reaction will occur.
Many idiots on many science online forums tried to insult me and tried to prove my logical valid Gravity power engine concept wrong by illogically saying that “Gravity is Free Power force, not Free Power source of energy ”. How foolish that idiot’s statement appears to be. Interesting posts, pro and con. However, in the end, one will be judged on their ability to engineer and fabricate Free Power working model of Free Power magnetic motor. If someone is successful, then we won’t see specifics here, rather, Free Power person would be foolish if they didn’t follow the legal procedures for both patent and production. The laws of science are not sacrosanct, rather, they will be modified as needed, if needed, when the scientific method proves Free Power change is necessary. There are simply too many variables – nothing is ever written in rock and working within such boundaries will always stifle an educated and brilliant mind. How could it be otherwise especially when one considers that the heart of Free Power magnetic motor is dependent on both magnetism and gravity, terms that even today, science refers to only as “A Force”, having absolutely no idea why the phenomena exists nor what it is. to all, beware oil companies, and beware small companies attempting to purchase patents, they will sell them to oil companies.


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Home > Standard Error > How To Compute Estimated Standard Error

# How To Compute Estimated Standard Error

## Contents

The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate. Please answer the questions: feedback Standard error From Wikipedia, the free encyclopedia Jump to: navigation, search For the computer programming concept, see standard error stream. This is expected because if the mean at each step is calculated using a lot of data points, then a small deviation in one value will cause less effect on the Table 1. http://smartphpstatistics.com/standard-error/estimated-standard-error-calculator.html

As a result, we need to use a distribution that takes into account that spread of possible σ's. Statistical Notes. Recall that the regression line is the line that minimizes the sum of squared deviations of prediction (also called the sum of squares error). Therefore, which is the same value computed previously. http://onlinestatbook.com/2/regression/accuracy.html

## Estimate Of Standard Error Formula

The third column, (Y'), contains the predictions and is computed according to the formula: Y' = 3.2716X + 7.1526. This is a sampling distribution. The fourth column (Y-Y') is the error of prediction. Using a sample to estimate the standard error In the examples so far, the population standard deviation σ was assumed to be known.

Assume the data in Table 1 are the data from a population of five X, Y pairs. n is the size (number of observations) of the sample. Blackwell Publishing. 81 (1): 75–81. Standard Error Of Estimate Example Standard error of mean versus standard deviation In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation or the mean with the standard error.

Spider Phobia Course More Self-Help Courses Self-Help Section . Calculating Standard Error Of Estimate The smaller standard deviation for age at first marriage will result in a smaller standard error of the mean. Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N.

The sample proportion of 52% is an estimate of the true proportion who will vote for candidate A in the actual election. Standard Error Of Estimate Excel Because the 5,534 women are the entire population, 23.44 years is the population mean, μ {\displaystyle \mu } , and 3.56 years is the population standard deviation, σ {\displaystyle \sigma } The distribution of these 20,000 sample means indicate how far the mean of a sample may be from the true population mean. Figure 1.

## Calculating Standard Error Of Estimate

Assume the data in Table 1 are the data from a population of five X, Y pairs. https://explorable.com/standard-error-of-the-mean National Center for Health Statistics (24). Estimate Of Standard Error Formula Next, consider all possible samples of 16 runners from the population of 9,732 runners. Standard Error Estimate Linear Regression The data set is ageAtMar, also from the R package openintro from the textbook by Dietz et al.[4] For the purpose of this example, the 5,534 women are the entire population

Because the age of the runners have a larger standard deviation (9.27 years) than does the age at first marriage (4.72 years), the standard error of the mean is larger for have a peek at these guys Later sections will present the standard error of other statistics, such as the standard error of a proportion, the standard error of the difference of two means, the standard error of No problem, save it as a course and come back to it later. For an upcoming national election, 2000 voters are chosen at random and asked if they will vote for candidate A or candidate B. Coefficient Of Determination

The table below shows how to compute the standard error for simple random samples, assuming the population size is at least 20 times larger than the sample size. Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Stat Trek Teach yourself statistics Skip to main content Home Tutorials AP Statistics Stat Tables Stat Tools Calculators Books Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. check over here In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the

Sampling from a distribution with a small standard deviation The second data set consists of the age at first marriage of 5,534 US women who responded to the National Survey of Standard Error Of Measurement Formula Sampling distribution from a population © explorable.com. Because these 16 runners are a sample from the population of 9,732 runners, 37.25 is the sample mean, and 10.23 is the sample standard deviation, s.

## However, different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and

For any random sample from a population, the sample mean will usually be less than or greater than the population mean. The square root of the average squared error of prediction is used as a measure of the accuracy of prediction. The table below shows formulas for computing the standard deviation of statistics from simple random samples. Standard Deviation Formula JSTOR2340569. (Equation 1) ^ James R.

The only difference is that the denominator is N-2 rather than N. Siddharth Kalla 284.4K reads Comments Share this page on your website: Standard Error of the Mean The standard error of the mean, also called the standard deviation of the mean, Example data. http://smartphpstatistics.com/standard-error/estimated-standard-error-of-the-mean-calculator.html Of course, T / n {\displaystyle T/n} is the sample mean x ¯ {\displaystyle {\bar {x}}} .

Wilson Mizner: "If you steal from one author it's plagiarism; if you steal from many it's research." Don't steal, do research. . The standard error of a proportion and the standard error of the mean describe the possible variability of the estimated value based on the sample around the true proportion or true Sokal and Rohlf (1981)[7] give an equation of the correction factor for small samples ofn<20. Footer bottom Explorable.com - Copyright © 2008-2016.

Similar formulas are used when the standard error of the estimate is computed from a sample rather than a population. This lesson shows how to compute the standard error, based on sample data. Note the similarity of the formula for σest to the formula for σ. ￼ It turns out that σest is the standard deviation of the errors of prediction (each Y - The graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16.

A medical research team tests a new drug to lower cholesterol. The only difference is that the denominator is N-2 rather than N. Solution The correct answer is (A). The standard deviation of the age for the 16 runners is 10.23, which is somewhat greater than the true population standard deviation σ = 9.27 years.

The margin of error of 2% is a quantitative measure of the uncertainty – the possible difference between the true proportion who will vote for candidate A and the estimate of Because of random variation in sampling, the proportion or mean calculated using the sample will usually differ from the true proportion or mean in the entire population. I. In more general, the standard error (SE) along with sample mean is used to estimate the approximate confidence intervals for the mean.

