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http://mathhelpforum.com/differential-equations/206950-showing-existence-nontrivial-solution.html
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# Thread: showing existence of nontrivial solution
1. ## showing existence of nontrivial solution
Show that the differential equation $x' = \frac{x \sin(e^x+t)}{1+(e^t \cos x+x)^2}, x(1) = 1$ has a nontrivial solution $\phi(t)$ defined on
$[0,2]$ such that $0 < \phi(t) < \frac{\pi}{2}$ for all $t \in [0,2]$.
I only know that x = 0 is a trivial solution. Then how do I proceed?
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https://nbickford.wordpress.com/2010/05/19/cellular-automata/
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# Cellular Automata
Cellular automata are simulations on a linear, square, or cubic grid on which each cell can be in a single state, often just ON and OFF, and where each cell operates on its own, taking the states of its neighbors as input and showing a state as output.
One of the simplest examples of these would be a 1-dimensional cellular automaton in which each cell has two states, ON and OFF, which are represented by black and white, and where each cell turns on if at least one of its neighbors are in the ON state. When started from 1 cell, this simply creates a widening black line. When the layers are shown all at once, though, you can see that it makes a pyramidal shape.
All layers at once
For example, in the figure above, the second line is generated from doing the rule for all cells in the first line, the third line from the second line, and so on. More complicated figures can be generated from different rules, such as a cellular automaton in which a cell changes to ON if either the cell to it’s top left or top-right is ON, but not if both are on. This creates a Sierpinski Triangle when starting from a single cell:
Stephen Wolfram developed a numbering system for all cellular automata which base only on themselves, their left-hand neighbor, and their right-hand neighbor, often called the elementary cellular automata, which looks something like this for the Sierpinski Triangle automata (Rule 18):
This code has all possible ON and OFF states for three cells on the top, and the effect that it creates on the cell below them on the bottom. Using this system, we can find that there are 256 different elementary cellular automata. We can also easily create a number for each automaton by simply converting the ON and OFF states at the bottom to 1s and 0s, and then combining them to make a binary number (00010010 in the Sierpinski Triangle example). Then, we convert the binary to decimal and so get the rule number. (128*0+64*0+32*0+16*1+8*0+4*0+2*1+1*0= 18 for the example). We can also do the reverse to get a cellular automata from a number. Using this method, we can create pictures of all 255 elementary cellular automata:
Some of these are rather interesting, such as Rule 30 and Rule 110:
Rule 30
Rule 110
Whilst some are rather boring, such as Rule 0, which is just white, or Rule 14, which is a single diagonal line.
There are many variations on this basic cellular automata type, such as an extension of the code where next-nearest neighbors are also included. This results in 4294967296 different cellular automata, a few of which appear to create almost 3-dimensional patterns such as the 3D Tetrahedrons cellular automata (rule 3283936144 ) which appears to show certain tetrahedral-ish shapes popping out of a plane.
There are also totalistic cellular automata, which are created by basing the next cell somehow on the average of the top-left, center, and top-right cells above it. These can have more than two states, and sometimes produce very strange-looking patterns, such as Rule 1599, a 3-state cellular automata:
As well as all these, there are continuous-valued cellular automata, which, instead of having cells that can only be in certain states, have the cells have real-number values. Then, at every step a function is applied to the cell that is to be changed as well as it’s neighbors. A good example of this is the Heat cellular automaton, in which the function is ((left_neighbor+old_cell+right_neighbor)/3+ a number between 0 and 1) mod 1). It produces a “boiling” effect, in which it resembles a pot of water slowly boiling on an oven.
There are tons more 1-dimensional cellular automata; Stephen Wolfram filled most of an entire (1200 page) book with these. However, there are essentially only 4 classes of cellular automata. The first type is the most boring; it is where the cellular automata evolves into a single, uniform state. An example of this would be the Rule 254 elementary cellular automata (the first example), which eventually evolves into all black. The second type, repetition, is a little more interesting, as it does not evolve into a single state but is instead repetitive. This can include a single line, simple oscillation, or fractal-like behavior, an example of which would be Rule 18. The third type is simply completely chaotic behavior- not very interesting, but definitely more than the previous two- such as in Rule 30. The last type, type 4, is where there are many individual structures that interact, sometimes passing right through, other times blowing up. An example of this would be Rule 110. This type is probably the most interesting to watch, as the eventual outcome is unknown.
These 4 types cover nearly any cellular automata, except for the ones which appear to be midway in between the types.
We can easily go past 1-dimension and study two-dimensional cellular automata. Probably the most famous of these is Conway’s Game of Life, invented by John Conway in 1970. In it, clusters of cells appear to grow, and then collapse as “gliders” move across the screen. It only uses 4 rules, and easily falls into the category of Class 4 cellular automata.
The rules are:
1. Any live cell with less than 2 neighbors dies. (starvation)
2. Any live cell with more than 3 neighbors dies. (overcrowding)
3. Any live cell with 2 or 3 neighbors stays alive.
4. Any dead cell with three live neighbors becomes alive (birth)
Here, the neighborhood of a cell is defined as the 8 cells that surround it.
When the Game of Life was first shown, tons of people went crazy writing programs for simulating it on computers, and supposedly thousands of hours of computer time were “wasted” simulating these patterns. One worker at a company even installed a “Boss” button for switching the display from Life to whatever he was supposed to be working on when his boss walked by! Conway had offered a \$50 dollar prize to whoever could find a pattern that expands infinitely. This could be a sort of glider gun, which shoots out gliders, a puffer, that leaves a trail of debris, or a spacefiller which expands out in all directions. The prize was claimed by Bill Gosper when he discovered the Gosper Glider Gun.
Since then, lots of new patterns have been discovered in the Game of Life, such as a puffer train, a hexadecimal counter, a fractal-generator, and even a “computer” which will do practically anything it is programmed to do.
Parts of the Life Computer
There are many other 2-dimensional cellular automata, which can be written in a certain notation which tells with which neighbor-numbers the dead cell turns alive, and for what neighbor-numbers the live cell stays alive. For example, Conway’s Game of Life could be written as B3/S23 . Many other cellular automata can be written using this notation. Some of the more interesting ones are Fredkin’s automaton (B1357/S02468) , which replicates any starting pattern. That’s all it does, no exceptions, so there’s no possibility of making anything like an adder in it. Another interesting one is the “Maze” rule (B3/S12345) , which produces maze-like patterns. Changing the rule to B37/S12345 creates dots that move through the shape. One of the most interesting of these, though, is 2×2 Life (B36/S125) , a rule that is similar in character to Life but has much different patterns. Gliders are also a bit more rare, although there are a lot of interesting oscillators. In rules like these, such as Day & Night (B3678/S3478) it makes almost no difference whether the colors are reversed. Day & Night also, at the end of patterns, has lots of oscillators.
Naturally, you can extend this form to allow multiple states. Brian’s Brain (/2/3) is an example of this, in which there are three states, and in which gliders and glider guns are very much common. In fact, Still Lifes are almost nonexistent! The notation above means that a cell in state 1 (and only in state 1) stays alive if it has (null) neighbors, that a dead cell becomes a state 1 cell if it has 2 neighbors, and that there are 3 states (0,1,2) .
A typical simulation
There are many modifications of this rule, one which causes scaffold-like structures to form, and even one which combines with Conway’s Game of Life!
You can easily make your own rules by simply choosing numbers to put in. Many of them appear to just be chaotic, but you can find rules which create rather interesting patterns. A good one is the Star Wars cellular automaton, 345/2/4 , which starts out like the Brian’s Brain rule but soon creates structures which shoot out gliders. A fun thing to do in this rule is to make “Train tracks” which let 1×3 rectangles move around them in both directions. Of course, you can also simulate all of the Life-ish rules by changing the number of states to be 2, so that there are only ON and OFF states.
As if all this weren’t enough, there’s even a generalization of the previous into arbitrarily many rules for arbitrarily many states, as a rule table. Basically, the rules are based on a large table that tells the cell in a certain state to change to a different (or the same) state if it has <this> many live neighbors. The different rules for each state makes it easy to get the cellular automaton to do exactly what you want it to do. A good example of this type of rule is the Wireworld cellular automaton, invented by Brian Silverman, in which electrons travel down wires simulating the connections in a computer. It’s easy to make a 1-way gate, an AND gate, a clock, a NOT gate… and nearly everything you’d need to create a computer. In fact, Mark Owen even made a wireworld computer that calculates and displays the prime numbers!
Amazing when actually run.
Rudy Rucker has also made a lot of Rule Table cellular automata, one of the most interesting being his Cars cellular automaton, which produces racing cars in several types, not usually something you’d expect to see from a cellular automaton. The cars also crash into each other, and, in the process, make rather strange cars.
I have also made an interesting cellular automaton, which only uses 2 states, but still shows interesting behavior on wrapped grids, called SkyscraperMakers. In it, large structures are easily made, and there is a very simple puffer which requires only 6 cells. Signals also appear to transfer through the structures, but mostly just lower the towers.
There are also cellular automaton rules where only 1 cell is actually active at any one time. An example of this is the Langton’s Ant cellular automaton, in which the moving cell has two rules:
1. If you are on a white square, turn right, flip the color of the square from white to black, and move forward one square.
2. If you are on a black square, turn left, flip the color of the square from black to white, and move forward one square.
Although this seems very simple, when the cellular automaton runs on a blank grid the pattern produced is rather chaotic. In fact, you have to wait around 11,000 steps until the “ant” produces a “highway” in which the ant repeats the same pattern over and over.
The first 200 steps of Langton's Ant
Naturally, there’s a generalization to multiple states and different rules, in which you simply tell the ant what to do when it touches a certain state. It is usually expressed using a string of Rs and Ls to show what direction the ant takes when it touches a certain-colored cell. For example, the classic Langton’s Ant rule could be expressed as RL, meaning that it turns right when it touches a cell of state 0 (white), and turns left when it touches a cell of state 1. Using this generalization, there are some rather interesting cellular automata. For example, LLRR makes a cardiod shape:
Whilst one of the longer rules, LRRRRRLLR fills space around itself in a square.
Naturally, the infinity of 1-dimensional and 2-dimensional cellular automata wasn’t enough for some people, who proceeded on to 3-dimensional cellular automata. The notation for these is similar to the normal Life notation (i.e., B (something)/S (something)), except that the numbers go from 0 to 26 instead of from 0 to 8. There are some interesting analogs of 2d cellular automata, such as Brian’s Brain, which have been discovered (B4/S) :
As well as some new rules, such as the “Clouds” rule (B13,14,17,18,19 /S13,14,15,16,17,18,19,20,21,22,23,24) in which random patterns quickly form cloud-like blobs and bridges between the blobs. The “clouds” eventually shrink down, sometimes to nothing but sometimes forming rather simple oscillators:
There has even been a version of Life in 3D, however, it turns to simple oscillators very quickly. Supposedly, gliders can be formed, but I haven’t seen any.
The problem with 3D cellular automata, though, is that computer screens are 2-dimensional. When a computer screen displays a picture of a 3D cellular automata, the front (that we see) may be rather dull, while the other side may be very chaotic, but we wouldn’t know the difference. Also, there may be lots of action inside a blob, but we can’t see what is happening inside.
An interesting way to make a 3-dimensional shape out of a cellular automata is to simply stack all the stages of a 2-dimensional cellular automata on top of each other. This makes the cellular automaton seem quite a bit different. Patterns like the Gosper Glider Gun in Conway’s Game of Life turn into a tower with suspension cables on one side, Langton’s Ant into a Sears Tower-like skyscraper, and Brian’s Brain I don’t even want to think about. It’s rather fun to construct these out of blocks (specifically ones that can be joined together) , as the results are often surprising.
Part of Wolfram’s book was devoted to designing and finding certain cellular automata that can do anything– calculate what 2+2 is, emulate other cellular automata- even display letters- called Universal cellular automata. The simplest of these to show universal would be Conway’s Game of Life, by making AND gates, OR gates, a memory cell, a 90 degree reflector ,and a NOT gate. Many of these base on bashing gliders together to form certain outcomes, and the NOT gate is the hard one- it needs to use a glider gun, or something to send out gliders, in order to actually be a NOT gate. Once that’s made, the rest is simple.
A similar method can be used to show that WireWorld is universal- by making the necessary logic components, various computers can easily be made, such as Mark Owen’s massive prime calculator. There are even constructions made by putting logic gates together such that 1-dimensional cellular automata can be made!
Von Neumann also designed a 2-dimensional cellular automata, the sole purpose of which was to show that computers were possible in cellular automata. The rules are quite complex, mostly operate on signals passing through wires and writing cells, and the cellular automaton has a whopping 29 states. Replicators are possible, but they use humongous “tapes” to store how the structure should be built.
Now here’s the amazing part: Even 1-dimensional cellular automata can be universal. In particular, Wolfram showed a certain 19-state next-nearest neighbor cellular automaton which, given the right setup, will emulate any other 1-dimensional cellular automata on a huge basis (20 cells per cell). Some examples of it emulating cellular automata are below:
Rule 90 and Rule 30, emulated
In particular, although it is hard to see, the 19-state cellular automaton is emulating rule 90 and rule 30, respectively.
Most amazing, though is that, though it is anything but straightforward to prove, Rule 110 is a universal cellular automaton. This was done by showing how it could emulate another 1-dimensional cellular automata class, the cyclic tag system, and working from there. Eventually, Wolfram shows it emulating other elementary cellular automata, computing, and even emulating Turing machines.
Quite a lot of cellular automata programs exist (many of them are listed at http://cafaq.com/soft/index.php), so I’ll simply list some of the best ones that I have found.
One of my favorite programs is Mirek’s Cellebration (MCell), made by Mirek Wojtowicz, which has quite a lot of cellular automata rules (200+), and even more cellular automata patterns. It has a large Life pattern database, as well as allows you to make your own rules and save them easily. Probably the only problems with this are that the speed of the automaton may vary depending on the number of life cells on the board, and that the software is no longer developed. However, you can add on small extensions and even change the source code of the online Java version. You can either download it here, or see the Java implementation.
Another program for simulating cellular automata is Five Cellular Automata, which simulates exactly 5 types of cellular automata: A small generalization of Life, using 4 parameters and q states; The Belousov-Zhabotinsky reaction, as a cellular automaton; a cellular automata in which blobs of colors try to meet with each other, and eventually take over the board; a probabilistic cellular automaton in which “viruses” break out among the population, kill everybody, and eventually die as the population regrows; and lastly, a DLA model. The program simulates all 5 rather well, but it only does those 5, and there are no manual editing features. This makes it so that the program is good for watching, but not useful for any experimentation. You can download it at the Hermetic Systems website.
The best of these which is being developed on would easily be Golly, a cellular automata program that has infinite universes, uses Bill Gosper’s speedy Hashlife algorithm, has hundreds of patterns, including a few Life lexicons, and even is scriptable (with examples!) in both Python and Perl. And it reads practically every CA file ever made. The only problem is that completely new rules, such as making a rule table cellular automaton, isn’t very easy unless it’s a Life-like cellular automaton (B something/S something). You can download it at the project’s Sourceforge page.
Lastly, there’s CAPOW by Rudy Rucker, which is a program for generating continuous-valued cellular automata. It supports 1D and 2D rules, as well as a number of discrete-valued cellular automata. It also has a mode in which the 2D cellular automata is extruded, based on what state the cell is at, into a 3D grid. It has quite a lot of cellular automata, can make up new ones, and includes a screensaver which shows various cellular automata animating. The only bad part is that it’s a bit confusing to make different rules or make new CA classes. You can download it at Rudy Rucker’s website.
There are tons more cellular automata that have not been studied, so the field of Cellular Automata is still an interesting field to explore in and find new and interesting rules.
## 12 thoughts on “Cellular Automata”
1. Another great post Neil – thanks.
You don’t seem to have written about the Sunday at G4G9 yet, is that coming?
Thanks for yet more interesting things to think about.
Colin.
• The very next blog post that I do is going to be about Sunday at G4G9, so I’m currently working on it.
Also, I actually had to leave out a bunch of cellular automata types, such as the Archean cellular automata, or the Gardens of Edens in Life, because I didn’t want the blog post to be as big as Wolfram’s book!
• Oh, I can understand that. There is a lot to write about with CA’s, I’m pretty sure even Stephen’s book isn’t complete. I still occasionally give proofs that Life is Turing equivalent, and it’s fun constructing things in Wire World.
BTW, I know Mark Owen. I must point him at your site when I get the chance.
Cheers!
3. Ian Sassoon
Cool stuff – gotta love fractals!
4. Mark Haslem
Hi Nick,
Would you mind making the black background you have at the sides a lighter color? Or make the print whiter and bigger? It’s just very hard to read if you’re in a light room, which, as it happens, I am, because my planet is currently tilted towards my local star.
I think you should add a self-photo too.
Good content, though. I can’t believe how much time you must spend on this thing.
Good work,
Mark.
5. Pabitra
This is really good and better understandable for everyone.
6. sara
can somebody give me the code or links for Langton’s Virtual Ant program that is shown above.
Thanks,
Sara
7. Brilliant article.
Over the past few years (on and off), I have concentrated my efforts in creating 1D binary (rule 2) patterns. Some can be seen here… http://the-orangery.weebly.com/ca.html
8. Pingback: Cellular Automata – Computart
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# Intuitions for Frobenius' generalization of characters to nonabelian finite group given the historical context
I'm reading about the history of character theory of finite group, especially about the invention of character theory by Frobenius.
According to most of the related papers (e.g. Pioneers of Representation Theory by C.W.Curtis), the character of abelian groups occurs earlier, which was defined as a homomorphism from an abelian group $$G$$ to $$C^{\times}$$. And it is the problem of factorization of group determinants which motivates Frobenius' generalization of characters to arbitrary finite group, as is described in the introduction of one of the Frobenius' papers Über Gruppencharaktere.
Here is Frobenius' original definition: Let $$G$$ be a finite group. For each conjugacy class $$C_{i}$$ of $$G$$, let $$h_{ijk}$$ be the number of solutions of the equation of the equations $$abc = 1$$ for $$a\in C_{i}, b\in C_{j}, c\in C_{k}$$. Denote as $$C_{i^{'}}$$ the conjugacy class consisting of the inverses of elements in $$C_{i}$$, and put $$a_{ijk}=\frac{h_{i^{'}jk}}{|C_{i}|}.$$ Via solving the 'structure equations' $$r_{j}r_{k}=a_{ijk}r_{i},$$ Frobenius got $$n$$ different solutions $$(r_{q1}, r_{q2}, ..., r_{qn})$$. Then he defined a character of a finite group $$G$$ as a class function (constant on every conjugacy class $$C_{i}$$ ) $$\chi$$ , taking for any $$g\in C_{i}$$ the value $$\chi_{q}(g) = \frac{fr_{qi}}{|C_{i}|},$$ where $${r_{qi}}$$ is given above by the structure equations and $$f$$ is used to normalize $$\chi_{q}$$ in order that it would satisfy the orthogonality relations.
This definition seems to be understandable in modern context of representation theory. What seems highly unnatural to me is that this definition is triggered by the attempt to factorize the group determinant.
First of all, the coefficients of linear factors of the group determinant are exactly the linear characters of the group, and any other factor of the determinant has a degree more than $$1$$ and can't be factored into linear forms, so there seems to be no reason to generalize characters to the nonabelian case given that the linear factors are the only to appear in the factorization.
Furthermore, I can't see how intuitively this problem of factorization could provide any clue leading to an explicit definition of group character like above. It seems that Frobenius' idea was to construct a homomorphism to $$C^{\times}$$ from an algebra whose elements are the conjugacy classes with the operation induced by usual multiplication law of a group. I do understand what this algebra is in the modern context, but I can't understand how the latter was motivated by the factorization problem.
Last but not least, Frobenius chose to base his definition on 'hypercomplexes' (seems equivalent to the modern concept of an algebra). I can't see the relation between the 'hypercomplexes' and the factorization problem.
Will anyone be so kind to explain the intuition behind this definition, given the context of group determinants? Thanks in advance!
We can’t redo the history. It just happens to be the case that Dedekind’s questions to Frobenius about group determinants were the original inspiration. It is not intuitive. Only later did Frobenius find a more conceptual approach, the one we use today. Group determinants were abandoned since they are not why we care about group characters and they are a strange motivation, as you have already seen. Maybe the article https://kconrad.math.uconn.edu/articles/groupdet.pdf will be helpful.
• Thank you very much. I can only express my appreciation by adding a comment because I'm new to this community and have no permission to vote for you.
– zyy
Apr 11, 2022 at 13:35
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10182
The adjusted winner procedure is a method of fair division for settling disputes between two parties when there are several items (or issues) involved. Each party independently allocates a total of 100 points to the items involved, with the allocation reflecting the relative importance of those items to that party. The adjusted winner procedure is then used to determine which party gets which items, and whether or not there must be a compromise on one of the issues. This Demonstration lets you allocate the points for both parties and illustrates the division procedure step-by-step. Several fixed examples of allocations are included for quick use.
### DETAILS
In the adjusted winner procedure, each item is initially given to the party who assigned it more points. In the event of a tie, the item is initially given to the party with the fewer total points from the items that were not tied. (See snapshot 4, where items 3 and 4 are given to B, who had 96 points to A's 97.) Items are then transferred from the party with more points to the party with fewer points in order of increasing point ratio: , where is the number of points assigned to the item by the party with more points initially and is the number of points assigned to the item by the party with fewer points initially. Items are transferred until each party has the same number of points. In the typical scenario, there will need to be a compromise on one item.
Reference
[1] COMAP, For All Practical Purposes, New York: W. H. Freeman and Company, 2009.
### PERMANENT CITATION
Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.
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Home > Root Mean > What Is A Good Root Mean Square Error
# What Is A Good Root Mean Square Error
## Contents
Three statistics are used in Ordinary Least Squares (OLS) regression to evaluate model fit: R-squared, the overall F-test, and the Root Mean Square Error (RMSE). What is the meaning of these measures, and what do the two of them (taken together) imply? The RMSD represents the sample standard deviation of the differences between predicted values and observed values. Any further guidance would be appreciated. Source
New York: Springer. This also is a known, computed quantity, and it varies by sample and by out-of-sample test space. Just using statistics because they exist or are common is not good practice. So a residual variance of .1 would seem much bigger if the means average to .005 than if they average to 1000. http://www.theanalysisfactor.com/assessing-the-fit-of-regression-models/
## What Is A Good Root Mean Square Error
The confidence intervals widen much faster for other kinds of models (e.g., nonseasonal random walk models, seasonal random trend models, or linear exponential smoothing models). Many types of regression models, however, such as mixed models, generalized linear models, and event history models, use maximum likelihood estimation. The MAPE can only be computed with respect to data that are guaranteed to be strictly positive, so if this statistic is missing from your output where you would normally expect It may be useful to think of this in percentage terms: if one model's RMSE is 30% lower than another's, that is probably very significant.
Perhaps that's the difference-it's approximate. Thus, before you even consider how to compare or evaluate models you must a) first determine the purpose of the model and then b) determine how you measure that purpose. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. Root Mean Square Error Value Range If there is evidence only of minor mis-specification of the model--e.g., modest amounts of autocorrelation in the residuals--this does not completely invalidate the model or its error statistics.
Ideally its value will be significantly less than 1. p.229. ^ DeGroot, Morris H. (1980). These distinctions are especially important when you are trading off model complexity against the error measures: it is probably not worth adding another independent variable to a regression model to decrease price, part 2: fitting a simple model · Beer sales vs.
share|improve this answer answered Mar 5 '13 at 14:56 e_serrano 111 add a comment| up vote 0 down vote RMSE is a way of measuring how good our predictive model is Rmse Example An equivalent null hypothesis is that R-squared equals zero. MSE is a risk function, corresponding to the expected value of the squared error loss or quadratic loss. However, when comparing regression models in which the dependent variables were transformed in different ways (e.g., differenced in one case and undifferenced in another, or logged in one case and unlogged
## Normalized Rmse
If there is any one statistic that normally takes precedence over the others, it is the root mean squared error (RMSE), which is the square root of the mean squared error. Consider starting at stats.stackexchange.com/a/17545 and then explore some of the tags I have added to your question. –whuber♦ May 29 '12 at 13:48 @whuber: Thanks whuber!. What Is A Good Root Mean Square Error Replace second instance of string in a line in an ASCII file using Bash A student takes a quiz (exam), a professor [verb]s a quiz, exam, etc Does the number of Interpretation Of Rmse In Regression So, even with a mean value of 2000 ppm, if the concentration varies around this level with +/- 10 ppm, a fit with an RMS of 2 ppm explains most of
For example, it may indicate that another lagged variable could be profitably added to a regression or ARIMA model. (Return to top of page) In trying to ascertain whether the error this contact form Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the CS1 maint: Multiple names: authors list (link) ^ "Coastal Inlets Research Program (CIRP) Wiki - Statistics". I will have to look that up tomorrow when I'm back in the office with my books. 🙂 Reply Grateful2U October 2, 2013 at 10:57 pm Thanks, Karen. Rmse Units
Hence, it is possible that a model may do unusually well or badly in the validation period merely by virtue of getting lucky or unlucky--e.g., by making the right guess about When the interest is in the relationship between variables, not in prediction, the R-square is less important. That is: MSE = VAR(E) + (ME)^2. http://mmoprivateservers.com/root-mean/root-mean-square-error-türkçe.html Unless you have enough data to hold out a large and representative sample for validation, it is probably better to interpret the validation period statistics in a more qualitative way: do
It is very important that the model should pass the various residual diagnostic tests and "eyeball" tests in order for the confidence intervals for longer-horizon forecasts to be taken seriously. (Return Root Mean Square Error Excel Retrieved from "https://en.wikipedia.org/w/index.php?title=Mean_squared_error&oldid=750249597" Categories: Point estimation performanceStatistical deviation and dispersionLoss functionsLeast squares Navigation menu Personal tools Not logged inTalkContributionsCreate accountLog in Namespaces Article Talk Variants Views Read Edit View history More Likewise, it will increase as predictors are added if the increase in model fit is worthwhile.
## Variance Further information: Sample variance The usual estimator for the variance is the corrected sample variance: S n − 1 2 = 1 n − 1 ∑ i = 1 n
If it is 10% lower, that is probably somewhat significant. When normalising by the mean value of the measurements, the term coefficient of variation of the RMSD, CV(RMSD) may be used to avoid ambiguity.[3] This is analogous to the coefficient of Why does Davy Jones not want his heart around him? What Does Rmse Mean RMSE is a good measure of how accurately the model predicts the response, and is the most important criterion for fit if the main purpose of the model is prediction.
The residual diagnostic tests are not the bottom line--you should never choose Model A over Model B merely because model A got more "OK's" on its residual tests. (What would you Lower values of RMSE indicate better fit. It indicates the goodness of fit of the model. http://mmoprivateservers.com/root-mean/root-mean-square-error.html Check out Statistically Speaking (formerly Data Analysis Brown Bag), our exclusive membership program featuring monthly webinars and open Q&A sessions.
Three statistics are used in Ordinary Least Squares (OLS) regression to evaluate model fit: R-squared, the overall F-test, and the Root Mean Square Error (RMSE). All three are based on two sums of squares: Sum of Squares Total (SST) and Sum of Squares Error (SSE). However, thinking in terms of data points per coefficient is still a useful reality check, particularly when the sample size is small and the signal is weak. (Return to top of The MSE has the units squared of whatever is plotted on the vertical axis.
The mean model, which uses the mean for every predicted value, generally would be used if there were no informative predictor variables. Even if the model accounts for other variables known to affect health, such as income and age, an R-squared in the range of 0.10 to 0.15 is reasonable. from trendline Actual Response equation Xa Yo Xc, Calc Xc-Xa (Yo-Xa)2 1460 885.4 1454.3 -5.7 33.0 855.3 498.5 824.3 -31.0 962.3 60.1 36.0 71.3 11.2 125.3 298 175.5 298.4 0.4 0.1 salt in water) Below is an example of a regression table consisting of actual data values, Xa and their response Yo.