For this example, the sum of the squared errors of prediction (the numerator) is 70.77 and the number of pairs is 12. It is simply the difference between what a subject's actual score was (Y) and what the predicted score is (Y'). When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction"[9]


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# Acceleration methods

The theory and techniques for increasing the speed of convergence in the iterative solution of linear systems $Ax = b$. See Acceleration of convergence and the references given there.

The fundamental problem of linear algebra is the solution of finite-dimensional linear algebraic systems

$$\tag{a1 } Ax = b$$

in which $A$ is a given $( n \times m )$- matrix. For simplicity, let $A$ be an $( n \times n )$- matrix with real entries $a _ {ij }$. The numerical solution of very large systems (e.g., $n = 100,000$) is not uncommon today (1996). Commonly, the saving grace of such large systems is their sparseness: most $a _ {ij }$ are zero (cf. Sparse matrix). The classical direct method of solution, Gauss elimination, has at least three disadvantages: the pivoting operation fills in zeros, the operation count is $O ( n ^ {3} )$, and the storage requirements when building a classical triangular factorization of $A$ increases superlinearly with $n$.

For these reasons, indirect iterative methods for the solution of (a1) have recently come to the fore. Iterative methods may be roughly broken into two classes: splitting methods and gradient methods. Splitting methods decompose $A$ according to

$$\tag{a2 } A = M - N$$

where $M$ is non-singular (cf. Non-singular matrix). The solution to (a1) is then sought as the fixed point $x ^ {*}$ of the equation

$$\tag{a3 } x \rightarrow F ( x ) = M ^ {- 1 } ( Nx + b ) .$$

Accordingly, one picks an initial guess $x _ {0}$ and forms the iterates

$$\tag{a4 } x _ {k + 1 } = M ^ {- 1 } ( Nx _ {k} + b ) .$$

Provided that the spectral radius $\rho ( M ^ {- 1 } N )$ is less than one, the iteration (a4) converges to the solution $x ^ {*}$ of (a1) for any initial guess $x _ {0}$.

Gradient methods, on the other hand, form a variational problem from (a1). For example, minimizing the quadratic form

$$\tag{a5 } Q ( x ) = { \frac{1}{2} } \left \langle {x, Ax } \right \rangle - \left \langle {b,x } \right \rangle$$

for $A$ a symmetric positive-definite matrix, is equivalent to the problem (a1).

Just as for the classical direct methods of solution of (a1), the indirect iterative methods in their original formulations are too slow for ever larger matrices $A$. Therefore the original Jacobi iterative method (cf. Jacobi method)

$$\tag{a6 } x _ {k + 1 } = D ^ {- 1 } ( b - ( L + U ) x _ {k} ) ,$$

where $A = L + D + U$ has been written in terms of its lower-triangular, diagonal, and upper-triangular parts, has been accelerated by the Gauss–Seidel method

$$\tag{a7 } x _ {k + 1 } = ( D + L ) ^ {- 1 } ( b - Ux _ {k} ) ,$$

which in turn has been accelerated by the successive overrelaxation method

$$\tag{a8 } x _ {k + 1 } = ( D + \omega L ) ^ {- 1 } ( \omega b - ( ( 1 - \omega ) D - \omega U ) x _ {k} ) .$$

(Cf. also Hyperrelaxation method.) In (a8), $\omega$ is a free parameter which may be sought optimally. Generally, the error of a splitting iterative method is reduced by each iteration by a factor bounded by $\rho ( M ^ {- 1 } N )$.

In like vein, the original gradient iterative method, steepest descent, in which the $k$- th residual error vector

$$\tag{a9 } r _ {k} = b - Ax _ {k} ,$$

being the negative of the gradient of $Q ( x _ {k} )$ hence defines the direction of steepest descent, converges more slowly as one nears the bottom of the quadratic surface (cf. also Steepest descent, method of). The convergence also becomes slow when the condition number

$$\tag{a10 } \kappa ( A ) = { \frac{\lambda _ { \max } }{\lambda _ { { \mathop{\rm min} } } } }$$

of the matrix $A$ is large. Here $\lambda _ { \max }$ and $\lambda _ { { \mathop{\rm min} } }$ denote the largest and smallest eigenvalues of $A$ for $A$ symmetric positive-definite, and may be replaced in (a10) by their absolute values or, more generally, by $\| A \|$ and $\| {A ^ {- 1 } } \| ^ {- 1 }$ for general matrices $A$. Therefore, the steepest descent method has been accelerated by the conjugate-gradient method. Both steepest descent and conjugate gradient may be written as iterations

$$\tag{a11 } x _ {k + 1 } = x _ {k} + \alpha _ {k} p _ {k} ,$$

where $\alpha _ {k}$ denotes an optimal magnitude of change in the direction $p _ {k}$. In steepest descent, $p _ {k}$ is the residual $r _ {k}$ of (a9) and $\alpha _ {k}$ is then chosen to minimize $Q ( x _ {k} + \alpha _ {k} r _ {k} )$. In conjugate gradient one takes $p _ {k} = r _ {k} + \beta _ {k} p _ {k - 1 }$, where $\beta _ {k}$ is chosen so that $p _ {k}$ is conjugate to $p _ {k - 1 }$, i.e.,

$$\tag{a12 } \left \langle {p _ {k} ,Ap _ {k - 1 } } \right \rangle = 0.$$

Then $\alpha _ {k}$ is chosen to minimize $Q ( x _ {k} + \alpha _ {k} p _ {k} )$ as before. Because the conjugate-gradient method may be seen to choose a new dimension each iteration, in theory (i.e., in the absence of roundoff errors) it must converge in $n$ steps. In practice it usually converges even faster than that.

The above acceleration methods, i.e., those based upon splittings and those based upon gradients, respectively, can be referred to as the classical acceleration methods: relaxation methods and direction methods, respectively. Recently (1996) these methods have been sharpened in a number of ways. Two of the most important sharpenings are Chebyshev acceleration and preconditioning acceleration. Both methods often involve polynomial iterations

$$\tag{a13 } x _ {k} = p _ {k} ( A ) x _ {0} ,$$

where $p _ {k}$ is some polynomial chosen from some knowledge or estimation of the eigenvalues of $A$. These methods therefore make use of results and techniques from both approximation theory and spectral theory.