International Journal of Forecasting. 8 (1): 69–80. H., Principles and Procedures of Statistics with Special Reference to the Biological Sciences., McGraw Hill, 1960, page 288. ^ Mood, A.; Graybill, F.; Boes, D. (1974). more hot questions question feed about us tour help blog chat data legal privacy policy work here advertising info mobile contact us feedback Technology Life / Arts Culture / Recreation Science if the concentation of the compound in an unknown solution is measured against the best fit line, the value will equal Z +/- 15.98 (?).
The mean square error represent the average squared distance from an arrow shot on the target and the center. The residuals do still have a variance and there's no reason to not take a square root. The goal of experimental design is to construct experiments in such a way that when the observations are analyzed, the MSE is close to zero relative to the magnitude of at Thus the RMS error is measured on the same scale, with the same units as .
Squaring the residuals, taking the average then the root to compute the r.m.s. am using OLS model to determine quantity supply to the market, unfortunately my r squared becomes 0.48. In many cases, especially for smaller samples, the sample range is likely to be affected by the size of sample which would hamper comparisons. The distance from this shooters center or aimpoint to the center of the target is the absolute value of the bias.
I need to calculate RMSE from above observed data and predicted value. Thank you and God Bless.
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# The number of arrangments of all digits of $12345$ such that at least $3$ digits will not come in its position is
Video Solution
Text Solution
Verified by Experts
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Step by step video, text & image solution for The number of arrangments of all digits of 12345 such that at least 3 digits will not come in its position is by Maths experts to help you in doubts & scoring excellent marks in Class 12 exams.
Updated on:21/07/2023
### Knowledge Check
• Question 1 - Select One
## The number of ways in which we can arrange the digit 1, 2, 3, …, 9 such that the product of five digits at any of the five consecutive positions is divisible by 7 is
A7!
B9P7
C8!
D5(7!)
• Question 2 - Select One
## How many 6-digit numbers have at least 1 even digit?
A884375
B3600
C880775
D15624
• Question 3 - Select One
## A two digit number is five times the sum if its digits. If 9 is added to the number, the digits interchange their positions. The sum of digits of the numbers is :
A11
B9
C7
D6
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Table 1 Participant characteristics
All Males Females p
N = 51 n = 31 n = 20
Race (%)
White 35 (69) 22 (71) 13 (65) 0.60
Black 11 (22) 7 (23) 4 (20)
Other 5 (9) 2 (6) 3 (15)
Means ± (SEMs)
Age 34.2 (1.6) 36.2 (2.0) 30.9 (2.5) 0.10
Education 14.4 (0.3) 14.3 (0.3) 14.7 (0.6) 0.52
Cigarettes per day 15.6 (0.8) 16.9 (1.0) 13.6 (1.2) 0.04*
Pack yearsa 12.5 (1.6) 15.2 (2.4) 8.4 (1.6) 0.04*
FTND scores 4.5 (0.2) 4.7 (0.3) 4.3 (0.5) 0.44
Craving scoresb 0.8 (0.2) 0.8 (0.3) 0.9 (0.4) 0.93
1. * p < 0.05.
2. aPack years calculation: Cigarettes per day (÷) cigarettes in a pack (X) years smoking.
3. bCraving scores calculation: Post-SC video craving score – Pre-SC video craving score.
4. FTND = Fagerstrom Test for Nicotine Dependence; FTND scores ranged from 1 to 9.
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J.Y.S
Member List
First Name Last Name wiki id Learn e-mail Phone
YuJin Jeong yujin.jeong yjeong 647 - 832 - 6771
James
Assignment 1
Due Date
Thursday, October 21st, 4:00pm Printout submission only( SELECT Statements + ALL outputs)
Solution
1.
YuJin
SELECT employee_id, RPAD(CONCAT(CONCAT(last_name, ', '),first_name), 25) "Full Name", job_id,
TO_CHAR(TRUNC(hire_date, 'MONTH'), 'fmMonth Ddspth "in the year" YYYY') "Start Date"
FROM employees
WHERE TO_CHAR(hire_date, 'Mon') IN ('May', 'Nov')
ORDER BY hire_date DESC;
James
2.
YuJin
SELECT 'Employee named ' || first_name || ' ' || last_name || ' who is '|| job_id || \
' will have a new salary of \$' || salary*1.15 "Happy Employees"
FROM employees
WHERE salary BETWEEN 5000 AND 12000
AND job_id IN ('IT_PROG', 'ST_CLERK')
ORDER BY salary;
James
3.
YuJin
SELECT last_name, salary, job_id,
NVL(TO_CHAR(manager_id), 'No Manager') "Manager #",
TO_CHAR(salary*12, '\$999,999') "Total Income"
FROM employees
WHERE (job_id LIKE 'MK%' OR NVL(manager_id, 0) = 0)
AND salary*(1 + NVL(commission_pct,0)) + 1000 > 10000
ORDER BY salary;
James
4.
YuJin
job_id, LPAD(TRIM(TO_CHAR(salary, '\$999,999')), 12, '=') "Salary", department_name
FROM employees JOIN departments
USING (department_id)
WHERE SUBSTR(job_id, 4) NOT IN ('PRES','VP','MAN','MGR')
AND SUBSTR(job_id, 1, 2) IN ('MK', 'SA')
ORDER BY 1;
James
5.
YuJin
SELECT last_name, salary, job_id
FROM employees
WHERE UPPER(job_id) NOt LIKE '%PRES'
AND salary >
(SELECT MAX(salary)
FROM employees
JOIN departments
USING (department_id)
JOIN locations
USING (location_id)
WHERE UPPER(city) IN ('TORONTO','OXFORD'));
James
6.
YuJin
SELECT last_name, first_name, job_id, hire_date
FROM employees
WHERE hire_date >
(SELECT MAX(hire_date)
FROM employees
WHERE department_id =
(SELECT department_id
FROM departments
WHERE UPPER(department_name) = 'IT'))
AND (department_id !=
(SELECT department_id
FROM departments
WHERE UPPER(department_name) = 'EXECUTIVE')
OR department_id IS NULL)
ORDER BY 4 DESC;
James
YuJin
James
YuJin
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Home > Sun & Moon > Day and Night World Map
# Day and Night World Map
The map below shows the current position of the Sun and the Moon. It shows which areas of the Earth are in daylight and which are in night.
= The Sun's position directly overhead (zenith) in relation to an observer.
= The Moon's position at its zenith in relation to an observer (Moon phase is not shown).
= Nautical Twilight
= Astronomical Twilight
= Night, no twilight (darkest shade)
See where the Moon is over the horizon
## Find the Sun at Another Time in a Location
/
/
:
:
Start typing name of city
## Position of the Sun
On Wednesday, October 31, 2012 at 13:18:00 UTC the Sun is at its zenith at Latitude: 14° 21' South, Longitude: 23° 36' West
The ground speed is currently 449.47 meters/second, 1618.1 kilometres/hour, 1005.4 miles/hour or 873.7 nautical miles/hour (knots). The table below shows position of the the Sun compared to the time and date above:
TimeLongitude DifferenceLatitude DifferenceTotal
LaterDegreesDistanceDirectionDegreesDistanceDirectionDistance
1 minute0° 15' 00.0"16.76 miWest0° 00' 00.8"0.02 miSouth16.76 mi
1 hour15° 00' 01.2"1005.27 miWest0° 00' 48.1"0.92 miSouth1005.24 mi
24 hours0° 00' 23.7"0.44 miWest0° 19' 09.0"21.94 miSouth21.95 mi
## Position of the Moon
On Wednesday, October 31, 2012 at 13:18:00 UTC the Moon is at its zenith at Latitude: 19° 27' North, Longitude: 175° 21' East
The ground speed is currently 423.58 meters/second, 1524.9 kilometres/hour, 947.5 miles/hour or 823.4 nautical miles/hour (knots). The table below shows position of the the Moon compared to the time and date above:
TimeLongitude DifferenceLatitude DifferenceTotal
LaterDegreesDistanceDirectionDegreesDistanceDirectionDistance
1 minute0° 14' 31.3"15.79 miWest0° 00' 04.1"0.08 miNorth15.79 mi
1 hour14° 31' 16.5"947.22 miWest0° 04' 01.3"4.61 miNorth947.03 mi
24 hours348° 26' 09.0"754.41 miWest1° 11' 07.1"81.54 miNorth756.01 mi
## Locations With the Sun Near Its Zenith
The following table shows 10 locations with Sun near zenith position in the sky.
Scroll right to see more
LocationLocal timeDistanceDirection
RecifeWed 10:18 AM1416 km880 miles765 nm WNW
SalvadorWed 10:18 AM1617 km1005 miles873 nm W
JamestownWed 1:18 PM1929 km1198 miles1041 nm E
FortalezaWed 10:18 AM2016 km1253 miles1088 nm NW
Rio de Janeiro *Wed 11:18 AM2273 km1413 miles1228 nm WSW
Brasilia *Wed 11:18 AM2615 km1625 miles1412 nm W
São Paulo *Wed 11:18 AM2627 km1632 miles1419 nm WSW
MonroviaWed 1:18 PM2686 km1669 miles1450 nm NNE
FreetownWed 1:18 PM2772 km1722 miles1497 nm NNE
ConakryWed 1:18 PM2855 km1774 miles1541 nm NNE
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# On Random Vectors and Eigenvectors of Symmetric Matrices
I have a question that might be answered with a pointer to some references or with some discussion. I did some searching, to no avail, but I realized that I might not have the vocabulary to form a successful search.
Suppose we have a random (or pseudorandom) vector $r\in\mathbb{R}^{n}$ with $\left\|r\right\|=1$, e.g. generated by the rand() function in Matlab and normalized. Let $S\in\mathbb{R}^{n\times n}$ be a symmetric matrix with the orthogonal decomposition $S=V^{T}DV$ where the columns of $V$ are orthonormal and $D$ is the diagonal matrix of eigenvalues. Let $v_{i}$ be a column of $V$. What is the probability that $\left|r^{T}v_{i}\right|< \alpha$ where $\alpha\in\left(0,1\right)$. In other words, what is the probability that $r$ has only a certain size component in a particular eigenvector?
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Note that using the rand() function in MATLAB to generate your $r$ vector won't generate a vector uniformly distributed over the surface of the unit ball- you want to use randn(). – Brian Borchers Jul 2 '12 at 14:15
Also note that it really doesn't matter where $v_{i}$ comes from. The question you want to answer is "Given a fixed vector $x$ and random vector $r$, what is $P(|r^{T}x| < \alpha)$?" – Brian Borchers Jul 2 '12 at 14:17
This depends on what the distribution of $r$ is. Also, is there some significance to $S$ that I'm missing here, or can you just replace $v_i$ with any unit vector? – Mark Meckes Jul 2 '12 at 14:18
Mark- I believe that $r$ is uniformly distributed over the surface of the hypersphere. The answer to the question will change dramatically if it isn't. – Brian Borchers Jul 2 '12 at 14:38
The only significance of $S$ is to the particular problem this arises from in my work. As has been stated, the question can be rephrased much more simply, making the answer clearer. – Kirk S. Jul 2 '12 at 14:40
As pointed out in the comments, the matrix has nothing to do with the question, and you are simply trying to compute the distribution of $x \cdot v$ as $x$ varies over the unit sphere, and $v$ is a fixed unit vector. By rotational invariance, you might as well assume that $v = (1, 0, \dots, 0),$ at which point the question becomes an easy integration exercise.
Edit if you don't want to bother integrating, the concentration of measure phenomenon (first observed by Boltzmann, I believe) is that the distribution of the areas of cross sections of the sphere is essentially normal for moderate $n.$ ( you can easily compute the variance), so then you can approximate the answer to your question by an inverse error function.
-
In this case, it's easy enough to compute the probability exactly in terms of the incomplete beta function.
Let $v$ be the fixed unit vector, and $r$ be the random unit vector, uniformly distributed over the $n$-dimensional hypersphere.
$P(| r^{T} v | < \alpha) = 1-P(| r^{T}v | \geq \alpha)$
$P(| r^{T} v | < \alpha) = 1-2P(r^{T}v \geq \alpha)$
Let $\theta$ be the angle between $r$ and $v$. Then $\cos \theta=r^{T}v$. Now, $r^{T}v \geq \alpha$ only if $r$ is in the spherical cap of the unit hypersphere centered around $v$, containing vectors within an angle $\cos^{-1}\alpha$ from $v$. The height of this spherical cap is $h=1-\alpha$.
The probability that $r^{T}v \geq \alpha$ is then given by the ratio of the surface area of this circular cap to the surface area of the hypersphere. Using standard formulas for the surface area of the hypersphere and of the spherical cap, we get that
$P(| r^{T}v | < \alpha)=1-I_{2h-h^{2}}(\frac{n-1}{2},\frac{1}{2})$
A quick test with a Monte Carlo simulation in MATLAB verifies this formula.
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# How to make a pyrometer, using an inductance measurement instrument
The pyrometers can be used to measure inductance, the amount of energy released by an electrical current.
Inductance can be measured in terms of how much of a current is flowing.
In an inductive system, the energy released is a measure of the amount the current is traveling through.
The energy of a voltage is a measurement of how quickly a voltage will travel from one point to another.
In a mechanical system, an electromotive force is the amount an electric current is pushing against an object.
An inductive voltage can be created by adding an inductor to a current source, like a battery, and then measuring how the voltage is affected by the current.
When you are trying to measure the inductance of an electric circuit, you want to make sure the inductor is very short, or that the current can’t cause the voltage to jump.
If you want an inductively-produced current to cause the current to jump, you need to measure how much the current changes as it moves.
You need to use the inductive measurement instrument as a measure to show how much current is being released by the electrical current flowing through the inducting coil.
When using inductive measurements, the measurement is done by placing a small electrical device inside a tube, and using a small amount of the current in the tube.
This measurement is known as the inductivity.
An average current of 100 mA will cause a voltage to rise as the current increases.
This is known in engineering as a inductive load.
When a voltage increases above 100 mAm, you can make the voltage jump.
In order to make the current jump, a voltage source will need to be very short.
To make this short, you use a very small amount in the circuit.
This can be done by using a wire to short a resistor.
The current from a wire will then jump as the wire travels through the circuit and is then used to increase the voltage.
You can use a short current to increase voltage by up to a factor of 4 or 5.
If a voltage jumps, you know the inductors current is low.
Induction is a common measure for the inductions inductance.
It is used to calculate inductance as well as other values such as inductance at a voltage.
When it comes to the inductances inductance and inductance ratio, the inductents inductance is measured as the square root of the inductence.
This means that the square of the number of inductors equals the number times the number.
The inductance in a circuit is the number divided by the inductent current.
For example, the circuit with two coils has a inductance that is 100 mAs, which is 3 times the inductency of the one coil.
The square root number of the coil is 2, and the inductancy is 1.7 mA.
If the inducted current in a coil is 100 milliamperes, the square roots will be 3, or 100 times the current of the single coil.
This shows that the inductant inductance equals the square (or squared) of the square and the square.
The two coils with the same inductance are both inductively active.
The only difference is the voltage will jump from the coil that is more active to the coil with less active.
If we want to measure an inductee’s inductance for a circuit with a small current, we need to create a very short current by placing an inducting device inside the tube with the current flowing very quickly through the tube, then measuring the voltage as the coil moves through the system.
This allows us to make measurements that are easy to compare to other measurements.
When the current drops below a certain point, the voltage jumps as the voltage rises.
This tells us that the voltage in the induction is low, and that the coil has very little inductance to it.
Inductive measurement instruments can be made in many different ways.
They can be a very simple electrical circuit, or they can be highly complex circuits, and have lots of components.
The most common method for making an induction instrument is by using an electric induction transformer.
An electric inductor, or coil, can be placed in the coil, and a small electric current will be drawn through the coil.
Once the coil reaches a certain current, it will start to spin.
The coil will then rotate in the direction of the electric current.
This motion causes the voltage across the coil to jump as it spins.
The voltage jump is the same as when you turn on the light, but when you switch off the light the voltage falls.
The circuit with more inductance has more inductors and a larger current.
The more inductive a circuit, the more voltage will be needed to cause a current to leap.
If your circuit is simple, the induction will be slow and there will be no current jump.
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# How to find equilibrium level of income. macroeconomics 2019-01-07
How to find equilibrium level of income Rating: 6,4/10 971 reviews
## Determination of Equilibrium Level of Income
That is, they cannot expand an economy, or its national income beyond a certain point in the long run. In this case, the short run equilibrium is at a higher level of real output than the economy can actually produce, as illustrated below. This quantiry inleters when multiplied with the temperature correction factor andthen converted into mass by multiplying with the density of fuelHence we get the fuel consumed per day. In other words the model produces a domestic equilibrium; if the external finance is forthcoming then it's an external equilibrium too. It's free to use and requires no registration! The increase in saving ΔS must be just enough to offset the increase in autonomous investment ΔI so that equilibrium is restored at point P after the initial disturbance at point E. An economy will grow if the value of injections is greater than the value of withdrawals, or shrink if the value of withdrawals is greater than injections.
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## How to Calculate the Equilibrium Level of Income
Step 3 Determine the cost of goods sold. The determination of equilibrium level of income can be better understood with the help of the following schedule and diagram: Table 8. Other models do not make that assumption. Diverging short run equilibrium and full employment Short run equilibrium may not coincide with the sustainable full-employment level of real output - the level at which the economy is achieving its economic potential. This change shows how demand-pull inflation might arise in an economy. The investment multiplier shows how shocks to one sector are transmitted throughout the economy.
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## Documented Problem Solving: Calculating Equilibrium Output
Aggregate Income-Expenditure Approach: In a two-sector Keynesian model, aggregate demand is composed of planned or desired consumption demand and planned investment demand. Power companies use kilowatt hours to charge you for electricity, to figure out how many kilowatt hours your appliance uses you would need to multiply the amperage your appliance uses by. As inventories decline, business firms step up production. Thus, there will be an unintended accumulation of inventories by producers. Home loans, auto loans etc. Step 4 Deduct the cost of goods sold from net receipts and add other income like fuel tax credits and you have your Gross Income. Thanks -- and Fool on! Figure 1 Macroeconomic equilibrium classical If nothing changes then the economy will be stable at this equilibrium, but any changes in aggregate supply and demand will lead to changes in output and the price level.
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## Determination of Equilibrium Level of Income
This is also consistent with planned aggregate demand equalling planned aggregate supply. Thinking about the Example's Solution At this point we have solved the problem by focusing on goods. All figures are in billions. In addition, students are asked to describe the process they used to calculate the equilibrium and analyze the state of the economy. Using this formula, an analyst can observe how a change in any of the factors will impact the level of income. Calculate this equilibrium using the function that derives consumption from aggregate income. The condition for stability is that the saving curve must be positively sloped.
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## How do you calculate the equilibrium level of income
Suppose the existing capital equipment is utilised fully but a quarter of the labor force is unemployed. According to equation 23 , it is the ratio of the two changes, viz. As a result, planned inventory would fall below the desired level. Changes in Equilibrium Income : Keynes pointed out that any change in autonomous spending would lead to a change in equilibrium income. Point E is the equilibrium point, since C + I̅ line cuts the 45° line at that point. A similar situation would arise if the government stimulated the economy via expansionary fiscal or monetary policy.
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## How to Calculate the Equilibrium Level of Income
In this case, government savings is zero. This amount is subtracted from the sell price to obtain your net gain or loss. For a large country, an increase in it imports should raise foreign incomes, and lead to higher exports. The current consumption will not be constant, so it must be averaged over time. Diagrammatically, this means that the C + I̅ line must cut the 45° line from above. Investment expenditure is assumed to be autonomous.
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## economics
Taxes do not vary with the level of income. What you are really looking for, and what banks want to know is your monthly debt to income ratio. Note that by assuming that X is exogenous, we are considering a small country case. The aggregate effect is very positive for the economy. We are told that the budget deficit is 300 billion or. The reason grams per square meter are used even in the U.
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## macroeconomics
Auto loan payment, rent, phone and every bill you can think of. We will look at these policies in more detail in the next section. They are thus appropriately treated as autonomous factors affecting aggregate demand. . If there is any deviation from the equilibrium level of income, i. This can be seen in the diagram.
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## economics
The company must decrease its production, potentially downsizing its employment base or cutting prices to unload the excess inventory. In this case, Government savings is negative. In this case, the new consumption expenditure of 80 units leads to an increase in production and thus generates a second round increase in income for some households of 80 units. Suppose labour and capital are used in fixed proportions in the production of output in a closed economy. With this information, we want to derive the equilibrium values of income, consumption saving, and investment. Public savings equations The public savings equation tells us how much the government is saving. Net Income average income and primary expenses for a stereotypical family of four in the mid-1970's and mid-2000's.
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# Inverse of a Matrix and Its Applications, Objectives, Determinant of a Square Matrix, Minors and Cofactors of the Elements of Square Matrix, Adjoint of a Square Matrix
Get unlimited access to the best preparation resource for IMO-Level-2 : fully solved questions with step-by-step explanation- practice your way to success.
## Objectives
After studying this lesson, you will be able to:
• Define a minor and a cofactor of an element of a matrix;
• Find minor and cofactor of an element of a matrix;
• Find the adjoint of a matrix;
• Define and identify singular and non-singular matrices;
• Find the inverse of a matrix, if it exists;
• Represent system of linear equations in the matrix form ; and
• Solve a system of linear equations by matrix method.
## Determinant of a Square Matrix
We have already learnt that with each square matrix, a determinant is associated. For any given matrix, say its determinant will be . It is denoted by .
Similarly, for the matrix , the corresponding determinant is
Example:
Determine whether matrix A is singular or non-singular where
Solution:
Here,
Therefore, the given matrix is a singular matrix.
## Minors and Cofactors of the Elements of Square Matrix
Consider a matrix
The determinant of the matrix obtained by deleting the row and column of , is called the minor of and is denotes by .
Cofactor of is defined as
For example, Minor of and Cofactor of
Example:
Find the minors and the cofactors of the elements of matrix
Solution:
For matrix ,
(minor of ) ;
(minor of ) ;
(minor of ) ;
(minor of ) ;
## Adjoint of a Square Matrix
Let be a matrix. Then
Let and be the minor and cofactor of respectively. Then
We replace each element of by its cofactor and get
The transpose of the matrix of cofactors obtained in above is
The matrix obtained above is called the adjoint of matrix . It is denoted by .
Thus, adjoint of a given matrix is the transpose of the matrix whose elements are the cofactors of the elements of the given matrix.
Example:
Solution:
Here, Let be the cofactor of the element .
Then,
We replace each element of by its cofactor to obtain its matrix of cofactors as
Transpose of matrix in is .
Thus,
Developed by:
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https://www.teacherspayteachers.com/Product/Kindergarten-Math-Journal-Prompts-for-Winter-173091
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DID YOU KNOW:
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# Kindergarten Math Journal Prompts for Winter
Teacher Tam
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Grade Levels
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Subjects
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Resource Type
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Pages
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#### Also included in
1. This BUNDLE of kindergarten math journal prompts includes all 5 units--enough for the entire year! Each of these kindergarten math journal sets are also available individually from my store, so do not purchase this set if you already have them. Each of these kindergarten math journal sets addresses
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Save \$4.99
### Description
Students will subtract snowmen, count snowflakes, compare and snow-covered trees, use shapes to draw snowmen and more with this set of 50 journal prompts. There are 2 versions of each prompt to allow for differentiation. That makes a total of 100 prompts! There are also at least 2 prompts for each standard. The standard(s) addressed are listed on every page.
Save \$5 when you purchase the COMBO pack! Click here to take a look!
Don't miss this Kindergarten Cut-and-Glue Math Set for Winter!
These journal prompts address ALL 22 of the Common Core Standards* for Kindergarten Mathematics in:
Counting and Cardinality
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
*Common Core Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Total Pages
100 pages
Answer Key
N/A
Teaching Duration
3 months
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### Standards
to see state-specific standards (only available in the US).
Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
Correctly name shapes regardless of their orientations or overall size.
### Questions & Answers
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# Matrix3x2.MultiplyMatrix3x2.MultiplyMatrix3x2.MultiplyMatrix3x2.Multiply Operator
## オーバーロード
Multiply(Matrix3x2, Matrix3x2) Multiply(Matrix3x2, Matrix3x2) Multiply(Matrix3x2, Matrix3x2) Multiply(Matrix3x2, Matrix3x2) 2 つの行列の乗算結果となる積行列を返します。Returns the matrix that results from multiplying two matrices together. Multiply(Matrix3x2, Single) Multiply(Matrix3x2, Single) Multiply(Matrix3x2, Single) Multiply(Matrix3x2, Single) 指定した行列のすべての要素をスカラー因子倍した行列を返します。Returns the matrix that results from scaling all the elements of a specified matrix by a scalar factor.
## Multiply(Matrix3x2, Matrix3x2)Multiply(Matrix3x2, Matrix3x2)Multiply(Matrix3x2, Matrix3x2)Multiply(Matrix3x2, Matrix3x2)
2 つの行列の乗算結果となる積行列を返します。Returns the matrix that results from multiplying two matrices together.
``````public:
static System::Numerics::Matrix3x2 operator *(System::Numerics::Matrix3x2 value1, System::Numerics::Matrix3x2 value2);``````
``public static System.Numerics.Matrix3x2 operator * (System.Numerics.Matrix3x2 value1, System.Numerics.Matrix3x2 value2);``
``static member ( * ) : System.Numerics.Matrix3x2 * System.Numerics.Matrix3x2 -> System.Numerics.Matrix3x2``
``Public Shared Operator * (value1 As Matrix3x2, value2 As Matrix3x2) As Matrix3x2``
#### パラメーター
value1
Matrix3x2 Matrix3x2 Matrix3x2 Matrix3x2
value2
Matrix3x2 Matrix3x2 Matrix3x2 Matrix3x2
2 番目の行列。The second matrix.
### 注釈
@No__t-0 メソッドは、Matrix3x2 オブジェクトの乗算演算子の演算を定義します。The Multiply method defines the operation of the multiplication operator for Matrix3x2 objects.
## Multiply(Matrix3x2, Single)Multiply(Matrix3x2, Single)Multiply(Matrix3x2, Single)Multiply(Matrix3x2, Single)
``````public:
static System::Numerics::Matrix3x2 operator *(System::Numerics::Matrix3x2 value1, float value2);``````
``public static System.Numerics.Matrix3x2 operator * (System.Numerics.Matrix3x2 value1, float value2);``
``static member ( * ) : System.Numerics.Matrix3x2 * single -> System.Numerics.Matrix3x2``
``Public Shared Operator * (value1 As Matrix3x2, value2 As Single) As Matrix3x2``
#### パラメーター
value1
Matrix3x2 Matrix3x2 Matrix3x2 Matrix3x2
スカラー倍演算の対象となる行列。The matrix to scale.
value2
Single Single Single Single
#### 戻り値
スケール調節された行列。The scaled matrix.
### 注釈
@No__t-0 メソッドは、Matrix3x2 オブジェクトの乗算演算子の演算を定義します。The Multiply method defines the operation of the multiplication operator for Matrix3x2 objects.
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# 5. Solve the recurrence relation: a) an = -6an-1 + 7an-2, n>...
## Transcribed Text
5. Solve the recurrence relation: a) an = -6an-1 + 7an-2, n> 2, given ao = 32, as = -17. 6. Express the generating function of each of the following sequences as a polynomial or as the quotient of polynomial: a) 0, 1, 4, 1, 0, 0, b) 3,3,3, 7. Consider the arithmetic sequence begins 5, 9, 13. a) Find the 32nd. and 100th. terms of this sequence.
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SUBROUTINE SORMQL( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC,
\$ WORK, LWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER SIDE, TRANS
INTEGER INFO, K, LDA, LDC, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), C( LDC, * ), TAU( * ),
\$ WORK( * )
* ..
*
* Purpose
* =======
*
* SORMQL overwrites the general real M-by-N matrix C with
*
* SIDE = 'L' SIDE = 'R'
* TRANS = 'N': Q * C C * Q
* TRANS = 'T': Q**T * C C * Q**T
*
* where Q is a real orthogonal matrix defined as the product of k
* elementary reflectors
*
* Q = H(k) . . . H(2) H(1)
*
* as returned by SGEQLF. Q is of order M if SIDE = 'L' and of order N
* if SIDE = 'R'.
*
* Arguments
* =========
*
* SIDE (input) CHARACTER*1
* = 'L': apply Q or Q**T from the Left;
* = 'R': apply Q or Q**T from the Right.
*
* TRANS (input) CHARACTER*1
* = 'N': No transpose, apply Q;
* = 'T': Transpose, apply Q**T.
*
* M (input) INTEGER
* The number of rows of the matrix C. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix C. N >= 0.
*
* K (input) INTEGER
* The number of elementary reflectors whose product defines
* the matrix Q.
* If SIDE = 'L', M >= K >= 0;
* if SIDE = 'R', N >= K >= 0.
*
* A (input) REAL array, dimension (LDA,K)
* The i-th column must contain the vector which defines the
* elementary reflector H(i), for i = 1,2,...,k, as returned by
* SGEQLF in the last k columns of its array argument A.
* A is modified by the routine but restored on exit.
*
* LDA (input) INTEGER
* The leading dimension of the array A.
* If SIDE = 'L', LDA >= max(1,M);
* if SIDE = 'R', LDA >= max(1,N).
*
* TAU (input) REAL array, dimension (K)
* TAU(i) must contain the scalar factor of the elementary
* reflector H(i), as returned by SGEQLF.
*
* C (input/output) REAL array, dimension (LDC,N)
* On entry, the M-by-N matrix C.
* On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.
*
* LDC (input) INTEGER
* The leading dimension of the array C. LDC >= max(1,M).
*
* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* If SIDE = 'L', LWORK >= max(1,N);
* if SIDE = 'R', LWORK >= max(1,M).
* For optimum performance LWORK >= N*NB if SIDE = 'L', and
* LWORK >= M*NB if SIDE = 'R', where NB is the optimal
* blocksize.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal size of the WORK array, returns
* this value as the first entry of the WORK array, and no error
* message related to LWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
INTEGER NBMAX, LDT
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
* ..
* .. Local Scalars ..
LOGICAL LEFT, LQUERY, NOTRAN
INTEGER I, I1, I2, I3, IB, IINFO, IWS, LDWORK, LWKOPT,
\$ MI, NB, NBMIN, NI, NQ, NW
* ..
* .. Local Arrays ..
REAL T( LDT, NBMAX )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL SLARFB, SLARFT, SORM2L, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
LEFT = LSAME( SIDE, 'L' )
NOTRAN = LSAME( TRANS, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
* NQ is the order of Q and NW is the minimum dimension of WORK
*
IF( LEFT ) THEN
NQ = M
NW = MAX( 1, N )
ELSE
NQ = N
NW = MAX( 1, M )
END IF
IF( .NOT.LEFT .AND. .NOT.LSAME( SIDE, 'R' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( K.LT.0 .OR. K.GT.NQ ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, NQ ) ) THEN
INFO = -7
ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
INFO = -10
END IF
*
IF( INFO.EQ.0 ) THEN
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
LWKOPT = 1
ELSE
*
* Determine the block size. NB may be at most NBMAX, where
* NBMAX is used to define the local array T.
*
*
NB = MIN( NBMAX, ILAENV( 1, 'SORMQL', SIDE // TRANS, M, N,
\$ K, -1 ) )
LWKOPT = NW*NB
END IF
WORK( 1 ) = LWKOPT
*
IF( LWORK.LT.NW .AND. .NOT.LQUERY ) THEN
INFO = -12
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SORMQL', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
NBMIN = 2
LDWORK = NW
IF( NB.GT.1 .AND. NB.LT.K ) THEN
IWS = NW*NB
IF( LWORK.LT.IWS ) THEN
NB = LWORK / LDWORK
NBMIN = MAX( 2, ILAENV( 2, 'SORMQL', SIDE // TRANS, M, N, K,
\$ -1 ) )
END IF
ELSE
IWS = NW
END IF
*
IF( NB.LT.NBMIN .OR. NB.GE.K ) THEN
*
* Use unblocked code
*
CALL SORM2L( SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK,
\$ IINFO )
ELSE
*
* Use blocked code
*
IF( ( LEFT .AND. NOTRAN ) .OR.
\$ ( .NOT.LEFT .AND. .NOT.NOTRAN ) ) THEN
I1 = 1
I2 = K
I3 = NB
ELSE
I1 = ( ( K-1 ) / NB )*NB + 1
I2 = 1
I3 = -NB
END IF
*
IF( LEFT ) THEN
NI = N
ELSE
MI = M
END IF
*
DO 10 I = I1, I2, I3
IB = MIN( NB, K-I+1 )
*
* Form the triangular factor of the block reflector
* H = H(i+ib-1) . . . H(i+1) H(i)
*
CALL SLARFT( 'Backward', 'Columnwise', NQ-K+I+IB-1, IB,
\$ A( 1, I ), LDA, TAU( I ), T, LDT )
IF( LEFT ) THEN
*
* H or H' is applied to C(1:m-k+i+ib-1,1:n)
*
MI = M - K + I + IB - 1
ELSE
*
* H or H' is applied to C(1:m,1:n-k+i+ib-1)
*
NI = N - K + I + IB - 1
END IF
*
* Apply H or H'
*
CALL SLARFB( SIDE, TRANS, 'Backward', 'Columnwise', MI, NI,
\$ IB, A( 1, I ), LDA, T, LDT, C, LDC, WORK,
\$ LDWORK )
10 CONTINUE
END IF
WORK( 1 ) = LWKOPT
RETURN
*
* End of SORMQL
*
END
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# How Many Decimals of Pi Do We Really Need? NASA Answers
Today is March 14, as in 3.14, which is why it is known as Pi Day. It is a good time to look at this question from a fan on Facebook who wondered how many decimals of the mathematical constant pi (π) NASA-JPL scientists and engineers use when making calculations:
Does JPL only use 3.14 for its pi calculations? Or do you use more decimals like say: 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360
Credit: NASA/JPL-Caltech
NASA/JPL posed this question to the director and chief engineer for NASA’s Dawn mission, Marc Rayman. Here’s what he said:
Thank you for your question! This isn’t the first time I’ve heard a question like this. In fact, it was posed many years ago by a sixth-grade science and space enthusiast who was later fortunate enough to earn a doctorate in physics and become involved in space exploration. His name was Marc Rayman.
To start, let me answer your question directly. For JPL’s highest accuracy calculations, which are for interplanetary navigation, we use 3.141592653589793. Let’s look at this a little more closely to understand why we don’t use more decimal places. I think we can even see that there are no physically realistic calculations scientists ever perform for which it is necessary to include nearly as many decimal points as you present. Consider these examples:
1. The most distant spacecraft from Earth is Voyager 1. It is about 12.5 billion miles away. [This answer was from 4 years back, and now Voyager 1 is over 13.8 billion miles away. ed.] Let’s say we have a circle with a radius of exactly that size (or 25 billion miles in diameter) and we want to calculate the circumference, which is pi times the radius times 2. Using pi rounded to the 15th decimal, as I gave above, that comes out to a little more than 78 billion miles. We don’t need to be concerned here with exactly what the value is (you can multiply it out if you like) but rather what the error in the value is by not using more digits of pi. In other words, by cutting pi off at the 15th decimal point, we would calculate a circumference for that circle that is very slightly off. It turns out that our calculated circumference of the 25 billion mile diameter circle would be wrong by 1.5 inches. Think about that. We have a circle more than 78 billion miles around, and our calculation of that distance would be off by perhaps less than the length of your little finger.
2. We can bring this down to home with our planet Earth. It is 7,926 miles in diameter at the equator. The circumference then is 24,900 miles. That’s how far you would travel if you circumnavigated the globe (and didn’t worry about hills, valleys, obstacles like buildings, rest stops, waves on the ocean, etc.). How far off would your odometer be if you used the limited version of pi above? It would be off by the size of a molecule. There are many different kinds of molecules, of course, so they span a wide range of sizes, but I hope this gives you an idea. Another way to view this is that your error by not using more digits of pi would be 10,000 times thinner than a hair!
3. Let’s go to the largest size there is: the visible universe. The radius of the universe is about 46 billion light years. Now let me ask a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places. If you think about how fantastically vast the universe is — truly far beyond what we can conceive, and certainly far, far, far beyond what you can see with your eyes even on the darkest, most beautiful, star-filled night — and think about how incredibly tiny a single atom is, you can see that we would not need to use many digits of pi to cover the entire range.
#### 2 Commentson "How Many Decimals of Pi Do We Really Need? NASA Answers"
1. Rahulakumar Subramaniam | August 8, 2021 at 2:11 pm | Reply
pi=22/7=3+1/7
1/7=0.14285714— in base 10
1/7=0.1 in base 7
If you want to get bounded number, you have to change base of numbers.
2. Rahulakumar Subramaniam | August 11, 2021 at 10:29 am | Reply
Base – 7
We should take the numbers 0,1,2,3,4,5,6
pi= 22/7=3+1/3 in base 10
we can convert the constant pi from base 10 to base 7
3 doesn’t change in base 7
1/7=0.1 in base 7 , therefor for pi=3.1 in base 7
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Sites:
##### From Diffraction, Fourier Optics and Imaging
7.1 INTRODUCTION
Approximations for computing forward and inverse diffraction integrals are of vital
significance in many areas involving wave propagation. As discussed in Chapters 4
and 5, approximations such as the Fresnel approximation, the Fraunhofer
approximation, and the more rigorous angular spectrum method (ASM) all involve
the Fourier transform, its discrete counterpart, the discrete Fourier transform, and its
fast computational routine, the fast Fourier transform (FFT).
The Fresnel approximation is valid at reasonable distances from the input plane
whereas the Fraunhofer approximation is valid in the far field. The ASM is a
rigorous solution of the Helmholtz equation; its numerical implementation is usually
done with the FFT, and possibly other digital signal processing algorithms, with their
related approximations [Mellin and Nordin, 2001; Shen and Wang, 2006].
When the sizes of the diffracting apertures are less than the wavelength, scalar
diffraction theory yields nonnegligible errors, and other numerical methods such as
the finite difference time domain (FDTD) method and the finite element method
(FEM) may become necessary to use [Kunz, 1993; Taflove and Hagness, 2005].
However, these methods are not practical with large scale simulations as compared
with methods utilizing the FFT. With diffracting aperture sizes of the order of...
More >>
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##### Topics of Interest
8.1 INTRODUCTION Geometrical optics (ray optics) involves approximate treatment of wave propagation in which the wavelength λ is considered to be infinitesimally small. In practice,...
5.1 INTRODUCTION Fresnel and Fraunhofer approximations of the scalar diffraction integral allow simpler Fourier integral computations to be used for wave propagation. They also allow...
3.1 OVERVIEW In addition to the Fourier and Hankel transforms, there are other integral transforms that are useful in optics and optical signal processing, which will be introduced here from a...
6.6 Evaluation of Reconstruction Algorithms To study the approximations involved in the reconstruction process it is necessary to calculate scattered data assuming the forward approximations are...
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+0
# Geometry Problem
+1
912
3
Triangle AHI is equilateral. We know BC, DE, and FG are all parallel to HI and AB = BD = DF = FH. What is the ratio of the area of the trapezoid FGIH to the area of triangle AHI? Express your answer as a common fraction.
Feb 13, 2018
#1
+569
+1
I will call the distance from B to C x.
Since all of the points are equal distance from eachother, $$FG = 3x \operatorname{and} HI = 4x$$.
Area of a trapezoid = $$^{1}\!\!/\!_{2}(b_1+b_2)\times h$$, where h = height, b1 = length of bottom, and b2 = length of top.
Area of FGIH = $$7x(\tfrac{\sqrt{3}}{2}x)=\tfrac{7\sqrt{3}}{2}x^{2}$$
Area of ABC = $$x(\tfrac{\sqrt{3}}{2}x)=\tfrac{\sqrt{3}}{2}x^{2}$$
Area ratio of FGIH to ABC = $$\tfrac{7\sqrt{3}}{2}x^{2}:\tfrac{\sqrt{3}}{2}x^{2}$$ --> $$7:1$$
.
Feb 13, 2018
#2
+103082
+1
Thaks, helperid...here's an alternative solution
Without a loss of generailty we can let AB = 1
So AHI = (1/2)(4)*2 sin (60) = 4√3 (1)
Area of AFG = (1/2)(3)^2 sin (60) = (9/4)√3 (2)
So area of trapezoid FGIH = √3 [ 4 - 9/4] = √ 3 [ 7/4]
So ratio of area of trapezoid FGIH to AHI =
√3 {7 /4] / [ 4√3 ] = 7 / 16
EDIT TO CORRECT AN IDIOTIC ERROR !!!!
Feb 13, 2018
edited by CPhill Feb 13, 2018
#3
+23059
+2
Triangle AHI is equilateral. We know BC, DE, and FG are all parallel to HI and AB = BD = DF = FH.
What is the ratio of the area of the trapezoid FGIH to the area of triangle AHI?
$$\text{Let FG = s } \\ \text{Let HI = c } \\ \text{Area of the triangle AHI = A_{AHI} } \\ \text{Area of the triangle AGF = A_{AGF} } \\ \text{Area of the trapezoid FGIH = A_{FGIH} } \\ \text{Let AK = h (height of the triangle_{AGF}) } \\ \text{Let AL = H (height of the triangle_{AHI}) } \\ \text{Let AF = \frac34 c }$$
1.
$$\begin{array}{|rcll|} \hline A_{AHI} &=& A_{AGF} + A_{FGIH} \\\\ \dfrac{cH}{2} &=& \dfrac{sh}{2} + \left( \dfrac{s+c}{2}\right)(H-h) \quad & | \quad \times 2 \\\\ cH &=& sh + (s+c)(H-h) \\\\ \not{cH} &=& \not{sh} + sH-\not{sh}+\not{cH}-ch \\\\ \mathbf{ch} &\mathbf{=}& \mathbf{ sH } \qquad (1) \\\\ &\text{or}& \\\\ \mathbf{\dfrac{s}{c}} &\mathbf{=}& \mathbf{\dfrac{h}{H}} \qquad (2) \\ \hline \end{array}$$
2.
$$\begin{array}{|rcll|} \hline \text{ratio} &=& \dfrac{ A_{FGIH} } {A_{AHI}} \\\\ &=& \dfrac{ \left( \dfrac{s+c}{2}\right)(H-h) } {\dfrac{cH}{2}} \\\\ &=& \dfrac{(s+c)(H-h)}{cH} \\\\ &=& \dfrac{sH-sh+cH-ch}{cH} \quad & | \quad ch=sH \qquad (1) \\\\ &=& \dfrac{cH-sh}{cH} \\\\ &=& 1-\dfrac{sh}{cH} \quad & | \quad \dfrac{s}{c} = \dfrac{h}{H} \qquad (2) \\\\ &=& 1-\dfrac{h^2}{H^2} \\\\ \mathbf{\text{ratio}} & \mathbf{=} & \mathbf{ 1-\left(\dfrac{h}{H}\right)^2} \\\\ && \boxed{\mathbf{3.}\\ \dfrac{h}{\frac34 c} = \dfrac{H}{c} \\ h = \frac34 c \cdot \dfrac{H}{c} \\ h = \frac34 \cdot H \\ \mathbf{\dfrac{h}{H} = \frac34} } \\\\ \mathbf{\text{ratio}} & \mathbf{=} & \mathbf{ 1-\left(\frac34 \right)^2} \\\\ &=& 1- \frac{9}{16} \\\\ &=& \frac{16-9}{16} \\\\ &\mathbf{=}& \mathbf{\dfrac{7}{16}} \\ \hline \end{array}$$
So the ratio of areas is $$\mathbf{\dfrac{7}{16}}$$
Feb 13, 2018
edited by heureka Feb 13, 2018
edited by heureka Feb 13, 2018
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7 th Grade Assignments
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# 7 th Grade Assignments - PowerPoint PPT Presentation
## 7 th Grade Assignments
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##### Presentation Transcript
1. 7th Grade Assignments T1 Week 1
2. 7th Grade Independent Study Assignments T1Week1Due September 8th • Physical Education Assignments: • ______P.E. Log (230 Minutes) • 5 PE Credits Due November 15th • Grade__________ • TW____________% Name__________ Total Work Completed: _______% Help: dbusk@psusd.us www.mrssyfert.wikispaces.com 760-404-7289 (1-3) *No Messages Please
3. Evaluate Algebraic Expressions-Pg. 6 Writing Algebraic Expressions-Pg. 10 Math Ch. 1 Lesson 1
4. Math • Assignments: • __________Do go.hrw.com lessons 1-1 and 1-2. __________________(P.S) • __________Ch.1 Lesson 1 Questions Pgs. 8-9 • __________Ch. 1 Lesson 2 Questions Pgs. 12-13 • __________Homework and Practice Workbook Lessons 1-1 and 1-2 • Grade_________ • TW___________% Objectives: -Students will substitute variables correctly in algebraic expressions -Students will translate between algebraic expressions and word phrases
5. 7th Grade Language ArtsChapter 1 Facing Your Fears!
6. My Fear…. What’s yours?
7. Character Conflict Local rite of passage is diving from a high bridge…will she do it? • Melly-Young Girl • Grandmother- The Dive-Pg. 50
8. Plot • Setting • Conflict • Resolution • Crinkling • Stubble • Caressed • Wafting Vocabulary
9. Why are you reading this? • What should you learn from it? • To learn how conflict makes stories/movies more interesting Purpose
10. Objectives: Students will…. • Be able to use selection vocabulary words in a cloze activity (Lang. Objective) • Analyze the structural elements of plot • Understand how conflicts are • Be able to identify and correctly use • Be able to use textbook 7th Grade Language ArtsT3-Week 1
11. Through(All Students) • ____Read “The Dive” Pgs. 50-60. Do Focus Questions A-O. • ____Reader-Writer Notebook Activities for “The Dive…” • ____Literature Response Pg. 61 • ____Workbook Packet • ____Fear Essay (Portfolio Pre) • Study for Dive Test (To Be Done in Class) • Beyond (Advanced and Extra Credit) • ____Watch “Jungle Book” the movie. Complete Movie Summary. • ____Read Novel…”Jungle Book” By Rudyard Kipling • ____Research Base Jumping or other extreme…Do PE Report/PPT • ____Choices Activities Pg. 63 Into (Strategic Students)____Interactive Reader Section for “The Dive”____Fluency Builder Activities (3 per week administered by parent)
12. Packet-To return To give to Guardians… PE Starts Oct. 1 Put on Post It Schedule All tutoring starts Oct. 1 Put on Post It Calendar Min. Days • Packet for work • Technology Use • Insurance??? • Return to me if you have insurance • Emergency Card • Grade and Attendance • PPT Start the year right!
13. Life Science Objectives: -Students will understand how measurements are used to conduct experiments -Students will understand the difference between Mass and Matter. -Students will understand what a solution is
14. Unit Intro Note Taking Title 2 Columns Terms Left-Blue Headings Right-Details • Book Walk • Related topics • What you know Life Science
15. Assignments: • Into/Through • _____Read Ch. 1 Lesson 1. Take Cornell Notes. • _____Do Ch. 1 Lesson 1 Questions Pg. 10 • _____Unit Intro Worksheet • _____Investigation Manual Lesson 1A (Use your family instead of a class) • Grade__________ • TW___________% Life Science Beyond: Find an experiment online that you would like to do as homework. Bring to class.
16. World History Objectives: -Students will factors that influenced the settlement of Rome -Students will explain the emergence of Christianity -Students will Use note taking skills to identify key information in the lesson
17. The World • Pretest in pencil • Map in Textbook • Continent Review • Test next week Geography Review
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# using the "Order" function to determine the US state with smallest population size, from a data set
I'm trying to order a data set using the following:
library(dslabs)
data(murders)
pop<- murders\$population
pop_min_index <- order(pop)
min(pop_min_index)
When I use this, R presents "pop_min_index" as all the values from the data set, when I just want the first value, i.e. the answer should be "51", where as my result is coming out as "51, 9, 46, 35, 2, 42", etc.
I've tried using:
min(pop_min_index)
but this doesn't seem to work either
Does anyone know what is wrong my code?
Thanks
in general if you have a vector and want only the first entry use
``````head(order(pop),n=1)
``````
will give 51
Thanks,
still seems to be keeping pop_min_index as the full list:
basically I am completing a practice exercise in R, and the question is as follows:
I have to match the answer that has already been input, and achieve the same answer
well, if you need to assign the result to a specific name then thats done with the `<-` operator. so in this case
``pop_min_index <- head(order(pop),n=1)``
THANK YOU
I hadn't even heard of "head", is this the only way to do this question, as we haven't been taught this yet, so just wondering if there is some content missing from the instructions i've been given?
you may have been taught how to arbitrarily index into a vector with square brackets.
In this case that would be
``pop_min_index <- order(pop)[1]``
That was exactly my issue. I kept inputting that as:
library(dslabs)
data(murders)
pop<- murders\$population
pop_min_index <- order(pop)
pop_min_index[1]
This topic was automatically closed 21 days after the last reply. New replies are no longer allowed.
If you have a query related to it or one of the replies, start a new topic and refer back with a link.
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Top 10 HOW FAR DOES A 22 BULLET TRAVEL Answers
# How Far Does A 22 Bullet Travel
Category: Travel
## 1. How Far Can A 22 Bullet Travel? – Aiming Expert
A 22 short round fired from a pistol will travel around 200 yards (170m) at sea level when aimed flat. A 22LR cartridge in a rifle will be accurate to 300m so (1)
Aug 10, 2018 · 15 answersGiven enough space for vertical drop, .22 Long Rifle bullets are capable of travelling about 2000 yards (~1830 meters) horizontally.What is the effective range of a .22 LR? – Quora8 answersJul 27, 2018What is range of .22 rifle? – Quora25 answersAug 29, 2018If you shoot a 22lr bullet at 90 degrees to the sky, can 7 answersFeb 8, 2018How fast does a 22 bullet travel? – Quora3 answersJun 20, 2019More results from www.quora.com(2)
So the answer is, a 22 round CAN kill as far out as you can hit a target which in near perfect conditions can go as far out as 440 yards or so.Nov 8, 2018 · Uploaded by Iraqveteran8888(3)
## 2. Expert: ‘The average person doesn’t realize how far a bullet …
Apr 2, 2019 — “The most common and easiest round to get is the .22 caliber. It can travel around 1.5 miles at a 12,000 foot altitude,” Paskiewicz said. Many (4)
Nov 8, 2009 — 22 LR fired at a 30-degree angle from a rifle will travel approx. a mile or so. The figure 1.14 miles or 2000 yards comes to mind. I think that (5)
Mar 22, 2021 — 22lr can travel for up to 100 yards. This is the effective range of the round. However, it is capable of going up to 2000 yards when fired (6)
## 3. How far can a 22 rifle bullet travel? – BoardGamesTips
What’s the difference between 22 long and 22 long rifle? How fast does a 223 bullet travel? What is the fastest bullet in mph? Can a bullet travel faster than (7)
May 29, 2008 — 1.25 miles for a high velocity .22 (according to the manufacturer) If you are at a higher elevation than the dogs (due to your height or the (8)
## 4. How far can a .22 shoot? – Factual Questions – Straight Dope …
Oct 6, 2002 — How far will the bullets go shot from a .22 ? Fired at a 45-degree angle above the horizontal, these bullets can travel over a mile.(9)
22 Long Rifle bullets are capable of travelling about 2000 yards (~1830 meters) horizontally. All four rounds cross their zero distance at 100 yards, then the (10)
How Far Does A 22 Pistol Bullet Travel. For a 22 caliber pistol bullet, the distance traveled will also depend on a number of variables.9 answers · Top answer: So yes, a 22LR bullet will travel out to 500m (half a kilometer) as long as your zero your (11)
‘Range’ is the distance your bullet will travel. Your safe and effective range is A .22 caliber bullet can travel over 1 ½ mile. A centerfire bullet can (12)
A 9mm-calibre Luger Parabellum round fired from a handgun travels at about 370m/s. To optimise its range, it would be fired at an angle of 45° and should (13)
## 5. Range of a Handgun Bullet – The Physics Factbook
### How To Buy A Vacation Rental Property?
At sea level, the flat-based bullet will travel a maximum of 3,800 m, A bullet almost never travels this far before it actually hits something.(14)
Take a look at the 22LR drop at 200 yards! The 22LR bullet drops roughly 32 inches at 200 yards. As you can see the by the 22 WMR Trajectory Chart which is (15)
Bullets fired from a rifle will have more energy than similar bullets fired from a Bullet travel through a gun barrel is characterized by increasing (16)
## 6. How far will a 40 caliber bullet go? | EveryThingWhat.com
Mar 30, 2021 — How far will a .22 bullet travel before dropping? The .22 LR is effective to 150 yd (140 m), thoughpractical ranges tend to be less. After 150 (17)
Sep 19, 2019 — Of course, handguns aren’t long-distance firearms. Rifles are built to be more accurate from greater distances. The notorious AR-15 rifle is the (18)
223 bullet travel in mph? Round, Caliber, Muzzle velocity .22LR, 0.223 in (5.7 mm), 1,500 ft/s (19)
How fast does a 22 bullet travel in mph? Hyper-velocity. Many . 22 LR cartridges use bullets lighter than the standard 40 gr, fired at even higher velocities.(20)
## 7. How far will a .22 magnum bullet travel? – Answers
Aug 1, 2011 — approx.1.5 to 2 miles,i’ve read information saying the effective range is 125 yard but i have multiple hits at 197 yards using hornady (21)
Sep 23, 2021 — If you’re wondering how far can a 223 bullet travel, then read through this article to find out. We discuss the bullet’s strength and (22)
How fast is a 22 bullet mph? — What is the fastest bullet? How far can a 9mm bullet travel? Does a bullet cause a sonic boom? Will a .22 stop an attacker?(23)
## 8. Quick Answer: How Far Will A .380 Bullet Travel? – pay for …
Is a 22 or 380 more powerful? — Is a 22 or 380 more powerful? Is .38 and .380 the same? What is the maximum distance a bullet can travel? Can a bullet hit a (24)
Jan 3, 2013 — I mean like if you just shoot a .22 short bullet, how long will it travel before it stops. Comment.(25)
So how far can a bullet travel? It turns out that in a test performed under optimum conditions and fired at a 40 degree up angle a 9mm bullet traveled 2405 (26)
## 9. Know Your Bullet Drop and Range When Hunting Whitetails
Knowing your bullet drop at various ranges, having a good rangefinder and 14.25 inches low at 300 yards – should you ever dare to shoot that far.(27)
Sep 23, 2009 — What I will do instead is make a numerical calculation of the motion Oh – also, they measured how far a 9mm bullet penetrated into the (28)
## 10. How fast is a bullet in mph – HowtoCreate.com
### How To Track Lost Luggage?
Nov 20, 2021 — A 22lr caliber bullet will exit the barrel between 1200 and 1750 feet per second (820 to How fast does a 50 cal bullet travel in mph?(29)
This article will deal with the primary forces on a bullet’s trajectory, velocity determines how far the bullet travels before it hits the ground.(30)
Feb 17, 2007 — All Distances are approximate using conventional ammunition. Rifle Ranges: .22 Short Sea Level – .75 Miles, 12,000 Ft – 1.5 Miles .22 LHV Sea (31)
Do bullets travel faster than sound? Do bullets travel in a straight line? a stopwatch and have someone time how fast it takes you to run that distance.(32)
Apr 21, 2015 — So it got me wondering, how far could bullets travel when fired must have shot a wheelbarrow full of .22 rimfire at various birds on the (33)
Jan 5, 2009 — Jack O’Connor, in The Complete Book of Rifles and Shotguns , stated that a normal velocity (about1,080fps at the muzzle) 40 gr. lead .22lr, (34)
I wondered, what’s the average distance the bullet would travel before hitting a tree? Assumptions. (1) Bullets travel about 2 miles. source.(35)
Feb 15, 2017 — Falling bullets can kill you, even if the guns are fired high into run into: a bullet from an AK-47 leaves the rifle traveling at over (36)
Sep 6, 2000 — I want to know to compare to a .22 cal pellet gun I can buy (I know its not as powerful as the real thing). Bib.25 posts · It depends on the round. Probably somewhere between 800-1200 f/s. amish (37)
This means the bullet takes just as long to fall to the ground as it would if it were dropped, despite it now travelling a large horizontal distance in the (38)
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# Trigonometric Substitution
posted by on .