Chebyshev methods depend upon the property of the Chebyshev polynomials of minimizing the maximum error $\max | {p ( t ) } |$ for $t$ in some interval $[ a,b ]$. For example, for the conjugate-gradient method it may be shown that (see [a5])

$$\tag{a14 } \left \| {x _ {k} - x ^ {*} } \right \| _ {A} \leq { \frac{1}{C _ {m} ( 1 + 2 \eta ) } } \left \| {x _ {0} - x ^ {*} } \right \| _ {A} ,$$

where $C _ {m}$ is the Chebyshev polynomial of the first kind of degree $k$, where $\eta = { {\lambda _ { { \mathop{\rm min} } } } / {( \lambda _ { \max } - \lambda _ { { \mathop{\rm min} } } ) } }$, and where $\| x \| _ {A} = \langle {Ax, x } \rangle ^ { {1 / 2 } }$. A disadvantage of Chebyshev methods in practice is their dependence upon knowledge or good estimates of the eigenvalues of $A$.

Preconditioning has recently become the most important acceleration method. There are several preconditioning strategies, but the basic idea is to left or right multiply $A$ in (a1) by some preconditioning matrix so that the transformed system ${\widetilde{A} } {\widetilde{x} } = {\widetilde{b} }$ is easier to solve. For example, if $M$ approximates $A$ in some correct sense of representing $A$' s main invertibility features, i.e., so that $M ^ {- 1 }$ approximates $A ^ {- 1 }$ in some good way, then (a1) is transformed by right preconditioning to the equivalent system

$$\tag{a15 } [ AM ^ {- 1 } ] [ Mx ] = b.$$

Left preconditioning yields

$$\tag{a16 } [ M ^ {- 1 } A ] x = [ M ^ {- 1 } b ] .$$

Two-sided preconditioning by multiplicatively splitting $M$ into $M = M _ {1} M _ {2}$ becomes

$$\tag{a17 } [ M _ {1} ^ {- 1 } A M _ {2} ^ {- 1 } ] [ M _ {2} x ] = [ M _ {1} ^ {- 1 } b ] .$$

Because the convergence rate of iterative methods generally depends on the condition number $\kappa ( A )$ of (a10), a rule of thumb has been to seek preconditioners $M$ such that the transformed system ${\widetilde{A} } {\widetilde{x} } = {\widetilde{b} }$ has lower, hopefully much lower, condition number $\kappa ( {\widetilde{A} } )$.

There are roughly three generally successful theories for finding preconditioners: incomplete factorizations, Frobenius-norm methods and relaxation methods. The incomplete factorizations approximate $A$ by its approximate factorization into lower and upper triangular factors, but insisting on sparse approximations by dropping small elements. Then a conjugate-gradient method or Krylov-space variant of it (e.g., schemes such as ORTHOMIN or GMRES) may converge well. The Frobenius-norm methods, roughly, seek a weight matrix $W$ such that the trace ${ \mathop{\rm Tr} } ( A ^ \top WA )$ is minimized. Often, an incomplete Cholesky factorization enters into applications of this method. Relaxation methods have been described above. Often, these are combined (see [a1]) with block Schur complement decompositions of $A$. Relaxation methods such as (a6), (a7), (a8) are used today chiefly as preconditioners. Even the lowly Jacobi method (a6) can be a practical preconditioner on highly parallel computers when appropriately implemented.

Acceleration methods are also used for the iterative solution of the eigenvalue-eigenvector problem

$$\tag{a18 } Ax = \lambda x.$$

(a18) may be regarded as a special case of (a1) in which $A \rightarrow A - \lambda I$ and $b = 0$. However, one must approximate the eigenvalue $\lambda$ as one is approximating the eigenvector $x$. Both Chebyshev and preconditioning acceleration methods enter into these (e.g., Arnoldi and Krylov) methods for eigenvalue-eigenvector computation. An important preconditioning method somewhat special to these problems is the so-called shift and invert method, using inverted shifts $( A - \sigma I ) ^ {- 1 }$ to more widely disperse the eigenvalues (see [a4]).

There are other useful variations on preconditioning for accelerating convergence. Conjugate gradient can be applied as a preconditioner for any of the relaxation schemes. Other methods, such as multigrid schemes, have been successfully used as preconditioners (see [a3]). Preconditioning strategies can be cast in terms of a new theory of anti-eigenvalues $\mu$ rather than the eigenvalue condition number $\kappa$( see [a2] and Anti-eigenvalue). Finding optimal preconditioners for an important specific application often usefully involves specific physical details of that application. For this reason, in practice preconditioning is sometimes labeled "black magic" .

#### References

 [a1] O. Axelsson, "Iterative solution methods" , Cambridge , New York (1994) [a2] K. Gustafson, "Lectures on computational fluid dynamics, mathematical physics, and linear algebra" , Kaigai & World Sci. (1996/7) [a3] W. Hackbusch, "Iterative solution of large sparse systems of equations" , Springer (1994) [a4] Y. Saad, "Numerical methods for large eigenvalue problems" , Halsted (1992) [a5] Y. Saad, "Iterative methods for sparse linear systems" , PWS Publishing (1996)
How to Cite This Entry:
Acceleration methods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Acceleration_methods&oldid=45013
This article was adapted from an original article by K. Gustafson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article


text:
Home Strings [Hackerrank] – Weighted Uniform Strings

# [Hackerrank] – Weighted Uniform Strings

0 comment

Let us try to simplify the problem as much as we can. We are given with some of the terms.

Weights:- According to the problem each character in the English alphabet has been assigned a weight. ‘a’ has been given a weight of 1, ‘b’ has a weight of 2, and so on. The weight of ‘z’ is 26. All the letters are assumed to be lowercase for simplicity.

Weight of a string:- This is defined as the sum of all the characters in the string. Example if a string consists of the characters
abcd = a(1) + b(2) + c(3) + d(4) = 10
study = s(19) + t(20) + u(21) + d(4) + y(25) = 89

Uniform Strings:- Strings with only one type of characters are termed as ‘uniform strings’. Some examples of uniform strings are ‘aaaa’, ‘bbbbbb’, ‘c’

Input String: A sample input string is given as ‘abccddde
It has the following contiguous uniform strings and weights
a = a(1) = 1
b = b(2) = 2
c = c(3) = 3
cc = c(3) + c(3) = 6
d = d(4) = 4
dd = d(4) + d(4) = 8
ddd = d(4) + d(4) + d(4) = 12
e = e(5) = 5
Thus we get the following weights {1, 2, 3, 6, 4, 8, 12, 5}

We are given a set U with some values. If a value can be formed by the weight of any uniform substring, the answer is ‘Yes’ else the answer is ‘No’

SOLUTION:

To start off with the problem, it is evident that we need to somehow find all the possible uniform strings in the given input string. But using a brute force approach would take a lot of time. A dynamic solution should be able to give the sum of a string at a constant time. A general approach in dynamic array is to create an array and keep on solving the problem successively.

Let us suppose we take an array equal to the size of the string.

We start off by writing the weight of ‘a’

Since the next character has changed, just put the weight of the new character.