Trigonometric Substitution
a. Explain why the substitution x = asin(Theta) guarantees that cos(Theta) is greater than or equal to 0 .
b. What sort of restriction must we put on x and a so that the substitution x = asec(Theta)will guarantee that tan(Theta) is greater than or equal to 0?
• Trigonometric Substitution - ,
x=1/2tan0 write f(X)= x/1+4
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Peter's Physics Pages
An Introductory Physics Course with Peter Eyland
Lecture 2 (Measurement)
In this lecture the following are introduced:
•Laboratory instruments: the balance, Vernier calipers, Micrometers, stopwatches, measuring cylinders,
•Uncertainty and accuracy: random and systematic errors,
•Accumulating errors in the laboratory,
•Graphing with errors,
•Volume and area formulae.
Laboratory instruments
The balance
There are balances available in the Laboratory that you can use to measure the mass of an object up to a few kilograms.
The balance should be "zeroed" before use by using the screw adjustments (this may take some time so if it's OK don't change anything).
If a liquid is to be weighed then the mass of the container must be subtracted from the total.
The accuracy of the balance should be estimated and this should be reflected in the number of significant digits.
Vernier Calipers
Vernier calipers are available in the Laboratory which you can use to measure lengths up to 200mm.
A vernier scale enables you to read to 1/10 of the smallest main scale division of 1mm, i.e. 0.1mm. Note that if the object is irregularly shaped the accuracy of a measurement may have to be determined by making measurements at a number of places.
Systematic errors may arise if the calipers are hotter or colder than its reference temperature. If the scale has expanded due to heating then the apparent measurements will be always be read as smaller than they really are.
Likewise if the scale has contracted because of the cold then the apparent measurements will always be read as larger.
Micrometers
Micrometers are generally available in the Laboratory which enable you to read to 1/100 of the smallest main scale division of 1mm, i.e. 0.01mm.
Each complete turn of the barrel moves the head by 0.5mm. The barrel scale has 50 divisions to divide the 0.5mm into 0.01mms.
The "zero" has to be established by using the ratchet to close the heads.
Again, the accuracy of the meter may exceed variations in the object.
Stopwatches
Stopwatches are used to measure time intervals.
There may be systematic errors if the clock runs fast or slow.
There will be random errors from estimating when to stop the clock and also from your reflexes that will slow as the day progresses.
This means that the error will usually be more than the smallest time measurement of 0.001s.
Measuring cylinders
Measuring cylinders are used to measure the volume of a liquid.
Different sizes enable you to measure a range of volumes.
Uncertainty and accuracy
The carpenter's rule is "measure twice and cut once", so in Physics we rely on taking a number of measurements to improve our confidence in the result. A number of measurements gives both a mean (or average) for the measurement and a variation for the uncertainty (or error).
A "mistake" is a human error that occurs by misreading a scale, or neglecting a zero error, etc.
An "error" is an estimate of measurement accuracy that is stated as an uncertainty.
Human errors (in the form of mistakes) are not accepted in Laboratory measurements but measurement error estimates are necessary.
The mean is calculated from a number of measurements by finding the sum and dividing by the number of measurements.
An estimate of the error is given by the largest difference from the mean, though this rule often has to be modified because it overestimates the error.
Random errors occur because of measurement technique, reaction time variations, variations in the system conditions between measurements (temperature, pressure etc), or simply variations in object dimensions. They have no pattern and cannot be completely eliminated.
Systematic errors always push the measurement higher or lower. They occur because the scale on the instrument may have changed with temperature, or some physical element within the measuring instrument (e.g. a spring) may have changed due to age or humidity etc. The latter are called "calibration" errors.
An overall uncertainty has to be calculated when a quantity is calculated by addition, subtraction, multiplication or division of measurements which have error estimates. The textbook does this is by working out the extreme cases.
Example:
A rectangular sheet of metal has
length measurements, 252mm, 247mm, 251mm, 247mm, 249mm ,and
breadth measurements, 100mm, 102mm, 99mm, 98mm, 101mm.
Find the area and its error.
There are 3 significant figures in the measurements so the mean is written as 249mm.
The largest difference from the mean is 3mm.
The result is then 249±mm, but this seems a bit too much because only one of the five measurements is 3mm from the mean.
Four out of the five are within 2mm.
It is then better to estimate it as 249±2mm.
This has the correct number of significant figures, and the largest variation is 2.
The result is 100±2mm
Two out of the five are within 2mm.
Three out of the five are within 1mm.
It is better to estimate it as 100±1mm.
The largest area is 251 x 101 = 25351mm2
The smallest area is 247 x 99 = 24453mm2
The final result is 24900±400mm2
Accumulating errors in the Laboratory
The laboratory manual (not included here) gives another way of accumulating errors, which you should use.
The absolute error of a quantity is simply the variation in that quantity [].
The relative error of a quantity is the fraction (or percentage) of the variation to that quantity [ or ].
For errors that are based on the same source of error, the following applies:
• When quantities are added or subtracted, the absolute errors are added.
• When quantities are multiplied or divided, the relative errors are added.
Example
The mass of an object was found to be 9.0±0.1g, and its volume from the difference between 98.00±0.05ml and 97.00±0.05ml.
Find the density.
The volume is given by the difference in the volume measurements: 98.00 - 97.00 = 1.00ml
The error in the volume is the sum of the absolute errors: 0.05±0.05 = 0.1ml.
The volume with its error is written as: 1.0±0.1ml
The density is given by the quotient: mass per unit volume.
The mass has a relative error of .
The volume has a relative error of
The relative error in the density (a quotient) is the sum of the relative errors, i.e.
=10% +1.1%= 11%
The absolute error in the density is then
The density and error is .
The number ranges from 8010 to 9990, but since the first digit changes, it is probably best to write the density as .
Graphing
In practice, all experimentally determined data are subject to the uncertainties in accuracy called "errors". When experimental data are presented in a graphical format, the measurements are drawn with "error bars" which give the range of uncertainty.
For example
In this case, there appears to be a linear (or straight line) relationship between the quantities.
Three slopes can be found. The two "worst lines of best fit", and then the line of best fit.
The steepest and shallowest slopes that go through all the error bars are first drawn.
Then using the point where the two worst lines cross, the best-fit line is drawn.
The best-fit line gives the mean slope and the error in the slope comes from (usually) the largest difference from the mean. If there is no line that can be drawn through all the error bars, then either the errors have been underestimated, or there is no straight line relationship. Most experimental relationships can be plotted to give a straight line (with sufficient ingenuity!) and this can give the constants of the relationship.
Example
Using a graphical technique, it is possible to determine the stiffness factor of a spring by hanging it vertically, suspending a mass from its lower end, setting the mass into vertical (up and down) oscillation, and measuring the period of that vibration.
Theory gives that for a spring of stiffness factor, k, and mass, m, if a mass, M, is suspended from it and set into vertical oscillation, the period of oscillation, T, is given by:
For each of a number of different masses suspended from the spring, the period of oscillation of the mass is measured.
To produce a straight-line graph, the relationship has to be squared and re-arranged:
A plot of M against T2 should yield a straight line graph.
• the slope of the graph would be
• the intercept on the vertical axis would be .
This would need to be projected in any graph because a zero mass is not allowed.
The following table, gives experimental data with estimates of the uncertainties.
M ΔM T ΔT T2 ΔT2 kg kg s s - s2 - s2 0.0500 0.0003 0.48 0.05 10 0.23 20 0.05 0.0700 0.0004 0.56 0.05 9 0.31 18 0.06 0.1000 0.0006 0.63 0.05 8 0.40 16 0.06 0.1500 0.0009 0.67 0.05 7 0.45 14 0.06 0.2000 0.0010 0.77 0.05 6 0.59 12 0.07 0.2500 0.0015 0.89 0.05 6 0.79 12 0.09 0.3000 0.0020 0.93 0.05 5 0.86 10 0.09 0.3500 0.0020 0.98 0.05 5 0.96 10 0.10 0.3700 0.0020 1.03 0.05 5 1.06 10 0.11
The errors in the mass are too small to worry about, but the errors in the time are significant.
Drawing the three slopes (note that "joining the dots" doesn't give any useful information).
The three slopes are 0.46, 0.41, and 0.35.
This means the slope and error is 0.41±0.05 kg.s-2.
The spring constant is
The error in the spring constant k, will have the same %error as the slope because the multipliers are pure numbers with no error.
The slope has a error.
The error in spring constant k, is
This shows that the "units" column is in doubt, so the answer is 16±2 N.m-1.
The error in the time has dominated.
Area and volume formulae
________________________________
________________________________
________________________________
Summarising:
In the Laboratory the following instruments are used:
mass balances, Vernier calipers, Micrometers, stopwatches and measuring cylinders.
Random errors occur because of measurement technique, reaction time variations, variations in the system conditions between measurements (temperature, pressure etc), or simply variations in object dimensions.
Systematic errors always push the measurement higher or lower. They occur when the scale on the instrument has changed with temperature, or some physical element within the measuring instrument (e.g. a spring) has changed due to age or humidity etc.
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# How do you write the number 9 in Chinese?
Contents
Once you’ve learned the characters that represent 4 (四 sì), 5 (五 wǔ), 6 (六 liù), 7 (七 qī), 8 ( 八 bā), 9 (九 jiǔ) and 10 (十 shí), Chinese numbers, up to 99 (九 十九 jiǔshíjiǔ), are quite easy to learn.
## What are numbers 1 to 10 in Chinese?
Chinese Numbers 1 to 10： 一到十
English Chinese Pinyin
seven
eight
nine jiǔ
ten shí
## How do you write numbers in Chinese?
Number System (數字系統)
1. 0: 〇(零)： líng.
2. 1: 一(壹) yī
5. 4: 四(肆) sì
6. 5: 五(伍) wǔ
8. 7: 七(柒) qī
## How do you write the number 0 10 in Chinese?
Chinese Lesson – Count from 0 to 10
• Zero (零): [líng]Ling with an upward inflection.
• One (一): [yī] Ee with a long “e” sound.
• Two (二): [èr] Are with a downward inflection for the letter “r”.
• Three (三): [sān] San without an inflection.
• Four (四): [sì] Suh. …
• Five (五): [wǔ] Woo with a downward-up inflection.
## How do you write Chinese characters?
Here are the essential stroke order rules for writing simplified Chinese characters:
1. Top to bottom. …
2. Left to right. …
3. First horizontal, then vertical. …
4. First right-to-left diagonals, then left-to-right diagonals. …
5. Center comes first in vertically symmetrical characters. …
6. Move from outside to inside and close frames last.
IT\'S FUNNING: Are there any bridges in China?
## How do you write Si in Mandarin?
Mandarin Chinese
Mandarin
Native to China, Taiwan, Singapore, and overseas communities
Native speakers 920 million (2017) L2 speakers: 200 million (no date)
Language family Sino-Tibetan Sinitic Mandarin
## How do you write 3 in Mandarin?
Once you’ve learned the characters that represent 4 (四 sì), 5 (五 wǔ), 6 (六 liù), 7 (七 qī), 8 ( 八 bā), 9 (九 jiǔ) and 10 (十 shí), Chinese numbers, up to 99 (九 十九 jiǔshíjiǔ), are quite easy to learn.
Chinese Numbers 0-10.
Number Hanzi Pinyin
1
2 Èr
3 Sān
4
## How do you count to 10 in Mandarin Chinese?
To count to 10 in Mandarin Chinese, say “yī, èr, sān, sì, wŭ, liù, qī, bā, jiŭ, shí.” If you want to count higher than 10, say 10, or “shí,” followed by the second number.
## How do you write in Chinese?
There are eight basic rules of stroke order in writing a Chinese character:
1. Horizontal strokes are written before vertical ones.
2. Left-falling strokes are written before right-falling ones.
3. Characters are written from top to bottom.
4. Characters are written from left to right.
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Mothers attempting to balance parenthood and academics.
Math Geek Mom: New Buildings
March 20, 2014 - 7:46pm
By
Recently, as we were studying probability in my Statistics class, a student asked me about the use of probability in math and economics. Actually “where will we ever use this?” might be a better way of explaining what was asked. I explained that probability is the foundation of all of statistics, and that economists, whose field is based on statistics, also use probability to study decisions made under uncertainty, when the best one can do is maximize profit or utility given the expected value of the outcome of a choice. One of the best examples of this is found in the study of how an insurance company sets the rates it charges customers. As many of us know, people with many traffic citations are often charged more for auto insurance than are those without any, since those who have had accidents or tickets are seen as higher risks for the company to insure. This is also why the issue of pre-existing conditions was so important in the recent overhaul of health insurance, as it was customary for those with medical conditions to be denied access to insurance or to be charged more for insurance, since they were seen as greater risks for expensive illnesses in the future. I found myself thinking of this as I opened the newspaper a few days ago to see a picture of the destruction on Ursuline’s campus only hours after the tornado tore through it in late July of last year. It seems that after months of negotiations, Ursuline College has come to a settlement with the insurance company. We now can continue the process of rebuilding our picturesque campus.
Although we are sad that the construction (temporarily, we hope) displaced the Math department’s offices, I must admit that the new buildings that will soon be springing up on campus look amazing. The new gym will be one worthy of our new-found NCAA Division II status, and the new building for the Creative and Healing Arts and Sciences will be a joy to have on campus. However, although the college has reached a settlement with the insurance company, it seems that quite a bit of money still needs to be raised. Some of the money was given to us long before the tornado, by donors who wanted to be part of the important role that Ursuline College plays in the healing professions in the Cleveland area. Indeed, it is currently the case that our Nursing School is the part of the college that is most recognizable to the general public. Alas, the Math major is not (yet!) the major that draws the most students to Ursuline.
While there is still money to be raised, construction and demolition can begin soon. This construction on campus is made possible only because the insurance company and Ursuline College were able to come to an agreement about a payoff on the insurance policy the college held. I imagine that, in a few years, when my students ask me what role probability plays in the “real world,” I will be able to point to the new classroom building on campus, or remind them of their sports practices which are held in a state of the art gym. This will give them an answer to their question that they can both walk into and touch with their own hands. Indeed, someday my own daughter may practice in or wander the halls of those same buildings.
Wishing everyone wonderful spring!
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# Resources tagged with: Graph sketching
Filter by: Content type:
Age range:
Challenge level:
### There are 15 results
Broad Topics > Coordinates, Functions and Graphs > Graph sketching
### Exploring Cubic Functions
##### Age 14 to 18 Challenge Level:
Quadratic graphs are very familiar, but what patterns can you explore with cubics?
### Up and Across
##### Age 11 to 14 Challenge Level:
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the dot affects its vertical and horizontal movement at each stage.
### How Far Does it Move?
##### Age 11 to 14 Challenge Level:
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.
### Speeding Up, Slowing Down
##### Age 11 to 14 Challenge Level:
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.
### Mathsjam Jars
##### Age 14 to 16 Challenge Level:
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
### Maths Filler
##### Age 11 to 14 Challenge Level:
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
### More Parabolic Patterns
##### Age 14 to 18 Challenge Level:
The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.
### Bio Graphs
##### Age 14 to 16 Challenge Level:
What biological growth processes can you fit to these graphs?
### Maths Filler 2
##### Age 14 to 16 Challenge Level:
Can you draw the height-time chart as this complicated vessel fills with water?
### Parabolic Patterns
##### Age 14 to 18 Challenge Level:
The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.
### Back Fitter
##### Age 14 to 16 Challenge Level:
10 graphs of experimental data are given. Can you use a spreadsheet to find algebraic graphs which match them closely, and thus discover the formulae most likely to govern the underlying processes?
### Immersion
##### Age 14 to 16 Challenge Level:
Various solids are lowered into a beaker of water. How does the water level rise in each case?
### Guessing the Graph
##### Age 14 to 16 Challenge Level:
Can you suggest a curve to fit some experimental data? Can you work out where the data might have come from?
### What's That Graph?
##### Age 14 to 16 Challenge Level:
Can you work out which processes are represented by the graphs?
### Fill Me Up
##### Age 11 to 14 Challenge Level:
Can you sketch graphs to show how the height of water changes in different containers as they are filled?
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## Free Lottery Wheeling System
Written by Noah Faulkner
A lottery wheeling system means you choose a base 'subset' of numbers, for example in a 6 number lottery, you might choose numbers 1 to 7, and then combine them in ways ensuring that all 7 numbers 'get a go' in 6 number lines. By playing combinations of a set of numbers, you are doing something similar to 'permutations' in football pools.
Let's look at a typical choose 6 numbers from 49 lottery game. Say your 'base' subset of favorite numbers are 1,2,3,4,5 and 6. However, you want to play 7 as well. How can you best 'spread out' your lines in order to maximize possibility of winning should any of your base numbers come up? By 'wheeling' them. Here's an example.
01 02 03 04 05 06 01 02 03 04 05 07 01 02 03 04 06 07 01 02 03 05 06 07 01 02 04 05 06 07 01 03 04 05 06 07 02 03 04 05 06 07
Notice how columns change as you move downwards. As you can see, there are 7 entries (at 7 dollars or 7 pounds cost). While these seven lines have same chance of winning jackpot as any other seven lines, it does increase your winnings should any of your numbers come up, because it increases probability of winning multiple prizes.
Don't believe me? Lets's imagine winning numbers are
01 02 03 04 48 49
Looks like a 4 ball win!. Actually, it's better than that! You have actually won...
## Baby Shower Guide - 10 easy steps
Written by Mrs. Party... Gail Leino
1. Who will plan Baby Shower
Baby showers can be hosted by anyone for new "parents to be." It can be a family member, a friend, or a co-worker. If it is a surprise shower, then family can help out with guest list and gift items.
2. Baby Shower Budget
To determine how much shower will cost, you will need to know number of guests, location of shower, menu, what type of party favors, and how many game prizes you will need.
3. Guest list
Organize your guest list by asking parents-to-be who they would like to have at shower. Usually this includes friends and family, but a few close co-workers may also be invited. Get address, phone numbers, and email addresses. Decide if it will be a "girl's only" shower or "couples".
4. Baby Shower Theme
Baby showers are fun when having baby shower games! Be a kid again with teddy bears and stuffed animals for decorations. Or, invite couples and do a buffet or outdoor party. Whatever theme you choose, remember this will be baby's first party. Baby shower party favors are fun to hand out at end of party. Have fun!
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# Middle School Physical Science: Homework Help Resource
• Course type: Self-paced
• Available Lessons: 63
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### Course Summary
Help your middle schooler master fundamental topics from their physical science class with this fun homework help course. Our short video lessons and quizzes clarify the challenging topics so your student can finish tough homework assignments or prepare for a class project.
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8 Chapters in Middle School Physical Science: Homework Help Resource
#### 8 chapters in Middle School Physical Science: Homework Help Resource
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Lesson 1 - What is Position in Physics? - Definition & Examples Score: {{cp.lessonAssetIdToProgress[45228].bestScoreCorrect}}/{{cp.lessonAssetIdToProgress[45228].bestScoreQuestions}} Take Quiz Optional
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Lesson 1 - What is An Atom? - Definition & Examples Score: {{cp.lessonAssetIdToProgress[45233].bestScoreCorrect}}/{{cp.lessonAssetIdToProgress[45233].bestScoreQuestions}} Take Quiz Optional
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Lesson 12 - Formation of the Moon: Theories Score: {{cp.lessonAssetIdToProgress[2965].bestScoreCorrect}}/{{cp.lessonAssetIdToProgress[2965].bestScoreQuestions}} Take Quiz Optional
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Lesson 1 - Valence Electrons and Energy Levels of Atoms of Elements Score: {{cp.lessonAssetIdToProgress[1615].bestScoreCorrect}}/{{cp.lessonAssetIdToProgress[1615].bestScoreQuestions}} Take Quiz Optional
Lesson 2 - Atomic and Ionic Radii: Trends Among Groups and Periods of the Periodic Table Score: {{cp.lessonAssetIdToProgress[1616].bestScoreCorrect}}/{{cp.lessonAssetIdToProgress[1616].bestScoreQuestions}} Take Quiz Optional
Lesson 3 - Ionization Energy: Trends Among Groups and Periods of the Periodic Table Score: {{cp.lessonAssetIdToProgress[1617].bestScoreCorrect}}/{{cp.lessonAssetIdToProgress[1617].bestScoreQuestions}} Take Quiz Optional
Lesson 4 - Electronegativity: Trends Among Groups and Periods of the Periodic Table Score: {{cp.lessonAssetIdToProgress[1618].bestScoreCorrect}}/{{cp.lessonAssetIdToProgress[1618].bestScoreQuestions}} Take Quiz Optional
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# Engineering Chemistry and Environmental Studies
UNIT I
(A)Atomic structure
(1). Write about the Fundamental particles in an atom?
Ans: They are three fundamental particles present in an atom.
They are
(1) Electron ( e )
(2) Proton (p)
(3) Neutron (n)
S.
No
1
Particle
Discovered
Electron (
J.J. Thomson
e )
2
Mass
Kgs
Proton (p)
Neutron (n)
Amu
9.1
0.005486
10-31
Gold stein
1.6
James
10-27
1.6
Coulombs
Charge
e.s.u
- 1.6
- 4.8
10-19
1.00728
1.008665
+ 1.6
10-19
0
Negative(-)
10-10
+ 4.8
10-10
0
Positive (+)
Neutral
10-27
## (2). What is Atomic Number and explain with an example?
Ans:
Atomic Number (Z): The number of protons or electrons present in the atom is called atomic
number.
This is denoted by the letter Z.
Atomic Number (Z) = Number of electrons (or) protons.
Example: Na = Z =11
i.e., sodium atomic number is 11.
So, sodium contains 11 electrons (or) 11 protons
Mosley proposed the relationship between atomic number and frequency of X-rays
= a(Z-b)
Where a, b = constants
= frequency of X-Rays
Z= atomic number
(3). what is Mass Number?
Ans.
Mass Number (A):
The sum of protons and neutrons present in an atom is called mass number.
It is denoted by A
Where A = No. of Protons + No. of Neutrons
A-Z = No. on Neutrons
(4). Explain the Bohrs Atomic Theory and write its Merits and Demerits?
Ans. Neils Bohr in 1913 proposed atomic model based on the Plancks quantum theory.
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
## Engineering Chemistry and Environmental Studies
UNIT I
He retained the Rutherfords concept that the atom contains positively charged nucleus at the
centre.
Postulates :
1. The electron in an atom revolves around the nucleus with high velocity in circular paths.
These circular paths are called orbits.
2. Each orbit is associated with certain amount of energy.
These orbits are called energy levels.
The energy levels are named as K, L, M, N . (or) numbered as 1,2,3,4
3. As long as the electron moves in a orbit, its energy is constant and does not emit any
electromagnetic radiation. Hence these orbits are called stationary orbits.
4. When electrons jumps from a higher energy level to a lower energy level, the difference
in energy is emitted as radiation. When electron jumps from a lower energy level to
higher energy level the difference is energy is absorbed as radiation.
E2 - E1 =h
Where E1 = Energy of first orbit
E2 = Energy of second orbit
h = Plancks Constant
= integer
5.
multiplies of
mvr =
h
2
nh
2
## where m = mass of electron
v= velocity of electron
h=Plancks constant
n=integer
## Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
## Engineering Chemistry and Environmental Studies
UNIT I
Merits:
1. Bohrs atomic model successfully explains the spectrum of hydrogen and hydrogen like
ions (He+, Li2+) which contains one electron
2. This model explains the stability of atom also.
3. This model also useful in calculating energy and radii of orbits.
Demerits:
1. This model fails to explain spectra of multi electron atoms
2. It is failed to explain the Zeeman effect.
Splitting of spectral lines in the applied magnetic field is called Zeeman effect.
3. It is failed to explain the Stark effect
Splitting of spectral line in the applied electric field is called Stark effect.
4. According Bohrs model electron revolving in definite orbits.
It is against to debrogles wave nature.
Quantum Numbers:
A set of numbers used to describe an electron completely with position and energy in an atom is
called quantum numbers.
There are four quantum numbers
They are
(1) Principle Quantum Number (n)
(2) Azimuthal Quantum Number (l)
(3) Magnetic Quantum Number (m)
(4) Spin Quantum Number (s)
(1). Principle Quantum Number (n):
(a) It is proposed by NeilsBhor
(b) It is denoted by the letter n
(c) n represents as 1,2,3,4 (or) K,L,M,N
(d) It gives the size and energy of stationary orbit
(e) The number of orbitals present in a given orbit as n2.
(f) The minimum number of electrons in a given orbit is 2n2.
(2). Azimuthal Quantum Number (l):
(a) It is proposed by Sommer field.
(b) It is also called Angular Momentum Quantum number
(c) It is denoted by the letter l
(d) It gives the shape of orbital
(e) l values are 0,1,2,. (n-1)
l = 0 indicates s orbital
l = 1 indicates p orbital
l = 2 indicates d orbital
l = 3 indicates f orbital
(3). Magnetic Quantum Number (m):
(a) It is proposed by Lande
(b) It is denoted by the letter m
(c) It gives the orientation of the orbital in space
(d) m values are -l.. 0 l
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
UNIT I
## (e) Total number of m values for a given l value is (2l+1)
(4). Spin Quantum Number (s):
(a) It is proposed by Goud smith and ulhen beck
(b) It is denoted by the letter s
(c) It gives spin of electron
(d) The spin may be clockwise S = +1/2 (or)
Or Anticlockwise S = -1/2 (or)
## (6) What is orbital ?
Ans:
Orbital:
The probability of finding electron maximum around the nucleus is called orbital.
The probability of finding electron 2) at that point can be calculated by Schrodinger wave
equation
2 2 2 8 2 m (
+ 2 + 2 + 2 EV ) =0
2
x y z
h
Where E = Total Energy of electron
V = potential energy
= Wave function of electron
7) Define and explain the Shapes of orbitals?
Ans:
Shapes of orbitals:
S orbital:
S - orbital has spherical shape is equal to all directions.
l value for s orbital is zero.
## Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
## Engineering Chemistry and Environmental Studies
UNIT I
p orbital:
p orbital has dumb bell shape.
l value for p orbitals is one.
m values are -1,0,+1
There are three orientations are possible to p orbital
There are represented as px, py, and pz.
Three p orbitals are mutually perpendicular to each other
These three orbitals have same energy called degenerate orbitals.
Each p orbital contains one nodel plane.
d orbital:
d orbital has double dumb bell shape
l value for d orbital is two
m values are -2,-1,0,+1,+2.
There are five orientations are possible to d orbital
These are represented as dxy, dyz, dzx, dx2-y2, dz2
dxy, dyz, dzx, dx2-y2 has four labels
dz2 has two labels
all d orbitals have same energy called degenerate orbitals each d orbital contains two nodel
planes
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
UNIT I
## 8) What is Electronic Configuration ?
Ans:
Electronic Configuration:
The arrangement of electrons in various orbitals in an atom is called electronic configuration.
(a) Aufbau Principle:Afbau means building up
Afbau principle states that electrons enter into the orbitals in the increasing order of their
energy
This means electron occupy the lowest energy orbital first.
The energy of orbital is given by (n+l)
n = Principle quantum number
l = Azimuthal quantum number
Energy orbitals shown by Moeller diagram
## 9) Write the Hunds Principle?
Ans:
(b) Hunds principle: Hunds rule states that pairing of e
## orbital is filled with one electron
Eg: Carbon electronic configuration = 6 = 1S2 2S2 2P2
## Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
1S2
UNIT I
2S2 2 P x 2 P y
## 10) Write Paulis exclusion principle?
Ans:
(c) Paulis Exclusion principle:
## can have the same set of 4 quantum
numbers in an atom.