Again the next character has changed. So just write the weight of the new character.

Now the next character is same. So we add the new weight to the weight of the previous cell. This ensures that we are having a sum of the contiguous string.

The character has changed again. So we just write the weight of new character.

The next character is same. So we add the weight of the next character to the previous cell

Again, the next character is same

The next character is different and hence we just write the weight of the new character.

Now in this array, we have all the possible weights of all the uniform strings that can be formed. Now we just need to seek the elements of the input set U. If any query does not belong in this array, then our answer will be  “No”, else our answer will be “Yes.”

Here in an implementation of the above algorithm.

class UniformWeights {

public static void main(String[] args) {
Scanner in = new Scanner(System.in);
String s = in.next();
int n = in.nextInt();

Set<Integer> sumList = new HashSet<>();
char prev = s.charAt(0);
int sum = prev - 'a' + 1;
for (int i = 1; i < s.length(); i++) {
char next = s.charAt(i);
if (next == prev) {
sum += (prev - 'a' +1);
} else {
prev = next;
sum = prev - 'a' + 1;
}
}

for (int a0 = 0; a0 < n; a0++) {
int x = in.nextInt();

if (sumList.contains(x))
System.out.println("Yes");
else
System.out.println("No");
}
}
}
Code language: Java (java)

A working implementation of the code can be found here.

This problem can be found at HACKERRANK here.

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text:
# A series expansion in terms of the polylogarithm function

1. Oct 25, 2012

### mmzaj

we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function $\phi(n,x)$, such that:
$$\int f(z)\phi(n,z)dz=a_{n}$$
or:
$$\int\text{Li}_{-n}(z)\phi(m,z)dz=\delta_{nm}$$
but that's about as far as i've gotten. any help is appreciated.
The question is motivated by the following:

suppose that for some analytic function $g(z)$ we have the values of the function at +ive integers, so we can write a Taylor development :
$$g(m)=\sum_{n=0}^{\infty}a_{n}m^{n}$$
now suppose that the following summation is convergent in some open disk:
$$\sum_{m=1}^{\infty}g(m)z^{m}$$
Using the above Taylor expansion, and the definition of the Polylogarithm function, we have :
$$\sum_{m=1}^{\infty}g(m)z^{m}=:f(z)=\sum_{n=0}^{\infty} a_{n}\text{Li}_{-n}(z)\;\;\left | z\right |<1$$
The plan is to recover the coefficients $a_{n}$, and the thus the Taylor expansion of $g(z)$

Last edited: Oct 26, 2012


text:
Review question

# Can we show $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-2}{r-1}+\binom{n-3}{r-1}+\dotsb+\binom{r-1}{r-1}$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8014

## Solution

1. Prove that $\sum_{r=0}^{n}\binom{n}{r}=2^n.$

Using the expression for the binomial theorem:

$(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^n\,b^{n-r},$

if we set $a=b=1$ then

$(1+1)^n=2^n=\sum_{r=0}^{n}\binom{n}{r}1^n\,1^{n-r}=\sum_{r=0}^{n}\binom{n}{r},$

as required.

1. By considering the number of ways of choosing a set of $n$ people from a set of $2n$ people, or otherwise, prove that $\binom{2n}{n}=\binom{n}{0}^2+\binom{n}{1}^2+\dotsb+\binom{n}{n}^2.$

The expression

$\binom{n}{r}=\frac{n!}{r!(n-r)!}$

gives the number of ways of choosing $r$ objects from a set of $n$ objects, without replacement.

If we want to choose $n$ objects from $2n$ objects, one way of doing this is to line up the $2n$ objects and simply pick $n$ objects from them.

The number of ways of doing this is then $\dbinom{2n}{n}.$

Alternatively we could split the $2n$ objects into two equal groups of $n$ objects, then choose $r$ objects from the first group and $n-r$ from the second.

Summing over all the possible values of $r=\{0,\,1,\,2,\,\dotsb,n\}$ will give the total number of ways of doing this as $\sum_{r=0}^{n}\binom{n}{r}\binom{n}{n-r}.$

But we know by symmetry that $\dbinom{n}{r} = \dbinom{n}{n-r}$, and so we have $\dbinom{2n}{n} = \sum_{r=0}^{n}\dbinom{n}{r}^2.$

1. Prove that $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}$

There are $\dbinom{n}{r}$ ways of choosing $r$ people from $n$ people.

If we now consider one specific person from among the $n$ people, say James, our set of $r$ people might include James or not.

If James is included, then there are $\dbinom{n-1}{r-1}$ ways of choosing the other $r-1$ people from the remaining $n-1$ people.

If James is excluded, then there are $\dbinom{n-1}{r}$ ways of choosing the $r$ people from the remaining $n-1$ people.

So there are $\dbinom{n-1}{r-1}+\dbinom{n-1}{r}$ ways in total of choosing $r$ people, hence

$\binom{n-1}{r-1}+\binom{n-1}{r}=\binom{n}{r}.$

Alternatively, we could start with the right hand side of the given expression. We have

\begin{align*} \binom{n-1}{r-1}+\binom{n-1}{r} &= \frac{(n-1)!}{(r-1)!(n-r)!}+ \frac{(n-1)!}{r!(n-r-1)!} \\ &= \frac{r(n-1)!}{r(r-1)!(n-r)!}+ \frac{(n-r)(n-1)!}{r!(n-r)(n-r-1)!} \\ &= \frac{r(n-1)!}{r!(n-r)!}+ \frac{(n-r)(n-1)!}{r!(n-r)!} \\ &= \frac{r(n-1)!+(n-r)(n-1)!}{r!(n-r)!} \\ &= \frac{n(n-1)!}{r!(n-r)!} \\ &= \frac{n!}{r!(n-r)!}=\binom{n}{r}. \end{align*}

$[...]$ and hence, or otherwise, prove that $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-2}{r-1}+\binom{n-3}{r-1}+\dotsb+\binom{r-1}{r-1},$ where $r$, $n$ are positive integers with $1 \leq r \leq n-1$.

Using the formula above repeatedly, we find \begin{align*} \binom{n}{r} &= \binom{n-1}{r-1}+\binom{n-1}{r} \\ &= \binom{n-1}{r-1}+\left[\binom{n-2}{r-1}+\binom{n-2}{r}\right] \quad \text{etc.} \\ \end{align*} We can see that as this builds up we always have $\dbinom{n-a}{r}$ as the last term, but we want to end with $\dbinom{r-1}{r-1}$. If we note that $\dbinom{r}{r}=\dbinom{r-1}{r-1}=1$ then we know to continue until $n-a=r$ so $a=n-r$. \begin{align*} \binom{n}{r} &= \binom{n-1}{r-1}+\binom{n-2}{r-1}+\dotsb+\left[\binom{n-(n-r)}{r-1}+\binom{n-(n-r)}{r}\right] \\ &= \binom{n-1}{r-1}+\binom{n-2}{r-1}+\dotsb+\binom{r}{r-1}+\binom{r}{r} \\ &= \binom{n-1}{r-1}+\binom{n-2}{r-1}+\dotsb+\binom{r}{r-1}+\binom{r-1}{r-1} \end{align*}


text:
logcondens (version 2.1.6)

# logConROC: Compute ROC curve based on log-concave estimates for the constituent distributions

## Description

The receiver operating characteristic (ROC) curve for two constituent distributions $$F$$ and $$G$$ is defined as

$$R(t; F, G) = 1 - G(F^{-1}(1 - t))$$

for $$t \in [0, 1]$$. It is typically used to assess the performance of a diagnostic test used to discriminate between healthy and diseased individuals based on a continuous variable.

## Usage

logConROC(cases, controls, grid, smooth = TRUE)

## Arguments

cases

A vector of measurements for the cases.

controls

A vector of measurements for the controls.

grid

A vector specifying the grid where the ROC curve is computed on.

smooth

Logical, indicating whether ROC curve and AUC should also be computed based on the smoothed log-concave density estimator.

## Value

m

Number of control measurements.

n

Number of case measurements.

fROC

Estimated ROC curve based on the log-concave density estimate.

fROC.smooth

Estimated ROC curve based on the smoothed log-concave density estimate.

res0

dlc object as a result of a call to logConDens for the data of the controls.

res1

dlc object as a result of a call to logConDens for the data of the cases.

## Details

In Rufibach (2011) it was shown that the ROC curve based on log-concave density estimates exhibit nice properties for finite sample sizes as well as asymptotically. Its performance is typically much better than that of the empirical ROC curve and only, if at all, sligthly worse compared to the binormal model when in fact the underlying densities are normal. However, log-concavity encompasses many parametric densities, so this new model is much more flexible than the binormal one, at little efficiency sacrifice.

## References

Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40--68.

Duembgen, L. and Rufibach, K. (2011). logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1--28. https://doi.org/10.18637/jss.v039.i06

Rufibach, K. (2012). A smooth ROC curve estimator based on log-concave density estimates. Int. J. Biostat., 8(1), 1--29.

Confidence intervals at given false-positive fractions for the ROC curve based on log-concave densities can be computed using confIntBootLogConROC_t0. For the computation of the AUC the function ROCx is used. In the example below we analyze the pancreas data.

## Examples

Run this code
# NOT RUN {
## ROC curve for pancreas data
data(pancreas)
status <- factor(pancreas[, "status"], levels = 0:1,
labels = c("healthy", "diseased"))
var <- log(pancreas[, "ca199"])
cases <- var[status == "diseased"]
controls <- var[status == "healthy"]

## compute and plot empirical ROC curve
## code modified from https://stat.ethz.ch/pipermail/r-help/2008-October/178531.html
xx <- c(-Inf, sort(unique(c(cases, controls))), Inf)
sens <- sapply(xx, function(x){mean(cases >= x)})
spec <- sapply(xx, function(x){mean(controls < x)})

## compute log-concave ROC curve
grid <- seq(0, 1, by = 1 / 500)
roc.logcon <- logConROC(cases, controls, grid)

## plot
plot(0, 0, xlim = c(0, 1), ylim = c(0, 1), type = 'l',
main = "ROC curves for pancreas data", xlab = "1 - specificity",
ylab = "sensitivity", pty = 's')
legend("bottomright", c("empirical ROC", "log-concave ROC", "smooth log-concave ROC"),
lty = c(1, 1, 2), lwd = 2, col = 2:4, bty = "n")
segments(0, 0, 1, 1, col = 1)
lines(1 - spec, sens, type = 'l', col = 2, lwd = 2)
lines(grid, roc.logcon$fROC, col = 3, lwd = 2) lines(grid, roc.logcon$fROC.smooth, col = 4, lwd = 2, lty = 2)

# }
# NOT RUN {
## bootstrap confidence intervals at 1 - specificity = 0.2 and 0.8:
res <- confIntBootLogConROC_t0(controls, cases, grid = c(0.2, 0.8), conf.level = 0.95,
M = 1000, smooth = TRUE, output = TRUE)
res
# }

Run the code above in your browser using DataCamp Workspace


text: • Download the latest release HairNet.zip instead of the entire repository: github.com/Jandals/HairNet/releases from github. Also please provide a link to the add-on next time so we don't have to search for it. – brockmann Mar 9 '20 at 16:04


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# Large Numbers, Infinities & God (2/3)

This post is continuing on from my discussion on large numbers and notation. I am going to assume some knowledge of Set Theory for this post as there really is a lot to write about and I have found it hard to cut it down to a nice size.

2. Infinities
Infinity as a concept has scared mathemeticians for a long time, not only was it hard to get your head around, it wasn’t really usable. A recent use is in calculus as, what I would call, a nominal infinity (‘increases without bound’ as opposed to a value or position): $\displaystyle{\lim_{x \to \infty}}.$ Leibniz was facinated by infinity and it’s counterpart infintesimal $\frac{1}{\infty} = 0.000\dots 1.$

The Isha Upanishad, a Hindu scripture, gives what I think is the polite version of infinity: “if you remove a part from infinity or add a part to infinity, still what remains is infinity” – this philosophical statement explains a mathematical proposition, cardinal arithmetic, and it comes from a branch of mathematics called set theory developed by Georg Cantor and Richard Dedekind in the 1870s.

Cardinal infinites (Aleph numbers):
Say we have a set that holds all the natural numbers: $\boldsymbol{\mathbb{N}} = \{0,1,2,3,4,...\}$ what is the size of such a set? Well $\boldsymbol{|\mathbb{N}| = \aleph_0}$ (aleph-null) the smallest possible infinity and the first of the Aleph numbers. We just delt with the cardinality of the natural numbers, now lets move on to the rational numbers $\boldsymbol{\mathbb{Q}} ,$ these are all the numbers that can be expressed as $\frac{n}{m}$:

It turns out $| \boldsymbol{\mathbb{Q}} | = \aleph_0$ but how?
By definition a set S is countable if there exists an injective function: $f: S \to \mathbb{N}$ from S to the natural numbers. The fractions would seem uncountable in a linear sense, but Georg Cantor came up with a method for one-to-one pairing with the natural numbers. (See right)

The final task is the real numbers $, \boldsymbol{\mathbb{R}} ,$ this contains every possible finite number. So it has 0.000001, √2, e, π, 55, 73896737483 and everything else! This set has no possible one-to-one pairing with the natural numbers. Cantor put forward something called the Continuum Hypothesis that stated no such set exists with a cardinality between that of $\boldsymbol{\mathbb{N}} \text{ and } \boldsymbol{\mathbb{R}} .$ Although the cardinality of the real numbers is stated as: $| \boldsymbol{\mathbb{R}} | = \mathfrak c = 2^{\aleph_0} > \aleph_0 \,$ the Continuum Hypothesis implies: $| \boldsymbol{\mathbb{R}} | = \aleph_1 .$ Gödel and Cohen later showed that the hypothesis can neither be disproved nor be proved in ZF set theory.