Eg: Helium =2 = 1S2
1S2
st
1 electron
n=1
l=0
m=0
s = +1/2
2nd electron
n=1
l=0
m=0
s = -1/2
## 11) Write the Electronic Configuration of Cr and Cu?
Ans:
Electronic Configurations:
Chromium (Cr)= Z =24 = 1S2 2S2 2P6 3S2 3P6 3d5 4S1
Copper (Cu) = Z =29 = 1S2 2S2 2P6 3S2 3P6 3d10 4S1
(12) Write the differences between orbit and orbital?
Ans:
Orbit
(1). Orbit is a circular path around the nucleus
## (2). Orbits are circular in shape
(3). The distance of the orbit from nucleus to
## around the nucleus
(2). Orbitals have different shapes
(3). It is impossible to know the exact position
an e
Orbital
is maximum
of electron in an orbital
is fixed
## 13) What is Chemical bonding ?
Ans:
Chemical bonding: The attraction force between two atoms (or) ions (or) molecules is known as
chemical bond.
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
UNIT I
## Electronic Theory of Valency:
Kossel and Lewis proposed a theory to explain chemical bond.
Postulates:
(1) The electron in the outermost orbit is called Valency electron. This orbit is called
Valency orbit.
(2) Valency electrons are participate in bond formation.
(3) Noble gas elements are stable and chemically inert due to octet (ns2 np6)
configuration.
(4) Atoms of other elements tend to attain 8 electrons in their Valency orbit to get
stability.
(5) The chemical bond is formed by losing electrons from Valency orbit to another atom
(or) gaining of electrons from other atom (or) sharing the electrons of the Valency
orbit with other atom.
(6) Valency electrons causes for chemical relativity.
Representation of Valency electrons by Lewis method?
## Types of chemical bonds
Mainly there are three types of chemical bonds
They are:
(1) Ionic bond
(2) Covalent bond
(3) Metallic bond
14) Define Ionic bond. Explain with examples NaCl, MgO?
Ans:
(1) Ionic bond:
The electrostatic attraction force between two opposite charged ions is called ionic bond.
The ionic bond is formed by transfer of electrons from one atom to the other atom.
The ionic bond is formed between two atoms which one atom is high electron affinity and other
atom is low ionization potential values.
Electro negativity difference is more than 1.9
Eg:
NaCl Sodium chloride.
MgO Magnesium oxide.
(1) Sodium Chloride formation (NaCl):
Sodium metal reacts with chlorine gas to form a crystalline sodium chloride solid.
Na = 11 = 1S2 2S2 2P6 3S1
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
## Engineering Chemistry and Environmental Studies
UNIT I
Na loses 3S1 electrons and gets nearest noble gas configuration i.e., Na+ ion is formed.
Na
Na+ + e
2,8,1
2,8
Chlorine = 17 = 1S2 2S2 2P6 3S2 3P5
Cl one electron is required to get Argon electron configuration.
Cl + e Cl
2,8,7
2,8,8
The oppositely charged Na+ and Cl ions are held together by electrostatic force of
attraction formed NaCl.
Na+ + Cl
NaCl
(2) Magnesium oxide formation (MgO):
Magnesium metal reacted with oxygen gas to form magnesium oxide.
Magnesium atomic number = 12.
Electronic Configuration of Magnesium= 1S2 2S2 2P6 3S2
Mg loses 3S2 electrons and gets nearest noble gas configuration formed Mg2+ ion
Mg
Mg2+ + 2 e
2,8,2
2,8
Oxygen atomic number = 8.
Electronic Configuration of oxygen = 1S2 2S2 2P4
Oxygen required two electrons to get nearest noble gas configuration.
O + 4 e
2O-2
2,6
2,8
The oppositely charged Mg2+ and O-2 ions are held together by electrostatic force of
attraction.
2Mg2+ + 2O-2
2MgO.
## 15) Write the Properties of Ionic Compounds?
Ans:
Properties of Ionic Compounds:
(1) Ionic compounds are crystalline solids due to strong electrostatic force of attraction
(2)
(3)
(4)
(5)
(6)
(7)
## between two opposite charged ions.
Ionic compounds are hard and brittle.
Ionic compounds have high M.P and B.P values due to strong electrostatic attraction.
Ionic compounds soluble in polar solvents Eg: water.
Ionic compounds acts as electric conductors in the form of solution and fused state.
The reactions between two ionic compounds in solution are very fast.
Ionic compounds do not exhibit isomerism due to ionic bond is non - directional.
## 16) Define Covalent bond. Explain with examples H2, O2, N2 ?
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
Fundamentals of chemistry
## Engineering Chemistry and Environmental Studies
UNIT I
Ans:
Covalent bond:
The attraction force is due to equal contribution and mutual sharing of electrons between two
atoms is called covalent bond.
Eg: H2, O2, N2
(1) Formation of H2
Atomic number of Hydrogen = 1.
Electronic Configuration of Hydrogen = 1S1
Hydrogen contains one Valency electron
Hydrogen is required one electron to get the configuration.
So two Hydrogen atoms contribute one electron each other to form bond pair. This bond
is shared by two Hydrogen atoms and two hydrogen atoms get the configuration.
## (2) Formation of Oxygen:
Oxygen atomic number = 8.
Electronic configuration of oxygen = 1S2 2S2 2P4.
Oxygen contains six Valency electrons.
Oxygen requires two electrons to get noble gas configuration.
So two Oxygen atoms contribute two electrons each to form two bond pairs.
These two bond pairs shared by two Oxygen atoms and double bond is formed between
two Oxygen atoms.
## (3) Formation of Nitrogen:
Oxygen atomic number = 7.
Electronic configuration of oxygen = 1S2 2S2 2P3.
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
10
Fundamentals of chemistry
UNIT I
## Nitrogen contains five Valency electrons.
Nitrogen required three electrons to get noble gas configuration.
So two nitrogen atoms contribute three electrons each to form three bond pairs.
These three bond pairs shared by two Nitrogen atoms and triple bond is formed between
two Nitrogen atoms
## 17) Write the Properties of Covalent Compounds:
Ans:
Properties of Covalent Compounds:
(1)
(2)
(3)
(4)
## Covalent Compounds are gases (or) volatile liquids (or) solids.
Covalent Compounds have low M.P and B.P values due to weak vanderwaals forces.
Covalent Compounds soluble in non polar solvents Eg: C6H6, CCl4, Ether.
Covalent Compounds are non conductors in fused state and also in solution state due to
absence of ions.
(5) The reactions between covalent compounds are slow due to participation of molecules.
(6) Covalent Compounds show isomerism due to covalent bond is directional.
18) Define Metallic Bond ?
Ans:
Metallic Bond:
The electrostatic attraction between positively charged metal ions and negatively charged sea of
electrons is called metallic bond.
Eg: Na metal, Fe metal etc.,
The nature of metallic bond can be explained by following theories.
(1). Free Electron Theory (or) Electron Sea Theory
(2). Valence Bond Theory (or) Resonance Theory
19) Write the Fee electron theory of metallic bond?
Ans:
(1). Free Electron Theory (or) Electron Sea Theory:
This theory was proposed by Prude and Lorentz.
According to this theory metal atoms lose their Valency electrons into crystal lattice to give
positive metal ions.
These ions are called kernels.
The Valency electrons freely move in the crystal lattice.
These electrons are called sea of electrons.
The electrostatic attraction arises between positive metal ions and sea of electrons.
Metallic properties depend upon valence electrons.
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
11
Fundamentals of chemistry
UNIT I
## 20) Write the valence bond theory of metallic bond?
Ans:
(2). Valence Bond Theory (or) Resonance Theory:
Valence bond theory proposed by Linus Pauling
Every atom in the metal lattice shares valence electrons with neighboring metal atoms.
The result is formation of bond pairs.
These bond pairs moves freely in vacant orbitals of metals equal structures are formed.
The equal structures are called resonance forms.
They are in ionic form as well as covalent form.
## 21) Write Difference between Covalent bond and Metallic bond?
Ans:
Covalent Bond
(1). Directional because valence electrons are
Metallic Bond
(1). Non - Directional because valence
localized
(2). Valence electrons strongly attracted
## electrons are delocalized
(2). Valence electrons weekly attracted towards
towards nucleus
nucleus
22) Write the Difference between Covalent bond and Ionic bond?
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
12
Fundamentals of chemistry
## Engineering Chemistry and Environmental Studies
UNIT I
Ans:
Ionic Bond
1). The Electrostatic attraction force between
Covalent Bond
1). The attraction force is due to equal
## 2). Ionic Bond is strong
3). Electronegativity difference between two
## two atoms is called Covalent Bond
2). Covalent Bond is weak
3). Electronegativity difference between two
## atoms is more than 1.9
atoms is less than 1.9
Eg:NaCl, KCl
Eg: H2, O2, N2
23) Write the Difference between Ionic Compound and Covalent Compound?
Ans:
Ionic Compounds
1). These are Crystalline solids
2). These are hard and brittle
3). These are having high MP & BP
4). These are solute in polar solvents
Eg: Water
5). These do not exhibit Isomerism
24) Define Oxidation and Reduction?
Covalent Compounds
1). These are gases (or) volatile liquids (or) solids
2). These are weak
3). These are having low MP & BP
4). These are soluble in non-polar solvents
Eg: Benzene
5). These are exhibit Isomerism
Ans:
Oxidation and Reduction:
Oxidation:
1) Removal of electron from an atom (or) ion (or) molecule
Na
Na+ + e
2) removal of H from a molecule or an ion
H2S + Cl2
2HCl + S
3) addition of oxygen to an atom (or) molecule (or) ion
C + O2
CO2
4) Increase in oxidation state of an atom
Fe+2SO4
Fe+32 (SO4)3
Reduction:
1) Gain of electrons
Na+ + e
Na
H2 + Cl2
2HCl
3) Removal of oxygen
CuO + H2
Cu + H2O
4) Decrease in oxidation state of an atom
Cu+2I2
Cu2+1 I2
Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
13
Fundamentals of chemistry
UNIT I
## 25) Define Oxidant ?
Ans:
Oxidant: The substrate which acts as electron acceptor
Eg: MnO4, K2Cr2O7, OSO4, O, F, Cl
26) Define Reductant ?
Ans:
Reductant: which acts as electron donor
Eg: Li, Na, K, Mg
27) Define Redox reaction?
Ans:
Redox reaction: A reaction which involves simultaneous oxidation and reduction is called redox
reaction
Zn + Cu+2
Cu + Zn+2
Zn + e
Reduction: Cu+2 + 2 e
Oxidation: Zn
Cu
Uses:
1) Important in biological process.
2) Important in electrochemical process.
28) Write the Difference between Oxidation State and Valency?
Ans:
Oxidation State
(1). The residual charge present on the atom or
Valency
(1). The strength of an element to combine
## ion is called Oxidation Number.
(2). It may be integral or fractional.
(3). It may be negative or positive.
(4). An atom may have zero oxidation number
## with other elements is called Valency.
(2). It is always integral only.
(3). No position or negative sign for Valency.
(4). An atom has never zero Valency in the
in the compounds.
compounds.
## Prepared by V Naga Surendra Reddy
Lecturer in Chemistry
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Fundamentals of chemistry
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# What city did the aztecs establish circa 1325 on a marshy island in lake texcoco?
###### Question:
What city did the aztecs establish circa 1325 on a marshy island in lake texcoco?
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### What are the four essential components of a UAS?
what are the four essential components of a UAS?...
### Patriotism and education are traditional American values. True False
Patriotism and education are traditional American values. True False...
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#### Re: 6cm rf?
Mark GM4ISM
Ben
160 C junction temperature is pretty standard for this technology, the data that is missing is the thermal resistance, junction to case measured in C/W, and power added efficiency. Without it you cannot calculate the permitted dissipation of the device and the expected output power WRT the junction temperature. Yes it can manage a P1dB of 2W but maybe only with a 20% duty cycle for thermal considerations.
If the value of thermal resistance is say 15C/W, then if you dissipate 5W in the junction, the junction rises to 75 degrees above the case, that would be manageable as the heatsink would have to keep the case at a max of 85C
if the value is 30 then the same 5W dissipation would put the junction at 170C even if you kept the case anchored at 20C with a fantastic heat sink.
we can guess from the figures.. at 400mW the data sheets states that the max case temperature is 85C and that this is at 5V and 600mA That is 3W in, 400mW out 2.6W dissipation ie about 14% efficiency
If you assume that at these values the case temperature is allowed to approach the max of 85C the the junction will also be OK below 160 C, some 75C above case. 75/2.6 is 28.8 C/W thermal resistance.
Lets say the efficiency reaches 30% at the P1dB point of 2w out, that would require a junction dissipation of 4.4W
With that thermal resistance which is a constant, the junction would be 127C above the case. Thus the case must not rise above 33C
Thats quite low, requiring a very good heatsink and likely a low duty cycle
These calculations make a few assumptions and may be quite a bit off, I may even have made a slip on the calculator but they look about right to me and support the supposition that 1W or maybe a bit more output, with good design and typical amateur duty cycles, is probably feasible.
There are a lot of devices out there that look good for really high output power at the P1dB point but being designed (thermally) for a lower mean power means they just cant realise that P1dB for a useful time. Not all data sheets tell you the whole story :(
Mark GM4ISM
On 20/05/2021 21:26, militaryoperator via groups.io wrote:
The data sheet from Skyworks does not give any thermal info and it may be that key down CW at 2W will overheat the junction.
160deg C Mark.
. if they work as advertised. If they dont, send em back!
Mark GM4ISM
I doubt worth trying to send back to China. Still you get either a nice little project box or a nice heatsink depending on which you buy, hi.
Ben,
Virus-free. www.avg.com
Join UKMicrowaves@groups.io to automatically receive all group messages.
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### Normal approximation to the binomial distribution
Assignment Help Basic Statistics
##### Reference no: EM1387479
Based on long experience, an airline found that about 9% of the people making reservations on a flight from Miami to Denver do not show up for the flight. Suppose the airline overbooks this flight by selling 272 ticket reservations for an airplane with only 255 seats. (Round your answers to four decimal places.)
a) What is the probability that a person holding a reservation will show up for the flight?
b) Let n = 272 represent the number of ticket reservations. Let r represent the number of people with reservations who show up for the flight. What expression represents the probability that a seat will be available for everyone who shows up holding a reservation?
P(r ≤ 255)
c) Use the normal approximation to the binomial distribution and part (b) to answer the following question: What is the probability that a seat will be available for every person who shows up holding a reservation?
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# The Mehta example solve game
#### Cubing Forever
##### Member
So, this is like the 3x3 example solve thread but only for the Mehta method.
If you don't know, the Mehta method is an algorithm-reliant method that solves the cube in the following order:
First block on the D layer.
3 edges of the E layer or 3QB for short.
Orient edges (Algorithmically) while solving the last edge or EOLE for short.
Orient last 6 corners (using 1 alg) or 6CO for short .
Permute last 6 corners (using 1 alg) or 6CP for short.
Algorithmic L5EP
(or there is a beginner variant which does the last 4 steps in 2 looks each and a ZB variant that algorithmically solves the DR block after EOLE thus, giving you a ZBLL finish)
There are 3 more variants with which you can do option select.
That's it for the introduction let's move on to the solves:
Scramble: U2 F2 R' U2 L2 F U' D2 R U2 L2 D' R2 L2 U' L2 U2 L2 F2
Solution:
73HTM 89QTM 65STM 68ETM
U2 L E2 L' E' z //FB
R E R2 u' R U R' u' R U R' E R U R' F R' F' R //Long belt
U' S' U' S U2 S' U' S //EO
R' U R U2 R2 U' R U2 R //6CO-1
U R2 D' R U2 R' D R U2 R //6CO-2
U' R2 U R2 //D layer corners+lucky DR block
U' x R2 D2 R U R' D2 R U' R x' // PLL Ab Perm
Beginners version(kinda) but got lucky.
Next: R D R2 D2 B2 D2 B R2 U2 B' U2 F' U2 D' L R2 D L' U' B'
This thread might be of interest to @Jam88 , @BenChristman1 @Devagio and @CuberStache and a few more people but please do try it.
#### Devagio
##### Member
So, this is like the 3x3 example solve thread but only for the Mehta method.
This is a great idea! This way the example solves page will not overflow with Mehta examples and we could digress into discussing particular solutions freely.
NEXT: R2 F D F B R U B2 U D2 R F2 U2 L2 D2 R L2 D2 F2 L2
Interesting observation: There is not a single anticlockwise quarter turn in this scramble. In a 20 move random-move scramble, the probability of this happening is 0.03%, roughly the same as a ZBLL skip.
x' y2 // Inspection
D2 R F2 L2 // FB (4/4)
R2 u R' u R // 3QB (5/9)
u F U R' U' F' // EOLE (6/15)
U R' U R' U' R U R' U' R U R // DCAL (12/27)
U2 R' U R U F' U R' U R U2 F // CDRLL (12/39)
u' S' U2 S U' // L5EP (5/44)
Next: B' U' R' B' R' U B' L D' U2 B' L2 F U2 F' D2 B R2 D2 R2 D2
#### ProStar
##### Member
Ok I'll try this out. Be warned though, I'm renowned for how bad I am at block building
Next: B' U' R' B' R' U B' L D' U2 B' L2 F U2 F' D2 B R2 D2 R2 D2
/* Scramble */
B' U' R' B' R' U B' L D' U2 B' L2 F U2 F' D2 B R2 D2 R2 D2
/* Solve */
B' D' R2 U2 L B' U2 B // FB (8)
u R' u U R2 U2 R u2 // 3QB (8, 16)
R2 U R' U' R' // EOLE (5, 21)
U R U' R' D' U' R U2 R' D // 6CO (10, 31)
U2 D' R2 U R2 U D R2 U2 D' R2 // 6CP (11, 42)
M2 U M2 U2 M2 U M2 // L5EP (7, 49)
U2 D // ADUF (2, 51)
// View at alg.cubing.net
NEXT: U F L2 D2 B2 F' D2 F U2 R2 B2 D F2 U' L F2 U' R' F
Last edited:
#### ProStar
##### Member
This is actually quite intriguing
NEXT: U F L2 D2 B2 F' D2 F U2 R2 B2 D F2 U' L F2 U' R' F
/* Scramble */
U F L2 D2 B2 F' D2 F U2 R2 B2 D F2 U' L F2 U' R' F
/* Solve */
z2 // Inspection
D' L F L' R' u2 R // FB (7)
y u' R E' R2 U2 R E2 // 3QB (7, 14)
u' F R F' // EOLE (4, 18)
U R U R' U R U2 R' // 6CO (8, 26)
U R2 U R2 U R2 // 6CP* (6, 32)
U2 E2 R E2 R2 E2 R D // Solved** (8, 40)
// View at alg.cubing.net
* Unsolve Belt by R2
** By fixing the Belt and solving the 2 remaining L5EP edges
NEXT: U' F' U2 B U2 R2 U2 B2 U2 B2 F R2 U' B' F L' D2 R' D' B' F2
Ninja'd
NEXT: F2 D' R2 F2 U' R2 D L2 D2 B' L D R' U2 R B U' L2 F
#### ProStar
##### Member
NEXT: F2 D' R2 F2 U' R2 D L2 D2 B' L D R' U2 R B U' L2 F
/* Scramble */
F2 D' R2 F2 U' R2 D L2 D2 B' L D R' U2 R B U' L2 F
/* Solve */
y2 // Inspection
R' F D L B' U2 B // FB (8)
U' R u' R E R // 3QB (5, 13)
F' S' U S U F // EOLE (6, 19)
R U D' R U R' U2 D R' // 6CO (9, 28)
R2 U' R2 U' R2 U2 R2 // 6CP (7, 35)
y M' U2 M' U M' U2 M U M2 // L5EP (9, 44)
U D // ADUF (2, 46)
// View at alg.cubing.net
NEXT: U' F' U2 B U2 R2 U2 B2 U2 B2 F R2 U' B' F L' D2 R' D' B' F2
#### Cubing Forever
##### Member
NEXT: U' F' U2 B U2 R2 U2 B2 U2 B2 F R2 U' B' F L' D2 R' D' B' F2
Beginners but with a 3 move 6CO1 and a 4 move APDR lol
Also, switched back to ACN from cubeDB for recons
U R' S2 U F' U L2 U B2 U' z //FB
D2 u R2 u' R u R2 U' R' u R2 U' R U R //Belt
U2 S' U' S //EO
R2 U R2 //6CO-1
U R U2 R2 U' R2 U' R2 U2 R //6CO2
U2 R2 U' R2 //APDR
U2 R U R' F' R U R' U' R' F R2 U' R' //PLL
U' //AUF
(64h, 85q, 61s, 62e)
Next: L2 D2 R2 U R2 B2 F2 U2 F2 L2 U B' D2 U B2 R U2 R' B2 F2
#### effperm
##### Member
Next: L2 D2 R2 U R2 B2 F2 U2 F2 L2 U B' D2 U B2 R U2 R' B2 F2
really easy red block
x' z // inspection
R F L' U L // fb
U R' u' R // 3qb
U' R U' R' // last edge
S' U S // eo
U R2 U' R2' // bottom corners
R U R' U R U' R' U R U2' R' // top corners
U' R2 U2' R2' // bottom corners
U S R2' S' R2 // bottom edge
U D' R U R' F' R U R' U' R' F R2 U' R' // pll
#### BenChristman1
##### Member
There's no next scramble, so I'll use Cubing Forever's again.
Next: L2 D2 R2 U R2 B2 F2 U2 F2 L2 U B' D2 U B2 R U2 R' B2 F2
6CO ---> 6CP ---> L5EP, 64 STM
z2 // Inspection
U' B2 U D R' D' L2 U L' U' L // FB+1E
R u' R2 U' R' U R E2 // 3QB
F R F' R' u' // EOLE
U R2 U R2 U2 R U2 R2 U' R2 U' R2 U2 R // 6CO
U2 R2 U' R2 U' D' R2 U R2 U' R2 D R2 U' D' // 6CP
M' U2 M' U M' U2 M U M2 U D // L5EP
alg.cubing.net
Next: R2 U' D2 L' U2 F2 L' B2 L2 U2 F B2 U2 B L2 D2 R2 L2 D' L'
#### effperm
##### Member
oh i forgot to put the next scramble lol
#### Devagio
##### Member
R2 U' D2 L' U2 F2 L' B2 L2 U2 F B2 U2 B L2 D2 R2 L2 D' L'
z2 // Inspection
U2D F D2 F' // FB (5/5)
U R' u R2 u R' // 3QB (6/11)
u'U2 R2 F R F' R2 // EOLE (7/18)
U' R U' R' D R' U R D' R U' R // TDR (12/30)
u R' U' F U' R2 U R2 U F' R U' R U' R' // ZBLL (15/45)
U2 // ABF (1/46)
Next: B2 L2 U L2 U' F2 R2 U' L2 U2 B2 L D2 F' U L2 B D2 L' D2 F2
#### ZB2op
##### Member
Interesting observation: There is not a single anticlockwise quarter turn in this scramble. In a 20 move random-move scramble, the probability of this happening is 0.03%, roughly the same as a ZBLL skip.
But it's not a random move scramble but instead a random state scramble.
#### carcass
##### Member
R' F2 D L2 B2 U F2 R2 U' F2 U R2 D' F L B' D' B F R2 D'
z' y2 //inspection
U2 R' U R' F' U2 B' U2 R U' R' D' //FB, very innefficent
u R U' u' R' U2 R2 u' F R' F' R u R' F R F' U' R' F R F' //Belt
U' R' F R2 F' R'//EO
U2 R2 U2 R2 U R U2 R' U' R U' R' //CO
R2 U' R2 U R2 U' R2 U R2 U2 R2 U R U R' U' R' F R2 U' R' U' R U R' F' //CP
D' U' M' U2 M D2 U M' U M2 U M2 U M' U2 M2 //L5EP
Next Scramble: B2 D2 B R2 B U2 R2 B' D2 L2 D L U' B2 F' L R' U' L' F
#### Cubing Forever
##### Member
Next Scramble: B2 D2 B R2 B U2 R2 B' D2 L2 D L U' B2 F' L R' U' L' F
x2 //Inspection
L2 U' L U L' U' L B' R2 D' //FB
U' R U2 u' R U' R U' R' //3QB
u' R U R' U' F' U F U' F U R U' R' F' //EOLE
R2 U2 R2 //6CO1
U R U2 R2 U' R2 U' R2 U2 R //6CO2
R2 U2 R2 //APDR
U2 D' R U R' U' D R2 U' R U' R' U R' U R2 D U2 //PLL
68h, 86q, 68s, 69e
Next: F2 R D2 R2 B' U2 F' L2 B U2 F R2 F R' D U' F2 L' D2 U' R2
#### Devagio
##### Member
F2 R D2 R2 B' U2 F' L2 B U2 F R2 F R' D U' F2 L' D2 U' R2
z // Inspection
L B L' B U B // FB (6/6)
u' R u R' U' R' // 3QB (6/12)
u2 U S' U' S R U R' // EOLE (8/20)
U2 R U2 R' D R' U2 R D' U R U R // TDR (14/34)
R U R' U R' U2 R2 U R2 U R2 U' R' // ZBLL (13-1/46)
U // ABF (1/47)
Next: R' U B2 R2 L' B2 D R2 B' F2 R2 B2 D' B2 U F2 D L2 U2 L2 U'
#### BenChristman1
##### Member
Next: R' U B2 R2 L' B2 D R2 B' F2 R2 B2 D' B2 U F2 D L2 U2 L2 U'
6CO > 6CP > L5EP, 74 STM
R U' R' U R U' R' D R D2 R' F' U2 F' // FB
E' R E' u R' U u R' u // 3QB
R F' U F U' R' // EOLE
U' R2 U R2 U' F (R U R' U')3 F' // 6CO
R2 U R2 U2 D' R2 U R2 U' R2 D R2 D' // 6CP
U' M' U2 M U M2 U M' U2 M U M2 U // L5EP
alg.cubing.net
6CO > APDR > PLL, 81 STM
R U' R' U R U' R' D R D2 R' F' U2 F' // FB
E' R E' u R' U u R' u // 3QB
R F' U F U' R' // EOLE
U' R2 U R2 U' F (R U R' U')3 F' // 6CO
U' R2 U' R2 U' R2 U R2 U' R2 U R2 U' R2 U' R2 // APDR
U' R' U R' U' R D' R' D R' U D' R2 U' R2 D R2 U2 D' // PLL
alg.cubing.net
Next: R B' D L D F' D2 L F' R2 B2 R D2 L2 F2 D2 L U2 F2 R
#### Cubing Forever
##### Member
Next: R B' D L D F' D2 L F' R2 B2 R D2 L2 F2 D2 L U2 F2 R
L2 U F2 U F' R' D L' x' D' //FB
R2 u' R U' R' //3QB
E R U' R' U' F U R U' R' F' //EOLE
U R2 U2 R2 //6CO1
U' R U2 R2 U' R2 U' R2 U2 R //6CO2
R2 U' R2 U2 R2 U' R U R' d' R' F' R2 U' R' U R' F R F//6CP
U' M' U2 M U2 R2 U' S' U2 S U' R2 U' D' //L5EP
78h, 99q, 73s, 74e
Next: L2 U2 L' B2 R U2 R B2 U2 R' B' L' F2 D2 R' B2 U F' D'
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# Percentages Fractions and Decimals Grid
This interactive activity explores the representation of fractions, decimals and percentages using either a square or circle divided into equal parts. When the activity is launched, a 100 square grid is shown. Click the random button to create a grid of different colors. The squares change so they are either orange or white. You can change the number of colors using the color number control, change it to 5 and press random again and there will now be 5 different colors in the grid. You can arrange the grid automatically by clicking arrange button, to reset the grid back to white click the trash button
To select your own fraction simply select one of the colors and click and drag your mouse over the either the grid or the circle.