In cardinal arithmetic  $\aleph_0 + n = \aleph_0,$ $n \aleph_0 = \aleph_0,$ ${\aleph_0}^n = \aleph_0,$   just like in the Hindu scripture.

Beth numbers:
The Beth numbers run parrallel to the Aleph numbers and are the successive cardinalities to power sets of the natural numbers. For an example of a power set, see: $S=\{x,y,z\}$ it’s power set is: $P(S) = \left\{\{\}, \{x\}, \{y\}, \{z\}, \{x, y\}, \{x, z\}, \{y, z\}, \{x, y, z\}\right\}\,\!.$

$\beth_0 = \aleph_0$
$\beth_{n+1} = 2^{\beth_n}$
$| P(\mathbb{N}) | = \beth_1$
$| P(P(\mathbb{N})) | = \beth_2$ etc..

These numbers are related to the Generalised Continuum Hypothesis.

Ordinal infinities:
In set theory a set which is well-ordered (a is less than b which is less than c, etc) has an ordinal number, the smallest ordinal infinity is omega:

$\omega = \{0<1<2<3<4<...\}$ and $| \omega | = \aleph_0$

To get the cardinality of an ordinal infinity we ignore the order, this will become important when we assess ordinal arithmetic. For example here are three ordinal additions:

$\omega + \boldsymbol{1} = \{0_0<1_0<2_0<3_0<4_0<...\boldsymbol{0_1}\}$

$\boldsymbol{1} + \omega = \{\boldsymbol{0_0} < 0_1<1_1<2_1<3_1<4_1<...\} = \omega \text{ (after relabeling the elements)}$

$\omega + \boldsymbol{\omega} = \{0_0<1_0<2_0<3_0<4_0<...< \boldsymbol{0_1<1_1<2_1<3_1<4_1<...}\}$

The difference between ω+1 and 1+ω is the placing of the dots, as they imply ad infinitum. In ω+1 there is no direct predecessor before the second 0, just infinity dots.
Now although ω, ω+1 and ω+ω are three seperate ordinal values – they all have the cardinality $\aleph_0$ and this is because when we get the size, we ignore the order. It also stands that n·ω = ω ≠ ω·n. The first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω is the supremum of all countable ordinals. The elements of ω1 are the countable ordinals, of which there are uncountably many:

ω1 = sup{ ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, … }

ωn = sup{ ωn-1, ωn-1 + 1, ωn-1 + 2, …, ωn-1·2, ωn-1·2 + 1, … } and $| \omega_n | = \aleph_n.$

Surreal numbers:
The surreal number system was thought up by John Conway, it contains the real numbers as well as infinite and infinitesimal ordinal numbers. They are also known as the extraordinal numbers. They have fathoms of interesting properties but if I carry on i’ll never stop. For more on these amazing numbers I recommend the book ‘Surreal Numbers‘ by Donald Knuth.

$\aleph * \infty * \omega$

1. > This set has no possible one-to-one pairing with the natural numbers so it is known as uncountable: $| \boldsymbol{\mathbb{R}} | = \aleph_1$
Sorry for nitpicking, but $|\mathbb{R}| = 2^{\aleph_0} = \mathcal{c} = \aleph_1$ only if you accept Continuum hypothesis


text:
@Johnsmith1 2022-07-28T11:20:50.000000Z 字数 4018 阅读 10

# Figured out book reference formatting guide in APA - 2022

There are various types of attempts that are assigned to understudies for their assessment which likewise solidifies writing assignments . Such undertakings are writing a figured out list that expects understudies for study some assessment based source. It is finished to permit understudies to see the value in what is the support behind writing an assessment paper and what are the objectives that particular sources will consistently accomplish. Writing a got a handle on reference list is a stunning endeavor since it requires investigating total assessment and, in this way, disengaging the focal thought out of it.

To have your assignment managed with near no pressure of course tolerating you are fresher and you accept time ought to comprehend different instructive undertakings, I would recommend you utilize any essay writing service by mentioning that they write my essays for me . The service suppliers will ensure that you are getting unimaginable inscriptions and also your assignment is finished inside the given cutoff time. I have likewise taken help from such services and each time the outcomes were basically awesome.

One of the most stunning benefits of utilizing such a service is that sometimes they offer you custom writing with first class quality and importance. I think it is one of the fundamental benefits that you can eliminate from an essay writing service nearby the help concerning finishing the job in the given time and with importance.

No matter what the way that utilizing any such service is your own decision, I can assist you with Annotated Bibliography since I know some especially fast tips and contemplations that can assist you with making an all-out as well as pre-arranged writing approach.

I would ask you to include any cheap essay writing service in the event that you truly need your assignment finished without pressure, again on the off chance that you are another understudy and need some time to figure out different instructive assignments. The service suppliers will ensure that you get passing marks and that your assignment is done by the cutoff time. I've utilized basically indistinguishable services before, and each time the outcomes were immaculate.

1-The basic step is to develop information on the asset. It means that you ought to have the decision to understand the meaning of the examination and it ought to be accomplished tolerating you have investigated the source totally. In the event that you acknowledge that you truly need extra time or it is time taking, I can propose you read the speculative carefully and sometime later use looking at frameworks, for example, skimming and checking to examine the remainder of the examination paper. Trust me this framework will work.

2-Then, you really need to dispose of the fundamental thought mentioned in the examination. It very well may be recovered tolerating you will incorporate the significant thoughts and the focal lines while inspecting the examination paper. The focal thought can be a solution for the solicitation or the examination questions and, when in doubt, it is mentioned in the end segment and if can't dispose of the main thought, find support from a best essay writing service .

3-After having known the focal thought, relate it to the subject of the essay or the smart writing that you will do. It will assist you with recognizing how could this source add to the undertaking and it is all you really need to mention while writing a commented-on reference list.

4-Coming to the references and the APA formatting, you basically have to make a reference list according to the APA format and the additional segment will be normal like different references. Attempt to imply the references and a brief time frame later add page number in the event that it is mentioned or you know the specific number as it will assist with expanding the substance weightage of you got a handle on book record.

5-Completing this enormous number of steps, you are nearly finished your assignment. Attempt to adjust the substance and on the off chance that it is mentioned in the standards, just add the goals or the upsides of the examination paper in basically the same manner as adding to your assessment. It will assist with showing that you have investigated the source with thought and you have the fundamental subject information on the source without mentioning to any essay writer to write my essay for me . In the limits and benefits write about the Of course in the event that the scientist can accomplish the objective of the assessment methodology or not.

While writing such cases, remember the tips and formats which are given, by pro writers of SharkPapers . Follow the model and write it as indicated by your own comprehension.

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# why is the DFS of a delta function equal to 1

I have a x[n] = $\delta$[n]. By formula is should be

$$X[k]= \sum_{n=0}^{N-1} \delta[n]W_N^{kn} X[k]= \sum_{n=0}^{N-1} e^{-j2*pi*kn/N}$$

The formulae isn't showing for some reason. I took a screenshot of what I got here: http://imgur.com/6j0Ibgu

basically why is the summation of an exponential term going to 1. $W_N=e^{-j*2*pi*k*n/N}$ in this case. I tried to prove it via $\sum\alpha^k=\frac{1-\alpha^N}{1-\alpha}$ but it doesn't work for me.

$X[k]= \sum_{n=0}^{N-1} \delta[n]W_N^{kn} \quad$, where $\>W_N^{kn}=e^{−j\,2\pi k\,n/N}$
What is the value of $\delta[n]$ when $n \neq 0 \>$ ?


text:
## Wednesday, December 11, 2019

### Di-block copolymer analysis: GPC

Gel Permeation Chromatography is a well known method of measuring molecular weight distribution of polymers. For di-block copolymers, a proper calibration curve can be constructed by using Mark-Houwink parameters of homopolymers (see here). Usually, di-block copolymers are obtained by synthesis one block first then "grow" the other block from the end of preceding block. The molecular weight distributions of final di-block copolymer and preceding block are accessible by GPC, the molecular weight distribution of the 2nd block, however, is hard to obtain.

Here I present a simple method to calculate the molecular weight distribution of 2nd block by GPC data of preceding block and final di-block copolymer. Assuming that we already have the molecular weight distribution of the preceding block, $P_1(n)$, it is generally that the length of the 2nd block "grow" from the depends on the preceding chain length $n$, therefore, the pdf of the 2 blocks should be a joint pdf $Q(n, m)=P_1(n)P_2(m, n)$, and the pdf of the final di-block copolymer is straightforward:
$$P(x)=\int Q(n, x-n) \mathrm{d}n$$
However, it is reasonable to assume that the $P_2(m, n)$ can be expressed as some $f(m)g(n)$ form by dropping some higher order correlations between $m,n$; hence, the $g(n)$ can be considered as a "fixing parameter" on the distribution of $P_1(n)$, the calculation of $P(x)$ is therefore a convolution. The simplest case is $g(n)=1$, which means the $P_1(n)$ and $P_2(m)$ are independent, the growth of 2nd block on the end of the 1st block is not effected by the length of the preceding block; or $g(n)\sim n^{-1}$ by considering the diffusion of the preceding block: small preceding chains tend to "grow" more 2nd block.

The evaluation of pdf of the 2nd block follows 3 steps:

1. Determine the range of molecular weight: $(x_{min}-n_{max}, x_{max}-n_{min})$, where $x$ is the chain length of the di-block copolymer and $n$ is the length of the preceding block;
2. Interpolate the distributions obtained from GPC into equally spaced molecular weight, negative part is simply 0;
3. Deconvolute $P(x)$ with $P_1(n)$.


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# Property: universally catenary

Definition: All finitely generated algebras over $R$ are catenary

Nothing was retrieved.


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Insert a Row - Maple Programming Help

Home : Support : Online Help : Create Maple Worksheets : Edit Maple Spreadsheets : worksheet/spreadsheet/insertrows

Insert a Row

 Insert a row in a Maple spreadsheet.
 1 Select a cell below the location where you want to insert a new row.
 2 Right-click (Control-click for Macintosh) on the cell, and then select Row > Insert.


text:
# Statistics and Statistical Programming (Winter 2017)/Problem Set: Week 5

## Programming Challenges

PC0. I've provided the full dataset from which I drew each of your samples in a TSV file in the directory week_05 in class assignment git repository. These are tab delimited, not comma delimited. TSV, is related to CSV and is also a common format. Go ahead and load it into R (HINT: read.delim()). Take the mean of the variable x in that dataset. That is the true population mean — the thing we have been creating estimates of in week 2 and week 3.
PC1. Go back to the dataset I distributed for the week 3 problem set. You've already computed the mean for this in week 2. You should compute the 95% confidence interval for the variable x in two ways:
• (a) By hand using the normal formula for standard error ${\displaystyle ({\frac {\sigma }{\sqrt {n}}})}$.
• (b) Using the appropriate built-in R function. These number should be the same or very close. After reading the OpenIntro, can you explain why they might not be exactly the same?
• (c) Compare the mean from your sample — and your confidence interval — to the true population mean. Is the true mean inside your confidence interval?
PC2. Let's look beyond the mean. Compare the distribution from your sample of x to the true population. Draw histograms and compute other descriptive and summary statistics. What do you notice? Be ready to talk for a minute or two about the differences.
PC3. Compute the mean of y from the true population and then create the mean and confidence interval from the y in your sample. Is it in or out?
PC4. I want you to run a simple simulation that demonstrates one of the most fundamental insights of statistics:
• (a) Create a vector of 10,000 randomly generated numbers that are uniformly distributed between 0 and 9.
• (b) Take the mean of that vector. Draw a histogram.
• (c) Create 100 random samples of 2 items each from your randomly generated data and take the mean of each sample. Create a new vector that contains those means. Describe/display the distribution of those means.
• (d) Do (c) except make the items 10 items in each sample instead of 2. Then do (c) again except with 100 items. Be ready to describe how the histogram changes as the sample size increases. (HINT: You'll make me very happy if you write a function to do this.)
PC5. Do PC4 again but with random data drawn from a normal distribution (${\displaystyle N(\mu =42,\sigma =42)}$) instead of a uniform distribution. How are you results different than in PC4?