## Display options
The initial set up is ideal for explaining percentages since there are 100 squares, but this can be changed by using the divisions selection. You can also change from the square to a circle by clicking the shape toggle control
## Hiding or Showing the values
On the right side of the screen our colored rectangles with ? labels. Click one of these to turn the rectangle over to display its decimal,percentage or fraction value. To change a whole column click one of the buttons above to toggle hide or show. Note values on display automatically change as the grid or circle is changed.
The fraction values by default are simplified, to change this click the cancel fraction toggle control.
## Related activities
This activity is closely related to the interactive fraction wall. Ihe interactive clock can also be used to display fractions using a clock face.
Select the number of parts for the fraction
Select the number of different colors in the grid
Choose between display type, a square grid or circle.
Click for a new random grid
Click to reset the grid so all parts are white
Click to arrange so the same colors are next to each other
Choose whether to simplify the fraction when possible
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https://qz.com/1941258/how-to-determine-the-accuracy-of-a-covid-19-test/
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CALCULATED RISK
# How to determine the accuracy of a Covid-19 test
By Amanda Shendruk
Visual journalist
Published Last updated
In the UK, university students are getting tested for coronavirus en masse. The nation-wide plan is to have all students receive a negative Covid-19 diagnosis before traveling home for Christmas holidays, often in other parts of the country.
However, with test results comes potentially false confidence. Because just as with antibody tests, there is always a chance the Covid-19 diagnostic test results will be incorrect.
Let us explain, with the help of some friends.
There are three factors that go into determining the likelihood you received an incorrect test result. They are the sensitivity and specificity of the test, as well as your own pre-test probability, or risk.
✅ Test sensitivity: How good the test is at correctly identifying someone who is infected.
❌ Test specificity: How good the test is at correctly identifying someone who is not infected.
Your pre-test probability is one of the most significant factors in determining whether you are likely to have received an incorrect result. But figuring out this risk is difficult, even for doctors. The calculation ultimately amounts to a GP’s educated guess, and it depends a lot on individual circumstances. Local rates of Covid-19, patient symptoms, likelihood of alternative diagnoses, and history of exposure to coronavirus can all affect pre-test probability.
A rough way to approximate your own risk is to start with the prevalence of Covid-19 in your area. You can often find this datapoint from local governments or media reports. Use it as your baseline. Now, increase your risk if you:
• 🧒 Have kids that go to school
• 💼 Work outside your home
• 👩⚕️ Work in a job where you come into contact with many people, or people likely to have Covid-19
• 🤭 Don’t follow local safety guidelines
• 🤒 Have developed symptoms of the virus
For example, someone who has no symptoms of Covid-19, lives in an area with very low rates of the virus, works from home, and rarely sees anyone outside their immediate bubble would have a very low pre-test risk. If they receive a negative test result, chances are good that it’s accurate.
Alternatively, a student sharing facilities in a dorm on a campus with high rates of coronavirus who develops a persistent cough and fatigue would have a much higher pre-test risk of infection. She would be wise to regard her negative test result with skepticism. The risks of acting on a false negative could be disastrous, especially as restrictions in many areas are temporarily easing for the holidays, and generations are planning to gather.
📬 Kick off each morning with coffee and the Daily Brief (BYO coffee).
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http://oeis.org/A164004
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A164004 Zero together with row 4 of the array in A163280. 6
0, 5, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..5000 Index entries for linear recurrences with constant coefficients, signature (3, -3, 1). FORMULA Conjectures from Colin Barker, Apr 07 2015: (Start) a(n) = n*(3+n) for n > 1. a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4. G.f.: x*(x^3 - 3*x^2 + 5*x - 5) / (x-1)^3. (End) E.g.f.: x*(x+4)*exp(x) + x. - G. C. Greubel, Aug 28 2017 MAPLE A033676 := proc(n) local a, d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a, d) ; fi; od: a; end: A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164004 := proc(n) if n = 0 then 0; else A163280(4, n) ; fi; end: seq(A164004(n), n=0..80) ; # R. J. Mathar, Aug 09 2009 MATHEMATICA Join[{0, 5}, Table[n*(n + 3), {n, 2, 50}]] (* G. C. Greubel, Aug 28 2017 *) PROG (PARI) x='x+O('x^50); concat([0], Vec(x*(x^3 -3*x^2 +5*x -5)/(x-1)^3)) \\ G. C. Greubel, Aug 28 2017 CROSSREFS Cf. A008578, A161344, A161345, A163280, A164000, A005563, A164005. Cf. A028552. Sequence in context: A092390 A035608 A091386 * A025010 A299144 A058555 Adjacent sequences: A164001 A164002 A164003 * A164005 A164006 A164007 KEYWORD nonn AUTHOR Omar E. Pol, Aug 08 2009 EXTENSIONS Extended beyond a(12) by R. J. Mathar, Aug 09 2009 STATUS approved
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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)
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https://backreaction.blogspot.com/2020/10/how-can-climate-be-predictable-if.html
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## Saturday, October 24, 2020
### How can climate be predictable if weather is chaotic?
[This is a transcript of the video embedded below. Some parts of the text may not make sense without the graphics in the video.]
Today I want to take on a question that I have not been asked, but that I have seen people asking – and not getting a good answer. It’s how can scientists predict the climate in one hundred years if they cannot make weather forecasts beyond two weeks – because of chaos. The answer they usually get is “climate is not weather”, which is correct, but doesn’t really explain it. And I think it’s actually a good question. How is it possible that one can make reliable long-term predictions when short-term predictions are impossible. That’s what we will talk about today.
Now, weather forecast is hideously difficult, and I am not a meteorologist, so I will instead just use the best-known example of a chaotic system, that’s the one studied by Edward Lorenz in 1963.
Edward Lorenz was a meteorologist who discovered by accident that weather is chaotic. In the 1960s, he repeated a calculation to predict a weather trend, but rounded an initial value from six digits after the point to only three digits. Despite the tiny difference in the initial value, he got wildly different results. That’s chaos, and it gave rise to the idea of the “butterfly effect”, that the flap of a butterfly in China might cause a tornado in Texas two weeks later.
To understand better what was happening, Lorenz took his rather complicated set of equations and simplified it to a set of only three equations that nevertheless captures the strange behavior he had noticed. These three equations are now commonly known as the “Lorenz Model”. In the Lorenz model, we have three variables, X, Y, and Z and they are functions of time, that’s t. This model can be interpreted as a simplified description of convection in gases or fluids, but just what it describes does not really matter for our purposes.
The nice thing about the Lorenz model is that you can integrate the equations on a laptop. Let me show you one of the solutions. Each of the axes in this graph is one of the directions X, Y, Z, so the solution to the Lorenz model will be a curve in these three dimensions. As you can see, it circles around two different locations, back and forth.
It's not only this one solution which does that, actually all the solutions will end up doing circles close by these two places in the middle, which is called the “attractor”. The attractor has an interesting shape, and coincidentally happens to look somewhat like a butterfly with two parts you could call “wings”. But more relevant for us is that the model is chaotic. If we take two initial values that are very similar, but not exactly identical, as I have done here, then the curves at first look very similar, but then they run apart, and after some while they are entirely uncorrelated.
These three dimensional plots are pretty, but it’s somewhat hard to see just what is going on, so in the following I will merely look at one of these coordinates, that is the X-direction. From the three dimensional plot, you expect that the value in X-direction will go back and forth between two numbers, and indeed that’s what happens.
Here you see again the curves I previously showed for two initial values that differ by a tiny amount. At first the two curves look pretty much identical, but then they diverge and after some time they become entirely uncorrelated. As you see, the curves flip back and forth between positive and negative values, which correspond to the two wings of the attractor. In this early range, maybe up to t equals five, you would be able to make a decent weather forecast. But after that, the outcome depends very sensitively on exactly what initial value you used, and then measurement error makes a good prediction impossible. That’s chaos.
Now, I want to pretend that these curves say something about the weather, maybe they describes the weather on a strange planet where it either doesn’t rain at all or it pours and the weather just flips back and forth between these two extremes. Besides making the short-term weather forecast you could then also ask what’s the average rainfall in a certain period, say, a year.
To calculate this average, you would integrate the curve over some period of time, and then divide by the duration of that period. So let us plot these curves again, but for a longer period. Just by eyeballing these curves you’d expect the average to be approximately zero. Indeed, I calculated the average from t equals zero to t equals one hundred, and it comes out to be approximately zero. What this means is that the system spends about equal amounts of time on each wing of the attractor.
To stick with our story of rainfall on the weird planet, you can imagine that the curve shows deviations from a reference value that you set to zero. The average value depends on the initial value and will fluctuates around zero because I am only integrating over a finite period of time, so I arbitrarily cut off the curve somewhere. If you’d average over longer periods of time, the average would inch closer and closer to zero.
What I will do now is add a constant to the equations of the Lorenz model. I will call this constant “f” and mimics what climate scientists call “radiative forcing”. The radiative forcing is the excess power per area that Earth captures due to increasing carbon dioxide levels. Again that’s relative to a reference value.
I want to emphasize again that I am using this model only as an analogy. It does not actually describe the real climate. But it does make a good example for how to make predictions in chaotic systems.
Having said that, let us look again at how the curves look like with the added forcing. These are the curves for f equals one. Looks pretty much the same as previously if you ask me. f=2. I dunno. You wouldn’t believe how much time I have spent staring at these curves for this video. f=3. Looks like the system is spending a little more time in this upper range, doesn’t it? f=4. Yes, it clearly does. And just for fun, If you turn f up beyond seven or so, the system will get stuck on one side of the attractor immediately.
The relevant point is now that this happens for all initial values. Even though the system is chaotic, one clearly sees that the response of the system does have a predictable dependence on the input parameter.
To see this better, I have calculated the average of these curves as a function of the “radiative forcing”, for a sample of initial values. And this is what you get. You clearly see that the average value is strongly correlated with the radiative forcing. Again, the scatter you see here is because I am averaging over a rather arbitrarily chosen finite period.
What this means is that in a chaotic system, the trends of average values can be predictable, even though you cannot predict the exact state of the system beyond a short period of time. And this is exactly what is happening in climate models. Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels.
This video was sponsored by Brilliant, which is a website that offers interactive courses on a large variety of topics in science and mathematics. In this video I showed you the results of some simple calculations, but if you really want to understand what is going on, then Brilliant is a great starting point. Their courses on Differential Equations I and II, probabilities and statistics cover much of the basics that I used here.
You can join the chat about this week’s video, tomorrow (Sunday, Oct 25) at 5pm CET, here.
1. Because climate is predictable to a certain extent, that's why we use scenarios.
2. As a graduate student I did something similar with the Lorentz equations. I added not a constant term, but an oscillating F = f sin(ωt) driving term. I did though find similar behavior. The system would exhibit a measure of resonance that depended on f. The system would enter an orbit that was less chaotic, and for f very large the path was an elliptical orbit. I did not numerically compute the average spent in the two loops, but I computed the Hausdorff dimension and it approached 1 as f became large. I never published anything on this, and I was just playing around with this.
I think it was Neil de Grasse Tyson in his Cosmos series who showed a man walking with a dog. The man would follow a fairly straight path while the dog would wander all around near that path. For weather watch the dog, but for climate watch the man.
On a more social dynamics note, it will not matter at this point whether one uses reason or data to make any point on global warming to a near majority. The social condition with politics has reached a sort of critical mass with Qanon and insane conspiracies to the point that 1/3 of people in the US see the world entirely through different mental lenses. I can only hope this does not cross a threshold into a majority, for once that happens, we, the meaning of the word we means not just Americans but the entire world, are in a whole lot of big trouble. Because of this a substantial percentage of people think this is a hoax. Look at sepp.com and wattsupwiththat,com to look as some of this. The last one touts itself as the most widely clicked on website devoted to climate. No matter how much you can debunk this stuff some people refuse to listen.
3. “Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels.”
I am not sure what you mean. If you delete “in 2079” or replace it with “in the future” I agree.
But to predict the average rainfall in 2079 is in my view still weather forecast.
In Germany we have dry years (600 mm/a) and wet years (1000 mm/a) and everything in between. I have some doubts that someone can predict that 2079 is a dry or wet or average year.
1. Yes, I am sorry, of course this refers to the average rainfall within a range of uncertainty.
4. But is it possible to predict the climate if not all causes for climate change are known and if the known causes are not known to even Lorenz's 3 decimal accuracy?
1. The answer is evidently yes, because it has been done.
2. Hehe. Ok, it has been done, but the future has not arrived yet so we don't know whether the predictions are correct!
3. They started making predictions in the 1960s. That was 60 years ago. These predictions were remarkably accurate, see eg here. The claim that climate models have not made predictions is a typical climate change denier argument. It is bluntly wrong and demonstrates that you probably didn't do as much as Google the question.
4. In 1983, I was a student and had to give a presentation about a paper describing an early one-dimensional climate model using extrapolated CO2 concentrations. All it's predictions about rising temperatures have come true.
But see this Science article for an evaluation:
www.sciencemag.org/news/2019/12/even-50-year-old-climate-models-correctly-predicted-global-warming
5. There are some underappreciated and potentially serious consequences of climate change. First, wind energy will be greatly diminished in the northern hemisphere as the Polar Regions warm. Next, increased cloud cover will reduce the efficiency of solar power. The location of cloud cover increase may not match up with solar power siting plant locations.
Global warming will weaken wind power, study predicts
https://www.theguardian.com/environment/2017/dec/11/global-warming-will-weaken-wind-power-study-predicts
Climate change itself may make the green power revolution ineffective as a solution to climate change.
1. I am not a panegyric for geo-engineering, but I suspect this will be forced on us. Methods to increase the atmospheric albedo, such as small particles in the upper atmosphere, will be a big part of this. We will also have to reduce carbonic acid in the oceans, which will involve using iron or other substances to increase phytoplankton activity, There may be other methods, even clouds of micron scale particles at the L1 point to Mie scatter light.
6. Le'usband: We have models, just as Dr. Hossenfelder was using. We can try, systematically, to see what the climate model does with a specific level of CO2 in the atmosphere, and then repeat that with a slightly varied level of CO2: Not wildly varied, but within the measurement error of CO2. Just as she was doing with her "+f" forcing constant. Doing that we can predict hundreds of models using values in a tiny "cloud" around a given CO2 level, and get the average result.
And as she demonstrated, the average can be quite stable and predictable, even if the specific values cannot be predicted, other than to say they are going to be in a small cloud centered on the average result.
If we then do the same for all the possible CO2 values, then we can predict, if CO2 continue to increase on its current trend, where the climate is going to be in 10 years, or 50 years, or 100 years, assuming the trend continues.
Of course, nothing can go on forever; at some point, if the trend continues, the disruptions to food chains, arability, storm severity, droughts, water shortages, forest fires, desert growth, and other phenomenon will disrupt society, cause mass extinctions, and end the technology age.
In other words, Global Warming may be a problem that sorts itself out, by killing us in sufficient numbers that we stop causing it. Of course we might also end up in a runaway heat trap like Venus, where no life-as-we-know-it can survive. But at least no one will be claiming it's all a hoax; we will have eliminated ignorance.
1. OK. But. The climate models have a large number of variables (not only CO2) few of which are exactly known (e.g. cloud cover) and not even all factors affecting climate are included. So how much confidence can you have in those models? And does the "attractor" idea apply to climate models given that they all lead to wildly different outcomes (the only common factor being that they all show warming)?
2. As you say correctly, they all show a common trend, which means the outcomes are not "wildly different." To quantify the confidence you can have in these predictions, the IPCC currently mostly relies on using the predictions from different models to generate a possible range. One can debate whether that's the best way to do it, but that's a debate which is better left to people who work in the field.
3. Le 'usband: We can have reasonable confidence; we aren't starting from scratch. We develop confidence by applying the new models we develop to past data that has known outcomes.
The same way we developed weather prediction in the first place.
The "science" is in developing the models, and seeing if, using data gathered in years past, how well they agree with the outcomes already known. That is one way to develop confidence levels.
If they work predictively, we believe the models are accurate (within the limits we found empirically, also informed by the theoretical mathematical limits of the statistical distributions we are using, such as the extreme value distribution).
To the extent that our models are NOT accurate on past data, we may investigate other variables we might use to improve the fit (subject to technical considerations about over-fitting the data).
But to the extent our models ARE accurate, we presume additional variables must not have been significant enough to collectively sway the result any more than the error we observe.
YES, the models can still have attractors; but as Dr. Hossenfelder notes, that doesn't mean the attractors are wildly different; particularly with regard to global temperature. There is no scenario where we triple the greenhouse gases in the atmosphere and stumble into some permanent springtime attractor; all the possible attractors will be unlivable.
Finally, it is important to note that chaotic systems do not have to be permanently chaotic, as Dr. Hossenfelder's experiments with the forcing factor demonstrate, an externality can kick a chaotic system with attractors into a single stable state. So that is another possibility; we may kick our climate system hard enough, by dumping methane and CO2 into our atmosphere, to knock it into a stable unlivable state, like Venus.
4. The models since the 1970s have indeed been checked against global. And 15 out of 17 models predicted temperature changes right.
www.scientificamerican.com/article/climate-models-got-it-right-on-global-warming/
Climate Models Got It Right on Global Warming
Even models in the 1970s accurately predicted the relationship between greenhouse gas emissions and temperature rise
7. Has the distribution of X been calculated rather than just the average? I could try to do this myself, but if it has already been done....
I have an idea that the average of X could be analagous to the polarisation vector [= "climate"] for an electron while that distribution (say, cos or sin curve) could be equivalent to the distribution of individual measured spin vectors [= "weather"]. Cosine or sine distribution here would, according to my calculations, conform to overall results of measurements complying with Malus's Law.
Austin Fearnley
8. You raise an interesting question "does the
state of Earth's atmosphere
display chaotic behaviour on climate time scales?"
But how can an analysis of an equation
that, according to you, "does not actually describe
the real climate", provide any explanation for
the claim that climate does not display chaotic
behaviour?!
1. We know that climate trends are predictable, so at least in the range that predictions have been made they are not chaotic. I merely provided an example for how this predictability is compatible with weather being chaotic.
2. On a time scale of years it still
seems chaotic, why is not also on
time scale of decades? You promised
an explanantion at the beginning
3. The question I asked is "how can they make predictions", and that's what I explained. Maybe read it again.
9. As a comment about the group chat, the discussion got mired in whether there is even global warming. One person in particular became very testy that I did not have data at hand showing CO_2 increase was human caused. This was not to be the focus of it.
Global warming tends to get people's dander up. Websites devoted to astronomy have largely dropped AGW, because anti-AGW types come out of the woodworks and pounce. The Universetoday site has not had an AGW topic in a long time, because the denier crowd would bomb it with posts.
1. Sorry, I couldn't make it yesterday. In any case, I am actually planning a video about exactly this question (though I'll probably not get there this year).
2. The skeptical science website answers the question of the origin of the CO2:
www.skepticalscience.com/co2-increase-is-natural-not-human-caused.htm
It will take some work to assemble the primary literature as this matter was settled decades ago. Also, human activities have altered the greenhouse gases during the whole Holocene by deforestation and land use changes.
10. "What this means is that in a chaotic system, the trends of average values can be predictable, even though you cannot predict the exact state of the system beyond a short period of time. "
The question that remains is if this is also true for dynamical systems other than the example you discussed here. In particular for the actual climate of Earth.
"And this is exactly what is happening in climate models. Scientists cannot predict whether it will rain on June 15th, 2079, but they can very well predict the average rainfall in 2079 as a function of increasing carbon dioxide levels. "
Why is that true? What are the references?
1. Search the IPCC report for precipitation.
2. I meant it more as a theoretical result. Are there any theorems that say that for some systems of PDE this is true. And specifically for the ones used in climate science.
IPCC stands for intergovernmental panel on climate change. I am not interested in climate change, just in climate predictability.
3. If there were any theorems they wouldn't help you because, needless to say, they can't actually integrate the equations exactly. In any case, I don't understand your problem because if they can make predictions, predictions that fit on the data, then that's better proof that they can make predictions than any theorem.
4. I don't know what the predictions are based on, that's why I ask the question. Is it based on analysis of the actual equations, or is it based on a model of best fitting? It would be a lot more interesting if it was analysis of PDE. And you don't need to integrate the equations in order the prove theorems like that. Take for example the stability of Minkowski space-time.
5. "Is it based on analysis of the actual equations, or is it based on a model of best fitting?"
Climate models are based on general circulation (air) models (equations) that include surface albedo and coupling with the oceans. They simulate the effects of changes in atmospheric composition (greenhouse gases).
Difficult parts where information is lacking are ocean mixing (removes heat and CO2) and cloud formation (changes the albedo).
Predictions in time are difficult as the release of greenhouse gasses depends on the economy, technology, and politics.
11. Of course anything including climate is predictable. And of course the average is less chaotic than the variable. And of course one can manipulate the outcome by adding constants to the model. And of course some of the predictions from the 60s are more accurate than others (likely discarded). And of course if a model is curve-fitted to the past, as they always are and should be, it will continue to be accurate as long as the past trend continues.
Here is what can still go wrong:
1. Trend may change in the future, thus diverging from the model.
2. Accurate models may be misinterpreted. IOW, the best way to go about changing the desired outcome may be different than thought.
3. What exactly are we trying to optimize? Total human population? Quality of life? Number of plants/animals?
4. How bad is the problem? So what if [say] Norway's climate turns into that of Sicily? If so, will Brazil be less livable than Greenland today? So what?
5. Who and how much gets affected? Living x%, children y%, grandchildren z%? Concept of climate "emergency" is not productive.
6. Should we trust older generations with solving our problems? Will grandchildren have better solutions than us?
7. What are the known unknowns and more importantly the unknown unknowns? Let's look at history: Malthus, peak coal, peak oil, peak humans, etc.
1. "If so, will Brazil be less livable than Greenland today? So what?"
If it is that bad, several billion people will have to emigrate. And there is no guarantee that agricultural production will stay at its current level
But the fact that you do not seem to care about even the population of Brazil (200M) tells us enough, I think.
2. @ Nonlin.org
Thank you.
One of the things I most despise about the climate debate, from either side,
Is the resistance to,
The defensiveness against,
And the aggression towards,
( Remember Socrates and Plato ? )
Best wishes
3. "Is the resistance to, The defensiveness against,"
Currently, there is now some 50 years of scrutinized research on Climate Change. However, the questions posed are still about the very earliest of problems. Even the most general searches on your favorite search engine will give you all the answers you will ever need. Which begs the question: Why are these people coming here for their questions? And are they even interested in the answers?
In many respects, most questions "against" climate change sound like "How do you know the earth is round? Where is the evidence?" or "Is COVID-19 just like a cold or the flu?".
And you are surprised scientists lose their patience after 50 years?
4. "Currently, there is now some 50 years of scrutinized research on Climate Change."
You might want to reflect on the events of 2008, when financial risk models, devised, in many cases by economic refugees from the physics community, proved catastrophically wrong. Why? The models were built on the prior 50 years of data and did not incorporate the most recent market disruption which occurred in 1929.
The point being that 50 years of data was insufficient in the context of modern financial markets which are at best a few hundred years old. In the context of the earth's climate 50 years isn't even a blip.
Certainty is a poison pill to science; doubt is the only trustworthy coin of the realm in scientific research.
5. Financial risk models were not "built on data". What the heck are you even talking about? Financial risk models are not models of the financial system, they are exactly what the name says, they're models to evaluate a supposed financial risk to help rich people become even richer.
6. bud rap: 50 years of scrutinized research on Climate Change, it has not been limited to the last 50 years of data. The research includes literally hundreds of thousands of years of data, using ice cores from both poles, soil samples and tree rings going back thousands of years, etc. We can measure or infer temperature, CO2 levels, oxygen levels and much more. The longest continuous historic record of temperature goes back two million years; but plant fossils and other ancient data gives us spot values going back to the dinosaurs.
You are right, 50 years by itself could be a blip, but 50 years in the context of a few million years is not a blip; that is enough to identify a rare-beyond-reason anomaly.
12. In March 2019, the German magazine SPIEGEL published an interview with the climate scientist Bjorn Stevens (Max-Planck-Institut für Meteorologie, Hamburg, Professor at University Hamburg and IPCC author)
(https://www.spiegel.de/wissenschaft/warum-die-vorhersagen-zur-erderwaermung-so-schwierig-sind-a-00000000-0002-0001-0000-000163037012).
He works in climate modelling for 20 years and claims that there is no real progress in the last 40 years (sounds like HEP). The uncertainty in the predictions was not improved. It is still 3K +-1.5K for double CO2 content. He mentioned that he cannot even predict whether glaciers in the Alpes will disappear or grow again. The problem is the modelling of the clouds.
That is not what you read and hear every day. Pretty confusing.
13. The Climatereanalyzer site, run by the University of Maine, provides a preview of what’s in store for us as the season progresses. What’s most revealing is the “Snow depth-MSLP” (Mean Sea Level Pressure, I think). I’ve been tracking it almost daily since mid-summer. Having gone a few days without checking it, I was startled by the enormous increase in snow cover both in Canada and the US. This could very well presage a brutal winter here. Come Friday the forecast is for a mixed rain and snow event with up to 4 inches of snow in southern New Hampshire. I’m hoping it doesn’t materialize as I need another 190 miles to reach this year's 2650 mile cycling season goal.
1. You talk about weather not climate.
Weather like this is not unusual for a continent where all the mountains run from north to south and not west to east like in Europe.
2. True, but the annual expansion/contraction of the snow cover in North America and Eurasia must surely be factored into long term climate models. That winter mantle of snow would increase the average albedo of our planet, functioning, I imagine, like a feedback mechanism in helping to cool our planet. Another feedback mechanism serving as a planetary air conditioner is the formation of giant storms like cyclones and hurricanes. These quite literally operate as giant heat engines transferring heat from the ocean surface to the upper atmosphere where it radiates out into space in the infrared band. In the process they perform work on the environment, imparting rotation to physical mass, just like any manmade heat engine – internal combustion, steam engine, etc.
I had firsthand experience of such ‘work’ performed by one of these atmospheric heat engines in 1991 when hurricane Bob plastered the south coast of New England. It’s a day I will never forget. I was living in an old estate in Falmouth, Massachusetts that was converted into multiple apartment dwellings. As the winds increased in intensity suddenly the top half of a huge pine tree toppled over just yards from my building. That was enough for me. I called my neighbors in the rear most building who had a basement. With winds topping out at 90 MPH I raced to their building as trees slammed down in front and behind me, leaping over and crawling under others to get to the safety of their basement. We all huddled fearfully in the dark basement for some time until the storm passed. When we emerged the transformation of our little complex was startling. What was once a thickly wooded old estate was now a shoulder level solid thicket of downed trees and brush covering the parking lots and grassy areas. Took us a week to clean up the mess and power was out for 11 days. But luckily only superficial damage was done to the buildings.
On the bright side, that hurricane cooled our planet by a little bit. But this year’s unusually large number of Atlantic hurricanes and tropical storms is likely our planet’s response to the ever rising accumulation of greenhouse gases with their heat trapping effect. That, of course, is a rather ominous trend for coastal regions around the world that are most impacted by tropical storms.
14. Talk about chaotic weather, what we are experiencing in the US for the last 6 or 7 days is truly remarkable. Temperature records dating back 100 years in 46 cities from the Midwest to the East coast have been either tied or outright shattered. Meanwhile a huge surge of arctic air has invaded the western part of the United States, bringing far below normal temperatures. This has been a godsend for us on the eastern half of the nation for outdoor activities and especially leaf cleanup in yards. I ended up with a huge yard, unintentionally, that I’ve been raking for 3 days now, about 2 to 3 hours each day before muscle exhaustion sets in. Today is the last day of this incredible record warmth between the high 60’s and low to mid 70’s Fahrenheit (20-23 C.). Hopefully, I’ll get the last leaves carted away today before the howling winds accompanied by whipping rain, snow squalls, and plunging temperatures arrives over the next few days.
15. Looking wistfully at the verdant green grass, enshrouded in fog, in the image Sabine posted in her tweet: “This is what a stereotypical German winter day looks like.”, all I could think of was how wonderful the Gulf Stream is for western Europe in maintaining a relatively stable, year round, temperature regime. In contrast, here in New England our climate is much more chaotic with an ancient, worn down mountain range unable to block the fierce arctic cold from the interior on northwesterly winds in winter, or oppressive heat and humidity from the southwest in summer. This morning with clear skies and a landscape blanketed with 24 inches (61 cm.) of snow in my SW New Hampshire town, and up to 44 inches (112 cm.) elsewhere, the temperature dropped to -8 Fahrenheit (-22 centigrade) in nearby Keene. On the bright side XC skiing will be great for a while, and there’s a beautiful Currier and Ives look to the scenery, just in time for Christmas.
16. Weather sure has been chaotic here in North America lately as a result of a polar vortex plunging south all the way to the Mexican border in Texas. Temps in Minnesota dropped to -42 F. (-41 C.), and in parts of the upper midwest century old records were shattered. Luckily, we've been relatively mild in my local area with temps bottoming out at only -9 F. (-22.8 C.). Nonetheless, for like a month my shared 375 foot, rather steep, driveway has been covered in rock hard, solid ice, most of the snow cover having been plowed off from multiple storms. We've had lower temps in the past. In the early 80's I well remember dangling in an open air chairlift on the slope of a local 2064 foot mountain, 30 feet above the ground, at -20 F. (-29 C.) due to a mechanical breakdown. Sat there swinging in a light breeze for 15 or 20 minutes, but fortunately my ski clothing kept me warm.
17. Whew, this is quite an outbreak of polar air targeting the US northeast. In the wee hours of this morning winds gusted to 46 mph (74 kph), while temperatures steadily nosedived to 5 F. (-15 C.) by 7 AM. The winds are still busy, gusting at 31 mph (50 kph), but at least the power is still on. This sort of chaotic weather during Springtime is pretty much the norm in this part of the world due to clashing air masses. Nonetheless, it’s quite a shock to the system to go from a pleasant high yesterday of 42 F. (5.5 C.) to this brutal temperature and wind chill.
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# Wk4 Q4 Online Learning Apr 20-24 Pre-Algebra
Teacher Stephanie Durrance 8th Grade Math (Pre-Algebra) 8th 4 Focus 12 : Frequency Tables
Standard(s) Taught
MAFS.8.SP.1.4: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.
Learning Targets and Learning Criteria
• understand the use of a two-way table to display bivariate categorical data. • create a two-way table to record the frequencies of bivariate categorical values. • determine the relative frequencies for rows and/or columns of a two-way table. • use the relative frequencies and context of the problem to describe possible associations between the two sets of data.
Classroom Activities
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https://www.eevblog.com/forum/programming/pointer-confusion-in-c-language/?wap2;PHPSESSID=sj9fjg3a4gqaachv99o6d115a4
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Computing > Programming
Pointer confusion in C -language
(1/37) > >>
Veketti:
Dear All,
https://www.geeksforgeeks.org/convert-floating-point-number-string/
I'm wondering why does pointer "res" work with this syntax. First of all variable "res" is not with * inside the ftoa -function. Then in main function, res is not called with & -prefix.
--- Code: ---// Converts a floating-point/double number to a string.
void ftoa(float n, char* res, int afterpoint)
{
// Extract integer part
int ipart = (int)n;
// Extract floating part
float fpart = n - (float)ipart;
// convert integer part to string
int i = intToStr(ipart, res, 0);
// check for display option after point
if (afterpoint != 0) {
res[i] = '.'; // add dot
// Get the value of fraction part upto given no.
// of points after dot. The third parameter
// is needed to handle cases like 233.007
fpart = fpart * pow(10, afterpoint);
intToStr((int)fpart, res + i + 1, afterpoint);
}
}
// Driver program to test above function
int main()
{
char res[20];
float n = 233.007;
ftoa(n, res, 4);
printf("\"%s\"\n", res);
return 0;
}
--- End code ---
To my understanding this should be:
--- Code: ---// Converts a floating-point/double number to a string.
void ftoa(float n, char *res, int afterpoint)
{
// Extract integer part
int ipart = (int)n;
// Extract floating part
float fpart = n - (float)ipart;
// convert integer part to string
int i = intToStr(ipart, *res, 0);
// check for display option after point
if (afterpoint != 0) {
*res[i] = '.'; // add dot
// Get the value of fraction part upto given no.
// of points after dot. The third parameter
// is needed to handle cases like 233.007
fpart = fpart * pow(10, afterpoint);
intToStr((int)fpart, *res + i + 1, afterpoint);
}
}
// Driver program to test above function
int main()
{
char res[20];
float n = 233.007;
ftoa(n, &res, 4);
printf("\"%s\"\n", res);
return 0;
}
--- End code ---
Are both examples actually the same, but the first is just some simplified version that compiler also understands? Testing in STM32CubeIDE
golden_labels:
--- Quote from: Veketti on June 20, 2021, 01:56:16 pm ---First of all variable "res" is not with * inside the ftoa -function.
--- End quote ---
Because:
--- Code: ---a[b] == *(a + b)
--- End code ---
Quite literally, including both + and the square brackets operators being commutative.
--- Quote from: Veketti on June 20, 2021, 01:56:16 pm ---Then in main function, res is not called with & -prefix.
--- End quote ---
Because in C arrays ot type T can be implicitly casted to T* pointing to the first element of such array.
More specifically:
In main there is a variable res. That variable is an object consisting of 20 elements of type char.
The ftoa function accepts a pointer to char as its second argument (char*).
Upon invocation, a pointer to the first element of res is taken and passed to that second argument. It is exactly equivalent of:
--- Code: ---char res[20];
char* ptr = &res[0];
ftoa(…, ptr, …);
--- End code ---
This should not be confused with &res, which would have a different type:
--- Code: ---&res[0] // `res[0]` is `char`, `&res[0]` is `char*`
&res // `res` is a `char[20]`, `&res` is `char(*)[20]`
--- End code ---
Due to how compilers are implemented, they may appear numerically equal if inspected and — ignoring compile errors/warnings — using them interchangeably may by pure coincidence seem to “work”. But they are not equal as they have different types. The latter type is also rarely seen in the wild, so don’t worry you don’t the notation now — quite likely you will not see it for the next few years.
Siwastaja:
Yes, as explained above, you can access a pointer like it was an array. Accessing something[0] is the same as *something. Then something[1] will be the next element, i.e., the compiler knows the size of the type and goes forward that many bytes.
You get used to it; it's very common to see
uint8_t single_variable;
uint8_t buffer[1024];
some_function(&single_variable);
some_function(buffer);
In such case, the latter would be equivalent to:
some_function(&buffer[0])
Veketti:
Thank you. One more thing. I noticed if I had in the main funtion the char array with different name eg:
--- Code: --- char notthesamename[20];
float n = 233.007;
ftoa(n, notthesamename, 4);
--- End code ---
It did still work. Should it, as ftoa -function is still handling char* res?
However if I changed the char array size to 33, it worked for a while and then the MCU started behaving strangely.
Thanks for helping
--- Quote from: Veketti on June 20, 2021, 04:03:21 pm ---It did still work. Should it, as ftoa -function is still handling char* res?
--- End quote ---
Yes, the function is independent of the calling code.
--- Quote from: Veketti on June 20, 2021, 04:03:21 pm ---However if I changed the char array size to 33, it worked for a while and then the MCU started behaving strangely.
--- End quote ---
How strangely? There is nothing inherently wrong with larger array size.
Your code does the work and exits, so I don't see how working "for a while" is even possible.
The only thing to keep in mind that this array is allocated on the stack, and the stack may overflow. But 33 bytes and this simple program should not cause stack overflow issues.
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# Cracking the Time Code: How Many Months Does 51 Days Equal?
The question of how many months 51 days equal to has been a puzzle to many people, particularly those involved in calculating time. The reason is that the time system is quite complex, and some rules must be followed in converting days to months accurately. This article will walk you through the steps you need to follow to crack the time code and get the correct answer to the question. So, let’s dive in!
## How Many Days are in a Month?
Before we delve into how many months 51 days equal, it is essential to understand the duration of each calendar month. While some months have 30 days, others have 31 days, except for February, which has 28 days in a regular year and 29 days in a leap year. The table below shows the number of days in each month:
Month Number of Days
January 31
February 28/29
March 31
April 30
May 31
June 30
July 31
August 31
September 30
October 31
November 30
December 31
## How Many Months Does 51 Days Equal?
With the knowledge of the number of days in each month, we can calculate how many months 51 days equal. To do this, we will divide 51 by the average number of days in a month. Since the average month has 30.44 days (365 days/12 months), we can find out how many months 51 days equal as shown below:
51 days ÷ 30.44 days/month = 1.68 months
Therefore, 51 days is equivalent to 1.68 months, but in practical terms, it is usually rounded up to 2 months.
## How Many Weeks is 51 Days?
Another question that people often ask when it comes to time conversion is how many weeks 51 days equal. To answer this question, we will divide 51 by seven, which is the number of days in a week:
51 days ÷ 7 days/week = 7.29 weeks
Therefore, 51 days is equivalent to 7.29 weeks. Since it is not possible to have a fraction of a week, we can round up the figure to 8 weeks, which means that 51 days is approximately 8 weeks.
## How Many Business Days is 51 Days?
Business days refer to weekdays, excluding weekends and public holidays. Therefore, the number of business days in 51 days will depend on the number of weekends and public holidays within the period. If the 51 days include seven weekends and three public holidays, the number of business days will be:
51 days – 7 weekends – 3 public holidays = 41 business days
Therefore, 51 days is equivalent to 41 business days.
## How Many Hours is 51 Days?
The number of hours in 51 days will depend on how many hours there are in a day. Since there are 24 hours in a day, we can calculate how many hours are in 51 days as follows:
51 days x 24 hours/day = 1224 hours
Therefore, 51 days is equivalent to 1224 hours.
## How Many Minutes is 51 Days?
To determine how many minutes are in 51 days, we need to know how many minutes there are in an hour. Since there are 60 minutes in an hour, we can calculate how many minutes are in 51 days as follows:
51 days x 24 hours/day x 60 minutes/hour = 73,440 minutes
Therefore, 51 days is equivalent to 73,440 minutes.
## How Many Seconds is 51 Days?
Finally, we can calculate how many seconds are in 51 days by knowing how many seconds there are in a minute. Since there are 60 seconds in a minute, we can calculate how many seconds are in 51 days as follows:
51 days x 24 hours/day x 60 minutes/hour x 60 seconds/minute = 4,406,400 seconds
Therefore, 51 days is equivalent to 4,406,400 seconds.
## The Bottom Line
Calculating time can be quite tricky, but with the information provided in this article, you can efficiently convert 51 days to months, weeks, business days, hours, minutes, and seconds. However, it is important to note that the number of days in a month may vary, depending on the month and the year. Therefore, it is advisable to check a calendar to confirm the exact number of days in a specific month.
Here are some of the most common questions and their answers related to the topic:
• Q: Can 51 days be rounded off to 1 month?
• A: Technically, 51 days is equivalent to 1.68 months, but it is usually rounded off to 2 months in practice.
• Q: How many weeks is 51 days?
• A: 51 days is equivalent to 7.29 weeks, which is usually rounded up to 8 weeks.
• Q: How many business days is 51 days?
• A: The number of business days in 51 days would depend on the number of weekends and public holidays within the period. If the 51 days include seven weekends and three public holidays, the number of business days will be 41.
• Q: How many hours is 51 days?
• A: 51 days is equivalent to 1224 hours.
• Q: How many minutes is 51 days?
• A: 51 days is equivalent to 73,440 minutes.
• Q: How many seconds is 51 days?
• A: 51 days is equivalent to 4,406,400 seconds.
## References
• https://www.timeanddate.com/date/durationresult.html?d1=51&m1=&y1=&d2=&m2=&y2=
• https://www.calculatorsoup.com/calculators/time/days-to-months-calculator.php
• https://www.calculatorsoup.com/calculators/time/days-to-weeks-months-years-calculator.php
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Cody
# Problem 719. Rotate a Matrix by 90 degrees
Solution 181192
Submitted on 24 Dec 2012 by Alfonso Nieto-Castanon
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
%% x = [1 2;3 4]; y_correct = [2 4;1 3]; assert(isequal(rotMatrix(x),y_correct))
``` ```
2 Pass
%% x = [1 2 3;4 5 6;7 8 9]; y_correct = [3 6 9;2 5 8;1 4 7]; assert(isequal(rotMatrix(x),y_correct))
``` ```
3 Pass
%% x=[1 2 3 4 5;6 7 8 9 0] y_correct = [5 0;4 9;3 8;2 7;1 6]; assert(isequal(rotMatrix(x),y_correct))
``` x = 1 2 3 4 5 6 7 8 9 0 ```
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$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$
## Section7.2Properties of Functions
### Subsection7.2.1Properties
Consider the following functions:
Let $A = \{1, 2, 3, 4\}$ and $B = \{a, b, c, d\}$, and define $f:A \rightarrow B$ by
\begin{equation*} f(1) = a, f(2) = b, f(3) = c \textrm{ and } f(4) = d \end{equation*}
Let $A = \{1, 2, 3, 4\}$ and $B = \{a, b, c, d\}$, and define $g:A \rightarrow B$ by
\begin{equation*} g(1) = a , g(2) = b, g(3) = a \textrm{ and } g(4) = b. \end{equation*}
The first function, $f\text{,}$ gives us more information about the set $B$ than the second function, $g\text{.}$ Since $A$ clearly has four elements, $f$ tells us that $B$ contains at least four elements since each element of $A$ is mapped onto a different element of $B\text{.}$ The properties that $f$ has, and $g$ does not have, are the most basic properties that we look for in a function. The following definitions summarize the basic vocabulary for function properties.
###### Definition7.2.1Injective Function, Injection
A function $f: A \rightarrow B$ is injective if
\begin{equation*} \forall a, b\in A, a\neq b \Rightarrow f(a) \neq f(b) \end{equation*}
An injective function is called an injection, or a one-to-one function.
Notice that the condition for an injective function is logically equivalent to $f(a) = f(b) \Rightarrow a = b$ for all $a, b\in A$. This is often a more convenient condition to prove than what is given in the definition.
###### Definition7.2.2Surjective Function, Surjection
A function $f: A \rightarrow B$ is surjective if its range, $f(A)$, is equal to its codomain, $B\text{.}$ A surjective function is called a surjection, or an onto function.
Notice that the condition for a surjective function is equivalent to
\begin{equation*} \textrm{For all } b \in B\textrm{, there exists } a\in A \textrm{ such that } f(a)=b.\text{.} \end{equation*}
###### Definition7.2.3Bijective Function, Bijection
A function $f: A \rightarrow B$ is bijective if it is both injective and surjective. Bijective functions are also called one-to-one, onto functions.
The function $f$ that we opened this section with is bijective. The function $g$ is neither injective nor surjective.
Let $A = \{1, 2, 3\}$ and $B = \{a, b, c, d\}$, and define $f:A \rightarrow B$ by $f(1) = b$, $f(2) = c$, and $f(3) = a$. Then $f$ is injective but not surjective.
The characteristic function, $\chi _S$ in Exercise 7.1.2 is surjective if $S$ is a proper subset of $A\text{,}$ but never injective if $\lvert A \rvert > 2$.
### Subsection7.2.2Counting
Let $A$ be the set of students who are sitting in a classroom, let $B$ be the set of seats in the classroom, and let $s$ be the function which maps each student into the chair he or she is sitting in. When is $s$ one to one? When is it onto? Under normal circumstances, $s$ would always be injective since no two different students would be in the same seat. In order for $s$ to be surjective, we need all seats to be used, so $s$ is a surjection if the classroom is filled to capacity.
Functions can also be used for counting the elements in large finite sets or in infinite sets. Let's say we wished to count the occupants in an auditorium containing 1,500 seats. If each seat is occupied, the answer is obvious, 1,500 people. What we have done is to set up a one-to-one correspondence, or bijection, from seats to people. We formalize in a definition.
###### Definition7.2.7Cardinality
Two sets are said to have the same cardinality if there exists a bijection between them. If a set has the same cardinality as the set $\{1,2,3,\ldots , n\}$, then we say its cardinality is $n\text{.}$
The function $f$ that opened this section serves to show that the two sets $A=\{1, 2, 3, 4\}$ and $B=\{a, b, c, d\}$ have the same cardinality. Notice in applying the definition of cardinality, we don't actually appear to count either set, we just match up the elements. However, matching the letters in $B$ with the numbers 1, 2, 3, and 4 is precisely how we count the letters.
###### Definition7.2.8Countable Set
If a set is finite or has the same cardinality as the set of positive integers, it is called a countable set.
The alphabet $\{A, B, C, . . . , Z\}$ has cardinality 26 through the following bijection into the set $\{1,2,3,\ldots ,26\}$.
\begin{equation*} \begin{array}{ccccc} A & B & C & \cdots & Z \\ \downarrow & \downarrow & \downarrow & \cdots & \downarrow \\ 1 & 2 & 3 & \cdots & 26 \\ \end{array}\text{.} \end{equation*}
Recall that $2\mathbb{P}= \{b\in \mathbb{P} \mid b= 2k \textrm{ for some } k \in \mathbb{P} \}$. Paradoxically, $2\mathbb{P}$ has the same cardinality as the set $\mathbb{P}$ of positive integers. To prove this, we must find a bijection from $\mathbb{P}$ to $2\mathbb{P}$. Such a function isn't unique, but this one is the simplest: $f:\mathbb{P} \rightarrow 2\mathbb{P}$ where $f(m) = 2m$. Two statements must be proven to justify our claim that $f$ is a bijection:
• $f$ is one-to-one.
Proof: Let $a, b \in \mathbb{P}$ and assume that $f(a) = f(b)$. We must prove that $a = b$.
\begin{equation*} f(a) = f(b) \Longrightarrow 2a = 2b \Longrightarrow a = b. \end{equation*}
• $f$ is onto.
Proof: Let $b \in 2\mathbb{P}$. We want to show that there exists an element $a \in \mathbb{P}$ such that $f(a) = b$. If $b \in 2\mathbb{P}$, $b = 2k$ for some $k \in \mathbb{P}$ by the definition of $2\mathbb{P}$. So we have $f(k) = 2k = b$. Hence, each element of 2$\mathbb{P}$ is the image of some element of $\mathbb{P}\text{.}$
Another way to look at any function with $\mathbb{P}$ as its domain is creating a list of the form $f(1),f(2), f(3), \ldots$. In the previous example, the list is $2, 4, 6, \ldots$. This infinite list clearly has no duplicate entries and every even positive integer appears in the list eventually.
A function $f:\mathbb{P}\to A$ is a bijection if the infinite list $f(1), f(2), f(3), \ldots$ contains no duplicates, and every element of $A$ appears on in the list. In this case, we say the $A$ is countably infinite, or simply countable
Readers who have studied real analysis should recall that the set of rational numbers is a countable set, while the set of real numbers is not a countable set. See the exercises at the end of this section for an another example of such a set.
We close this section with a theorem called the Pigeonhole Principle, which has numerous applications even though it is an obvious, common-sense statement. Never underestimate the importance of simple ideas. The Pigeonhole Principle states that if there are more pigeons than pigeonholes, then two or more pigeons must share the same pigeonhole. A more rigorous mathematical statement of the principle follows.
Assume no such element exists. For each $y \in Y\text{,}$ let $A_y = \{x\in X \mid f(x) =y \}\text{.}$ Then it must be that $\lvert A_y \rvert \leq n\text{.}$ Furthermore, the set of nonempty $A_y$ form a partition of $X\text{.}$ Therefore, $\lvert X \rvert = \sum_{y\in Y}{\lvert A_y \rvert} \leq n \lvert Y \rvert$ which is a contradiction.
Assume that a room contains four students with the first names John, James, and Mary. Prove that two students have the same first name. We can visualize a mapping from the set of students to the set of first names; each student has a first name. The pigeonhole principle applies with $n = 1$, and we can conclude that at least two of the students have the same first name.
### SubsectionExercises for Section 7.2
###### 1
Determine which of the functions in Exercise 7.1.1 of Section 7.1 are one- to-one and which are onto.
The only one-to-one function and the only onto function is $f$.
###### 2
1. Determine all bijections from the $\{1, 2, 3\}$ into $\{a, b, c\}$.
2. Determine all bijections from $\{1, 2, 3\}$ into $\{a, b, c, d\}$.
###### 3
Which of the following are one-to-one, onto, or both?
1. $f_1:\mathbb{R} \rightarrow \mathbb{R}$ defined by $f_1(x) = x^3 - x$.
2. $f_2 :\mathbb{Z} \rightarrow \mathbb{Z}$ defined by $f_2(x)= -x + 2$.
3. $f_3:\mathbb{N} \times \mathbb{N}\to \mathbb{N}$ defined by $f_3(j, k) =2^j3^k$.
4. $f_4 :\mathbb{P} \rightarrow \mathbb{P}$ defined by $f_4(n)=\lceil n/2\rceil$, where $\lceil x\rceil$ is the ceiling of $x\text{,}$ the smallest integer greater than or equal to $x\text{.}$
5. $f_5 :\mathbb{N} \rightarrow \mathbb{N}$ defined by $f_5(n)=n^2+n$.
6. $f_6:\mathbb{N} \rightarrow \mathbb{N} \times \mathbb{N}$ defined by $f_6(n)= (2n, 2n+1)$.
1. $f_1$ is onto but not one-to-one: $f_1(0)=f_1(1)$.
2. $f_2$ is one-to-one and onto.
3. $f_3$ is one-to-one but not onto.
4. $f_4$ is onto but not one-to-one.
5. $f_5$ is one-to-one but not onto.
6. $f_6$ is one-to-one but not onto.
###### 4
Which of the following are injections, surjections, or bijections on $\mathbb{R}\text{,}$ the set of real numbers?
1. $f(x) = -2x$.
2. $g(x) = x^2- 1$.
3. $h(x)=\left\{ \begin{array}{cc} x & x < 0 \\ x^2 & x\geq 0 \\ \end{array} \right.$
4. $q(x)=2^x$
5. $r(x) =x^3$
6. $s(x) = x^3-x$
###### 5
Suppose that $m$ pairs of socks are mixed up in your sock drawer. Use the Pigeonhole Principle to explain why, if you pick $m + 1$ socks at random, at least two will make up a matching pair.
Let $X=\{\textrm{socks selected}\}$ and $Y=\{\textrm{pairs of socks}\}$ and define $f:X \to Y$ where $f(x) =$the pair of socks that $x$ belongs to . By the Pigeonhole principle, there exist two socks that were selected from the same pair.
###### 6
In your own words explain the statement “The sets of integers and even integers have the same cardinality.”
###### 7
Let $A =\text{ }\{1, 2, 3, 4, 5\}$. Find functions, if they exist that have the properties specified below.
1. A function that is one-to-one and onto.
2. A function that is neither one-to-one nor onto.
3. A function that is one-to-one but not onto.
4. A function that is onto but not one-to-one.
1. $f(n)=n$, for example
2. $f(n)=1$, for example
3. None exist.
4. None exist.
###### 8
1. Define functions, if they exist, on the positive integers, $\mathbb{P}\text{,}$ with the same properties as in Exercise 7 (if possible).
2. Let $A$ and $B$ be finite sets where $|A|=|B|$. Is it possible to define a function $f:A \rightarrow B$ that is one-to-one but not onto? Is it possible to find a function $g:A \rightarrow B$ that is onto but not one-to-one?
###### 9
1. Prove that the set of natural numbers is countable.
2. Prove that the set of integers is countable.
3. Prove that the set of rational numbers is countable.
1. Use $s:\mathbb{N}\to \mathbb{P}$ defined by $s(x)=x+1$.
2. Use the function$f:\mathbb{N}\to \mathbb{Z}$ defined by $f(\text{x0}=x/2$ if $x$ is even and $f(x)=-(x+1)/2$ if $x$ is odd.
3. The proof is due to Georg Cantor (1845-1918), and involves listing the rationals through a definite procedure so that none are omitted and duplications are avoided. In the first row list all nonnegative rationals with denominator 1, in the second all nonnegative rationals with denominator 2, etc. In this listing, of course, there are duplications, for example, $0/1=0/2=0$, $1/1=3/3=1$, $6/4=9/6=3/2$, etc. To obtain a list without duplications follow the arrows in Figure 7.2.13, listing only the circled numbers.
We obtain: $0,1,1/2,2,3,1/3,1/4,2/3,3/2,4/1,\ldots$ Each nonnegative rational appears in this list exactly once. We now must insert in this list the negative rationals, and follow the same scheme to obtain: $0,1,-1,1/2,-1/2,2,-2,3,-3,1/3,-1/3, \ldots$ which can be paired off with the elements of $\mathbb{N}\text{.}$
###### 10
1. Prove that the set of finite strings of 0's and 1's is countable.
2. Prove that the set of odd integers is countable.
3. Prove that the set $\mathbb{N}\times \mathbb{N}$ is countable.
###### 11
Use the Pigeonhole Principle to prove that an injection cannot exist between a finite set $A$ and a finite set $B$ if the cardinality of $A$ is greater than the cardinality of $B\text{.}$
Let $f$ be any function from $A$ into $B$. By the Pigeonhole principle with $n=1$, there exists an element of $B$ that is the image of at least two elements of $A$. Therefore, $f$ is not an injection.
###### 12
The important properties of relations are not generally of interest for functions. Most functions are not reflexive, symmetric, antisymmetric, or transitive. Can you give examples of functions that do have these properties?
###### 13
Prove that the set of all infinite sequences of 0's and 1's is not a countable set.
The proof is indirect and follows a technique called the Cantor diagonal process. Assume to the contrary that the set is countable, then the elements can be listed: $n_1,n_2,n_3,\ldots$ where each $n_i$ is an infinite sequence of 0s and 1s. Consider the array: $\begin{array}{c} n_1=n_{11}n_{12}n_{13}\cdots\\ n_2=n_{21}n_{22}n_{23}\cdots\\ n_3=n_{31}n_{32}n_{33}\cdots\\ \quad \vdots\\ \end{array}$
We assume that this array contains all infinite sequences of 0s and 1s. Consider the sequence $s$ defined by $s_i=\begin{cases} 0 & \textrm{ if } n_{\textrm{ii}}=1 \\ 1 & \textrm{ if } n_{\textrm{ii}}=0 \end{cases}$
Notice that $s$ differs from each $n_i$ in the $i$th position and so cannot be in the list. This is a contradiction, which completes our proof.
###### 13
Prove that the set of all functions on the integers is an uncountable set.
###### 14
Given five points on the unit square, $\{(x,y) \mid 0 \leq x, y \leq 1 \}$, prove that there are two of the points a distance of no more than $\frac{\sqrt{2}}{2}$ from one another.
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Symbol
Problem
EXAM Find the value of x which satisfies each of the following equas in each case verify the solution. 2. $3x-5=0$ 1. $7x-4=17$ 3. $2x+15=23$ 4. $5x-9=21$ 5. $7x=18-2x$ 6.3x $3x=25-2x$ $5x+2=6x-1$ 7. $4x-3=2x+1.$ 8.5x $9.3x$ $3x+2=4x-3$ 10. $4x-3=3x+4$ $11.8x-9=33-4x$ 12. $5x+3=15-x.$ 13. $2x+15=27-4x$ $14.$ $7x+11=3x+27$ 15. $15-5x=24-8x$ 16. $9x+21-4x=46$
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### Capillarity
Home -> Lecture Notes -> Fluid Mechanics -> Unit-I
Rise or fall of a liquid in a capillary tube is caused by surface tension and depends on the relative magnitude of cohesion of the liquid and the adhesion of the liquid to the walls of the containing vessel.
Liquids rise in tubes if they wet (adhesion > cohesion) and fall in tubes that do not wet (cohesion > adhesion).
Wetting and contact angle
Fluids wet some solids and do not others.