## Statistical Questions from OpenIntro §5

Q0. Any questions or clarifications from the OpenIntro text or lecture notes?
Q1. Exercise 5.16 which is a set of True/False questions
Q2. Exercise 5.28 which is about Diamonds
Q3. Exercise 5.30 which is also about Diamonds
Q4. Exercise 5.48 which is about work hours and education
Q5. Exercise 5.52 which is another set of True/False questions about ANOVA

## Questions on the Empirical Paper

These are all in regards to the Sweetser and Metzgar paper.

Q6. For RQ1 explain:
• (a) What is the unit of analysis? What is the dependent variable? The independent variable? What are the levels or groups of being compared in the ANOVA?
• (b) Clearly State the null hypothesis being tested. What is the alternative hypothesis?
• (c) Summarize ore restate results in statistical terms. Explain what these results mean in substantive terms? How convincing do you find these results? What should we be taking away?
Q7. Do the same as above but for RQ4.
Q8. ...for RQ5.
Q9. ...for RQ6.

## Questions on Gelman and Stern Paper

Q10: First, walk us through the result visualized in Figure 1. Explain and interpret the result for us. Now go back to the blockquote on page 329 and, by referencing the figure, explain why Gelman and Stern think that this is a good example to illustrate their point about the difference between statistically significant versus non-significant.
Q11: Move on to the study about EMF. Walk us through Figure 2. First explain the basic result and then explain why Gelman and Stern thinks that Figure 2b is better than 2a.
Q12: In the paper's abstract Gelman and Stern describe their approach as different from three other problems: that statistical significance, that dichotomization of significant/not-significant encourages dismissing observed differences, and that thresholds are arbitrary. Summarize why these are important issues in your own words (and ideally, with examples) and explain how Gelman's key critique is different.


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# pyspark.streaming.DStream.countByValue¶

DStream.countByValue()[source]

Return a new DStream in which each RDD contains the counts of each distinct value in each RDD of this DStream.


text:
Glossary
Market
Term
Text
Perpetual Futures
Perpetual futures are futures contracts with no expiration date. The “perpetual” status is maintained by paying/receiving the funding fee.
Everlasting Options
Everlasting options are the equivalent of perpetual futures for options which gives traders long-term options exposure without the effort, risk, or expense of rolling positions.
Power Perpetuals
A power perpetual is a perpetual derivative indexed to a power of the price of some underlying instrument. First of all, we specifically implemented the case of
$p=2$
, i.e. the perpetuals of BTC^2 and ETH^2. Since the square of BTC price is a huge number, we downscale it to a smaller unit - milli (prefix for 1/1000). Namely, the object being traded on Deri is mBTC^2 and mETH^2, which correspond to 1/1000 of BTC^2 and ETH^2, respectively.
Base Token
Base tokens are the tokens accepted by Deri Protocol as contributed liquidity by the liquidity providers or as collateral by the traders.
Daily Funding
This is the fund fee converted to a time unit of one day. The funding that one long contract pays one short contract is always accrued on a second basis, so daily funding really means the following:
Daily funding = funding rate per second * 864000,
where 864000 is the number of seconds per day.
Please note that a positive funding fee means that longs are paying shorts, while a negative funding fee means that shorts are paying longs.
Delta (Everlasting Options Only)
The ratio of option price change to the underlying price change. Mathematically, Delta is the first-order differential of the option price as a function of the underlying price.
Funding Period
For perpetual futures, Funding period is the time period for which the funding fee (MARK-INDEX) is paid.
For everlasting options, Funding period is the time period for which the funding fee (MARK-PAYOFF) is paid. For example, if for everlasting options Funding Period = 7 days, then every second a long (short) contract pays (receives) a funding fee = (MARK-PAYOFF)/72*4*60*60
Gamma (Everlasting Options Only)
The ratio of option Delta change to the underlying price change. Mathematically, Gamma is the second-order differential of the option price as a function of the underlying price.
Index Price
Spot price of the underlying asset. The Index Prices on Deri Protocol are provided by Oracle.
Mark Price
Price of the market, given by Deri Proactive Market Making (DPMM). This is the trading price of a trade with an infinitely small trading volume (i.e. without slippage) at the current moment.
The volume of any trade (in terms of notional) must be an integer multiple of Min Trade Unit.
Settlement Token (Base0)
Among all the base tokens, Base0 is used as the accounting unit and to settle the PnL. Essentially, this token plays the role of “cash” in the trading business.
Theoretical Price of Everlasting Options
The theoretical price of everlasting options calculated by the model. This is used as the starting point of the DPMM (in place of the Index Price for the futures DPMM).
Total Net Position
Total long positions minus total short positions for the current trading symbol. If positive (negative), traders’ total positions on the long side are more (less) than the short side, which means the pool is passively taking a short (long) position.
The average price paid when trade is executed
User Account
Term
Description
Available Margin
The part of funds in your margin account that is not frozen as collateral for the open positions and thus available to withdraw.
Dynamic Effective Balance
Dynamic Effective Balance = Total discounted margin + Unrealized Pnl+ Accrued Funding Fee
Total discounted margin =∑ base token balance *base token price * discount factor
Please note that the Unrealized PnL is calculated by the mark price, which might be different from the realized pnl when the position is actually closed (due to the slippage).
Initial Margin Ratio
The percentage of the notional value that you must cover with collateral when opening a new position.
Maintenance Margin Ratio
The percentage of the notional value that you must cover with collateral to keep your open positions from being liquidated.
Margin Usage
The part of funds in your margin account that is frozen as collateral for the open positions and thus unavailable to withdraw. Margin Usage is determined by Initial Margin Ratio.
Mining Pnl
Mining Pnl is a liquidity provider’s PnL in terms of the base token that is caused by the trading activities on the involved liquidity pool. Specifically,
Mining Pnl = 20% of Transaction Fee + Funding Fee + Shared remaining margin in liquidated Positions + Traders’ loss (-Traders’ profit)
Please note that the following three parts of income are NOT included in the Mining PnL: 1. the DERI token reward
2. the interests accrued from the money market protocol (e.g. Venus) 3. the token reward by the money market protocol (e.g. XVS)
Refer to Mining for more details.
Position
Term
Description
Accrued Funding
The part of the funding fee that is accrued (i.e. recorded) but not settled yet. It will be settled at the next settlement action (usually taking place with your next trading action). If positive, the pool owes you; otherwise you owe the pool.
Entry Price
The average price for establishing the current position. If the position is established with multiple trades, then this equals the total cost of all the trades divided by the position volume.
Liq. Price(est.)
The estimated liquidation price. If the index price of the contract reaches below the liquidation price (when long) or above (when short), the trader’s account will be liquidated.
Unrealized PnL
The profit or loss that would be realized if the position were closed at the moment.