The figure shows some of the possible wetting behaviors of a drop of liquid placed on a horizontal, solid surface (the remainder of the surface is covered with air, so two fluids are present).
Fig.(a) represents the case of a liquid which wets a solid surface well, e.g. water on a very clean copper. The angle q shown is the angle between the edge of the liquid surface and the solid surface, measured inside the liquid. This angle is called the contact angle and is a measure of the quality of wetting. For perfectly wetting, in which the liquid spreads as a thin film over the surface of the solid, q is zero.
Fig.(c) represents the case of no wetting. If there were exactly zero wetting, q would be 180o. However, the gravity force on the drop flattens the drop, so that 180o angle is never observed. This might represent water on teflon or mercury on clean glass.
We normally say that a liquid wets a surface if q is less than 90o and does not wet if q is more than 90o. Values of q less than 20o are considered strong wetting, and values of q greater than 140o are strong nonwetting.
Capillarity is important (in fluid measurments) when using tubes smaller than about 10 mm in diameter.
Capillary rise (or depression) in a tube can be calculated by making force balances. The forces acting are force due to surface tension and gravity.
The force due to surface tesnion,
Fs = pdscos(q), where q is the wetting angle or contact angle. If tube (made of glass) is clean q is zero for water and about 140o for Mercury.
This is opposed by the gravity force on the column of fluid, which is equal to the height of the liquid which is above (or below) the free surface and which equals
Fg = (p/4)d2hgr,
where r is the density of liquid.
Equating these forces and solving for Capillary rise (or depression), we find
h = 4scos(q)/(rgd)
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http://www.tutorcircle.com/arc-length-calculator-ca1gblc.html
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# Arc Length Calculator
Arc length can be defined as length of an arc which is a segment of a curve or it can also be segment of a Circle. Formula for calculating Arc Length when value of central angle and radius are known is given as shown below:
Arc length = 2 ∏ r (C / 360), where 'r' is the radius of arc, 'C' is the central angle of arc which is measured in degrees, value of '∏' is approximately equals to 3.14.
Arc length calculator is an online tool or calculator which can be used to calculate arc length when values of Radius and Central angle are known.
Steps to use arc length calculator are shown below:
Step 1: Enter radius of Arc in first text box.
Step 2: Enter Central angle of arc in degrees in second text box.
Step 3: Click on 'Solve' button and result will be displayed in 'Arc Length' text box.
Let us take an example:
Step 1. Enter radius of Arc as 10 inch in first text box.
Step 2. Enter Central angle as 450 in second text box.
Step 3. Click on 'Solve' button and result will be 7.854 inches.
Math Topics
Top Scorers in Worksheets
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Home > Standard Error > How To Calculate Standard Error Of Regression Coefficient
# How To Calculate Standard Error Of Regression Coefficient
## Contents
Identify a sample statistic. Loading... Identify a sample statistic. The standard error of the forecast is not quite as sensitive to X in relative terms as is the standard error of the mean, because of the presence of the noise Source
Naturally, the value of a statistic may vary from one sample to the next. I love the practical, intuitiveness of using the natural units of the response variable. Test Your Understanding Problem 1 The local utility company surveys 101 randomly selected customers. The error that the mean model makes for observation t is therefore the deviation of Y from its historical average value: The standard error of the model, denoted by s, is this page
## How To Calculate Standard Error Of Regression Coefficient
Is there a succinct way of performing that specific line with just basic operators? –ako Dec 1 '12 at 18:57 1 @AkselO There is the well-known closed form expression for Some regression software will not even display a negative value for adjusted R-squared and will just report it to be zero in that case. There’s no way of knowing. This means that the sample standard deviation of the errors is equal to {the square root of 1-minus-R-squared} times the sample standard deviation of Y: STDEV.S(errors) = (SQRT(1 minus R-squared)) x
• Statistics II for DummiesDeborah J.
• Example data.
• My home PC has been infected by a virus!
• The standard deviation is computed solely from sample attributes.
• In a multiple regression model in which k is the number of independent variables, the n-2 term that appears in the formulas for the standard error of the regression and adjusted
• However, in the regression model the standard error of the mean also depends to some extent on the value of X, so the term is scaled up by a factor that
• The critical value is the t statistic having 99 degrees of freedom and a cumulative probability equal to 0.995.
• Formulas for a sample comparable to the ones for a population are shown below.
The S value is still the average distance that the data points fall from the fitted values. S becomes smaller when the data points are closer to the line. Thanks for the beautiful and enlightening blog posts. How To Calculate Standard Error In Regression Analysis Conversely, the unit-less R-squared doesn’t provide an intuitive feel for how close the predicted values are to the observed values.
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For any given value of X, The Y values are independent. Standard Error Of Regression Coefficient For example, if the sample size is increased by a factor of 4, the standard error of the mean goes down by a factor of 2, i.e., our estimate of the Check out the grade-increasing book that's recommended reading at Oxford University! The dependent variable Y has a linear relationship to the independent variable X.
## How To Calculate Standard Error Of Regression In Excel
III. http://www.statisticshowto.com/find-standard-error-regression-slope/ The accuracy of a forecast is measured by the standard error of the forecast, which (for both the mean model and a regression model) is the square root of the sum How To Calculate Standard Error Of Regression Coefficient Compute alpha (α): α = 1 - (confidence level / 100) = 1 - 99/100 = 0.01 Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.01/2 How To Calculate Standard Error Of Regression Slope temperature What to look for in regression output What's a good value for R-squared?
Fitting so many terms to so few data points will artificially inflate the R-squared. this contact form Is 8:00 AM an unreasonable time to meet with my graduate students and post-doc? The correlation between Y and X is positive if they tend to move in the same direction relative to their respective means and negative if they tend to move in opposite S is 3.53399, which tells us that the average distance of the data points from the fitted line is about 3.5% body fat. How To Calculate Standard Error In Regression Model
Are there any saltwater rivers on Earth? How to Find the Confidence Interval for the Slope of a Regression Line Previously, we described how to construct confidence intervals. Take-aways 1. have a peek here Suppose our requirement is that the predictions must be within +/- 5% of the actual value.
The standard deviation cannot be computed solely from sample attributes; it requires a knowledge of one or more population parameters. Standard Error Of Estimate Interpretation Smaller values are better because it indicates that the observations are closer to the fitted line. price, part 4: additional predictors · NC natural gas consumption vs.
## Can you show step by step why $\hat{\sigma}^2 = \frac{1}{n-2} \sum_i \hat{\epsilon}_i^2$ ?
Why I Like the Standard Error of the Regression (S) In many cases, I prefer the standard error of the regression over R-squared. In a simple regression model, the standard error of the mean depends on the value of X, and it is larger for values of X that are farther from its own What is missing from a non-afterburning engine to prohibit the use of afterburning? Standard Error Of Estimate Calculator In a simple regression model, the percentage of variance "explained" by the model, which is called R-squared, is the square of the correlation between Y and X.
Find the margin of error. What if I want to return for a short visit after those six months end? Check out our Statistics Scholarship Page to apply! http://xvisionx.com/standard-error/standard-error-formula-regression-coefficient.html My hard disk is full - how can I determine what's taking up space?
S provides important information that R-squared does not. Sign in to report inappropriate content. The standard error is a measure of variability, not a measure of central tendency. For example, the standard error of the estimated slope is $$\sqrt{\widehat{\textrm{Var}}(\hat{b})} = \sqrt{[\hat{\sigma}^2 (\mathbf{X}^{\prime} \mathbf{X})^{-1}]_{22}} = \sqrt{\frac{n \hat{\sigma}^2}{n\sum x_i^2 - (\sum x_i)^2}}.$$ > num <- n * anova(mod)[[3]][2] > denom <-
Is there a different goodness-of-fit statistic that can be more helpful? You can choose your own, or just report the standard error along with the point forecast. price, part 1: descriptive analysis · Beer sales vs. The accuracy of the estimated mean is measured by the standard error of the mean, whose formula in the mean model is: This is the estimated standard deviation of the
The critical value that should be used depends on the number of degrees of freedom for error (the number data points minus number of parameters estimated, which is n-1 for this Sign in to add this to Watch Later Add to Loading playlists... Here is an Excel file with regression formulas in matrix form that illustrates this process. Frost, Can you kindly tell me what data can I obtain from the below information.
But still a question: in my post, the standard error has $(n-2)$, where according to your answer, it doesn't, why? –loganecolss Feb 9 '14 at 9:40 add a comment| 1 Answer A good rule of thumb is a maximum of one term for every 10 data points. The standard error of a coefficient estimate is the estimated standard deviation of the error in measuring it. Unable to use \tag in split equation "ON the west of New York?" Is this preposition correct?
regressing standardized variables1How does SAS calculate standard errors of coefficients in logistic regression?3How is the standard error of a slope calculated when the intercept term is omitted?0Excel: How is the Standard The system returned: (22) Invalid argument The remote host or network may be down. e) - Duration: 15:00. For example, select (≠ 0) and then press ENTER.
Estimation Requirements The approach described in this lesson is valid whenever the standard requirements for simple linear regression are met.
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# How far is Ranong from Myitkyina?
The distance between Myitkyina (Myitkyina Airport) and Ranong (Ranong Airport) is 1076 miles / 1732 kilometers / 935 nautical miles.
The driving distance from Myitkyina (MYT) to Ranong (UNN) is 1449 miles / 2332 kilometers, and travel time by car is about 28 hours 4 minutes.
1076
Miles
1732
Kilometers
935
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2 h 32 min
155 kg
## Distance from Myitkyina to Ranong
There are several ways to calculate the distance from Myitkyina to Ranong. Here are two standard methods:
Vincenty's formula (applied above)
• 1076.343 miles
• 1732.207 kilometers
• 935.317 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth's surface using an ellipsoidal model of the planet.
Haversine formula
• 1081.303 miles
• 1740.188 kilometers
• 939.627 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## How long does it take to fly from Myitkyina to Ranong?
The estimated flight time from Myitkyina Airport to Ranong Airport is 2 hours and 32 minutes.
## What is the time difference between Myitkyina and Ranong?
The time difference between Myitkyina and Ranong is 30 minutes. Ranong is 30 minutes ahead of Myitkyina.
## Flight carbon footprint between Myitkyina Airport (MYT) and Ranong Airport (UNN)
On average, flying from Myitkyina to Ranong generates about 155 kg of CO2 per passenger, and 155 kilograms equals 343 pounds (lbs). The figures are estimates and include only the CO2 generated by burning jet fuel.
## Map of flight path and driving directions from Myitkyina to Ranong
See the map of the shortest flight path between Myitkyina Airport (MYT) and Ranong Airport (UNN).
## Airport information
Origin Myitkyina Airport
City: Myitkyina
Country: Burma
IATA Code: MYT
ICAO Code: VYMK
Coordinates: 25°23′0″N, 97°21′6″E
Destination Ranong Airport
City: Ranong
Country: Thailand
IATA Code: UNN
ICAO Code: VTSR
Coordinates: 9°46′39″N, 98°35′7″E
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### HolkinPV's blog
By HolkinPV, 11 years ago, translation,
285A - Slightly Decreasing Permutations
As the answer you can print such permutation: n, n - 1, ..., n - k + 1, 1, 2, ..., n - k. For example, if n = 5, k = 2, then the answer is: 5, 4, 1, 2, 3. If k = 0, you should print 1, 2, ..., n. Such solution can be written in two loops.
285B - Find Marble
It is known that a permutation can be considered as set of cycles. The integer i moves to p[i] for all i (1 ≤ i ≤ n). You can start moving from integer s along the cycle. If you find integer t, then print the length of the path. If you return to s, then print - 1.
285C - Building Permutation
The solution of the problem is rather simple. Sort all integers a and then make from integer a[1] integer 1, from integer a[2] integer 2 and so on. So, integer a[i] adds to the answer the value |a[i] - i|. The answer should be count in 64-bit type. You can simply guess why such solution is correct.
285D - Permutation Sum
For a start, describe bruteforce solution. Firstly, we will always assume, that a is identity permutation, that is a[i] = i. In this case, the answer should be multiplied by n!. Or in other way your bruteforce will not be counted. Secondly, using our bruteforce we can see, that for even n the answer is 0.
What do you also need to get accepted? First case is to calculate answers for all n on your computer and write them in constant array. In other words you can make precalc. Second case is to make you solution faster. The soltuion using meet-in-the-middle idea works fast for n ≤ 15. If you remember that for even n answer is 0 then you can get accepted using such solution. But other simple bruteforces and dynamic programmings on maps work slower than 3 seconds.
285E - Positions in Permutations
• -29
| Write comment?
» 11 years ago, # | +29 We want E, please :(
• » » 11 years ago, # ^ | +3 State Compress Dynamic Programming ... O(n^2 2^3) ...
• » » » 11 years ago, # ^ | +2 What dynamic? I don't know what to do with positions, that we haven't taken.
• » » » » 11 years ago, # ^ | ← Rev. 9 → +36 Just do res[i] *= (n-i)!, where res[i] is a result with i good position without some position (such that, as you say, "we haven't taken"). Now res[i] may contain some permutations with more than i good positions. But consider that you have correctly caclulated res[i+1], res[i+2], ..., res[n]. Then each of res[i+1] permutations was overcounted in res[i] exactly С(i+1, i) times, each of res[i+2] — C(i+2, i) times and so on. After that you will also have res[i] caclulated correctly.
• » » » » » 11 years ago, # ^ | +5 My idea in problem E is the same as witua :D
• » » » » » 11 years ago, # ^ | +6 Thanks for such a nice solution. It's sad that I came so close but didn't figure out how to exclude the overlapping part.
• » » » » » 11 years ago, # ^ | ← Rev. 2 → 0 Could You explain dynamic programming part more precisely, please. I mean part where your in process of creating res[] array.(as I understood, according to your code smth likeres[i] ~ R[n][i][][] , right?).Else seems more-less understandable =)Thank you
• » » » » » » 11 years ago, # ^ | +1 Can you explain it a bit more? I really don't understand the english from the sentence:Just do res[i] *= (n-i)!, where res[i] is a result with i good position without some position (such that, as you say, "we haven't taken")
• » » » 11 years ago, # ^ | +2 could you explain your solution ? and what is State Compress Dynamic Programming ?
• » » 11 years ago, # ^ | +9 no offence bro, but i'm still scratching my head reading D!!
» 11 years ago, # | 0 Can you proof D-Permutation Sum ? Pls...
• » » 11 years ago, # ^ | -7 If you mean in the "proof" why answer for even n-s is 0, I will explain it here. First of all, let's subtract 1 from each permutation. Than our problem becomes: How many permutations a,b exist for {0,1,...n-1} such that the sequence ((a[i]+b[i])%n, 1<=i<=n) is also a permutation of (0, 1, ... , n-1). Assume we have permutations a,b. Than, if ((a[i]+b[i])%n) is also a permutation, considering modulo n, we have: n(n-1)/2 = 1+2+...+n = (sum_of_all (a[i]+b[i])%n ) = (sum_of_all(a[i]) + (sum_of_all(b[i])) = (0+1+...+(n-1)) + (0+1+...+(n-1)) = n(n-1) = 0 (mod n). So, n(n-1)/2 = 0 (mod n), which simply implies that n should be odd.
» 11 years ago, # | ← Rev. 2 → +13 Can Anyone Describe Meet-In-The-Middle Idea For D, Please?
• » » 11 years ago, # ^ | ← Rev. 2 → 0 can anyone please give a link to some problems related to 'Meet in the middle' idea?
• » » 11 years ago, # ^ | +1
• » » » 11 years ago, # ^ | 0 I'm sorry. I thought just about idea "Meet-In-The-Middle"
• » » 11 years ago, # ^ | -12 have you taken Cryptography course? if yes. then it is so easy for you :)
• » » » 11 years ago, # ^ | 0 would you explain it in more detail?? thanks!!
• » » 11 years ago, # ^ | +4 Are you sure that meet in the middle solution exist for problem D? I thought about that problem this way but didn't succeed.
• » » » 11 years ago, # ^ | ← Rev. 4 → +7 yes there exist meet in the middle solution with O(C(n,n/2)*(n/2)! ). for maximal N (15) its about 32 432 400 . but my solution needs 7 sec for 15 (I think its because of constant factor)edit: it does about 518 918 400 . but even that is not too mach for codeforces if constant is small..
• » » » » 11 years ago, # ^ | +3 Thanks a lot :)
• » » » » » 11 years ago, # ^ | +6 if you cant get my code I can say you how it works but not in english..
• » » » » » » 11 years ago, # ^ | 0 No thanks, it's clear :)
• » » » » 11 years ago, # ^ | 0 Oh! i had that solution O(C(n, n/2)*(n/2)!) but i thought it takes too long for N=15
» 11 years ago, # | +2 At A, if k = 0, shouldn't it be 1, 2...n-1, n? :o
• » » 11 years ago, # ^ | 0 Yes, fixed)
» 11 years ago, # | ← Rev. 2 → 0 I can't understand how one can use brute-force for n = 15 because it'll take O(n!) operation which for n = 15 it'll need almost three hours to be done. Is there another approach?Edit: It seems there is another solution for this problem. The answer for odd n when we fix our first permutation, is equal to " Number of toroidal semi-queens on a (2n+1) X (2n+1) board". May somebody explain why?
• » » 11 years ago, # ^ | 0 Set numbers one by one and every step check if the answer still can be correct.
• » » 11 years ago, # ^ | 0 I guess you looked up sequence A006717 in OEIS. It's also "the number of good permutations on 2n+1 elements", which seems more related to this problem. Thats how I solved the problem. Just computed the first few examples by bruteforce, looked up in OEIS and copied the sequence into my code. No need to optimze the brute-force in any way for N=15 ... actually I only calculated until n=7 and had enough trust, that i found the right sequence.
» 11 years ago, # | +5 For problem D, it is mentioned that answer would be equal to 0 if n is even. Can someone tell my why this is true ? I am thinking on the lines that there would be some number that we won't be able to generate when the length of permutations is even but haven't succeeded as of yet.
• » » 11 years ago, # ^ | ← Rev. 2 → +23 Consider the permutations on set 0,...,n-1, and let's work modulo n. The sum of any permutation is 0+1+...+n-1 = (n-1)n/2 (mod n), which is n/2 because n is even. But if a,b,c are permutations and c = a+b (mod n), then sum of elements of c is also 2*(n-1)n/2 (mod n) = 0, a contradiction.
• » » » 5 years ago, # ^ | ← Rev. 2 → 0 Can you please explain this — "c = a+b (mod n), then sum of elements of c is also 2*(n-1)n/2 (mod n) = 0"
• » » » » 5 years ago, # ^ | 0 Ok i got it Thanks
» 11 years ago, # | ← Rev. 2 → +8 I got a wrong answer on D just because of printf !!how come does this line printf("%I64d\n", 0); produce 1174031664003678208 !!I don't know what kind of compilers are you using, I'm really frustrated :(3373779
• » » 11 years ago, # ^ | +19 0 has type int, but your printf was trying to print long long. So the behavior is undefined.
• » » » 11 years ago, # ^ | 0 :'( :'(guess I will never use printf again, thank you :-)
• » » » » 11 years ago, # ^ | ← Rev. 3 → +1 No , its better to use printf. It's a lot faster than cin . If you want to print a constant just do it like this : printf("0 \n");
• » » » » » 11 years ago, # ^ | +5 cin isn't so slower as you think, if you use it in right way.
• » » » » » » 11 years ago, # ^ | +4 Well it sure makes a big difference if the output has many lines and you use endl instead of "\n": it flushes the output after every line, and that can easily get TLE.
• » » 11 years ago, # ^ | ← Rev. 2 → -29 printf("%I64d\n",0ll); is expected.
» 11 years ago, # | 0 I coded problem 'C' correctly, but it seems that the C# Sort function was not fast enough to finish within the 2 second time limit for test #23. Is it possible to appeal this, or will I be penalized for using C# language?
• » » 11 years ago, # ^ | +1 anti-quicksort test
• » » 11 years ago, # ^ | +1 As I understood from here http://msdn.microsoft.com/en-us/library/aa311213(v=vs.71).aspx C# uses QuickSort. QuitSort does O(n2) in worst case. So you should randomly shuffle array before executing Arrays.sort().
• » » » 11 years ago, # ^ | 0 Thank you! I'll make a note of that, yet it still seems that using C# here was a disadvantage.
• » » » » 11 years ago, # ^ | 0 Nah, its the same for Java. Just remember now for future contests :)
• » » » » » 11 years ago, # ^ | 0 Same problem with java. But I found a really interesting solution in my contest room, just swap some elements and it runs in about 300-400ms. Code here: http://codeforces.com/contest/285/submission/3367901
• » » » » » 11 years ago, # ^ | 0 i did nothing about the original sequence, and still got ac in java 6
• » » » » » 11 years ago, # ^ | ← Rev. 2 → 0 for the problem C i used long array in java 6. but it took TLE. Arrays.sort() took so much time to sort long array. then I saw someone used Long array instead of long array. I used it and it finished under 1.5s. Can anyone explain why this different ??
• » » » » » » 11 years ago, # ^ | 0 Arrays.sort on Objects uses merge sort which has a wost case of O(n*log(n)). Arrays.sort on primitives uses quicksort which has a worst case of O(n^2) This happens if you build the array in a certain manner otherwise quicksort on average is also O(n*log(n)).
• » » » 11 years ago, # ^ | +1 Yeah. you're right. My solution http://codeforces.com/contest/285/submission/3376657
» 11 years ago, # | +6 Solution for problem C: «You can simply guess why such solution is correct.»Can you give me any hint on how to prove this? I had to take that for granted during the contest, but was wondering what is the most elegant way to prove it...
• » » 11 years ago, # ^ | -29 since all numbers from 1 to n must be there (in any order), the number of changes done are minimized when the lowest input is converted to the lowest output, and so on, till the highest input to the highest output! if you are not yet convinced, try it for test cases for upto n=10 or 15...
• » » » 11 years ago, # ^ | ← Rev. 2 → +17 This is not answering his request for a proof.If you have |a[x] — i| + |a[y] — j| in your sum, where i > j but a[x] < a[y], then swapping i and j will make the sum not larger (try it). So if you have any sequence which isn't sorted, then it you can transform it into a sorted sequence using these inversions, and you will get a result which is no worse.
• » » 11 years ago, # ^ | ← Rev. 2 → +3 Take a random i, the amount needed to add to the total sum is abs ( i + 1 — a[i] ), assume that it is not the minimum, and try instead to use some other possibility of a[i], if you take smaller number, you immediately see that you obtain bigger sum, in the case a[k] = a[i] for some k < i, it's still true that the minimum is the firstly calculated one because the amount is the same). Now suppose you take a number of the right of a[i] say a[k] for some index k > i (greater one), the differences abs((i+1)-a[k])) and abs((k+1)-a[i]) in the best case will be equal ( otherwise the first calculated sum is always less) and still we don't have improvement. So we conclude that the best choice is to use abs ( i + 1 — a[i] ), keeping in eye that the elements of a[] are from [1:n] and a[] is sorted.
» 11 years ago, # | 0 For Problem B: There is no need that we return to s. For example: 5 1 5 2 3 4 2 5
• » » 11 years ago, # ^ | +3 yeah even i thought about this and felt a bit disappointed after i had submitted and locked my solution, but the problem statement says that all Pi's are distinct so he cant loop around any cycle containing neither the start nor the end point!!
» 11 years ago, # | +8 Could someone please give me some idea on how to generate the array using meet in the middle approach for problem D? Thank you.
» 11 years ago, # | 0 can anyone please explain how to precalculate the answers for odd n in Problem D
• » » 11 years ago, # ^ | 0 Just search.. Notice that you only need to calculate b array for a[]={1,2,3,...,N} Then multiply this result with N! I was wondering how to solve this problem without precalculation but got no clue...
» 11 years ago, # | ← Rev. 2 → -10 deleted
» 11 years ago, # | +19 "The soltuion using meet-in-the-middle idea works fast for n ≤ 15" " But other simple bruteforces and dynamic programmings on maps work slower than 3 seconds." Could you explain them?:) I think the tutorial should be more detailed and accurate
• » » 11 years ago, # ^ | +2 Agree with you!!!
» 11 years ago, # | +3 Can you explain problem D in detail ?
• » » 11 years ago, # ^ | 0 Agree!!
» 11 years ago, # | ← Rev. 2 → +12 I am seriously hoping that codeforces does something for better tutorials . I am scratching my head for problem 4 and 5 with only precomputed codes in front of my eyes. Something like codechef tutorials would be very good .
» 11 years ago, # | 0 I don't think problem D is suitable for algorithm competition. However, thank the author of the problem. It's a really good mathematic problem.
• » » 11 years ago, # ^ | 0 How is problem D a mathematic problem ?
• » » » 11 years ago, # ^ | 0 Maybe we'll have a faster solution by using combinatorial mathematics theory. Or somebody proves it's impossible.
» 11 years ago, # | +1 Problem D: how i can get ll arr[]={1, 3, 15, 133, 2025, 37851, 1030367, 36362925}; please anyone help in post details; every coder do the same way,but i can't understand. Al..helal
• » » 11 years ago, # ^ | 0 Just do the normal brute force recursion 2^n*2^n*n (memoize in a map if u want) compute the answer offline for all 1<=n<=15 and u will get a similar array with the actual answers. I believe that the array u presented doesn't account for the factorial thing so u should account for it somehow before u print the answer.
» 11 years ago, # | 0 WRITE PROBLEM E TUTORIAL !!!
» 11 years ago, # | 0 I would appreciate it if someone explain the details of using DP or meet-in-the-middle approach to solving problem D.
» 11 years ago, # | ← Rev. 3 → +1 Where is the official solution for E?this is the worst editorial I've ever seen no official solution for E and no explanation how to use meet in the middle for D.
» 9 years ago, # | ← Rev. 2 → 0 E can also be solved more generally using rook polynomials.First, compute the rook polynomial R(x) = r[n]x^n + r[n-1]x^(n-1) + ... + r[0]. The coefficients of R can be computed with an O(N^2) dp. We define the hit number h[k] = number of ways to place the rooks such that exactly k of them are in the restricted positions (i.e. number of permutations such that there are exactly k good poistions). In other words, h[k] is the answer to this problem. Define the hit polynomial H(x) = h[n]x^n + h[n-1]x^(n-1) + ... + h[0]There's the following relationship between H(x) and the rook coefficients: H(x) = sum(r[i] * (n-i)! * (x — 1)^i, i = 1..n)There's a nice proof for the above relationship here http://www.math.ucsd.edu/~remmel/files/Book.pdf.Given R(x), H(x) can be computed in O(N^2) using the above identity.
» 7 years ago, # | 0 How should we precompute aray for odd numbers in problem D? I am getting a TLE if done naively and have no clue how to optimise it
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The Rudolf Steiner Archive
a project of Steiner Online Library, a public charity
Second Scientific Lecture-Course: Warmth CourseGA 321
Lecture I
1 March 1920, Stuttgart
My dear friends,
The present course of lectures will constitute a kind of continuation of the one given when I was last here. I will begin with those chapters of physics which are of especial importance for laying a satisfactory foundation for a scientific world view, namely the observations of heat relations in the world. Today I will try to lay out for you a kind of introduction to show the extent to which we can create a body of meaningful views of a physical sort within a general world view. This will show further how a foundation may be secured for a pedagogical impulse applicable to the teaching of science. Today we will therefore go as far as we can towards outlining a general introduction.
The theory of heat, so-called, has taken a form during the 19th century which has given a great deal of support to a materialistic view of the world. It has done so because in heat relationships it is very easy to turn one's glance away from the real nature of heat, from its being, and to direct it to the mechanical phenomena arising from heat.
Heat is first known through sensations of cold, warmth, lukewarm, etc. But man soon learns that there appears to be something vague about these sensations, something subjective. A simple experiment which can be made by anyone shows this fact.
Imagine you have a vessel filled with water of a definite temperature, $$t$$; on the right of it you have another vessel filled with water of a temperature $$t-t_1$$, that is of a temperature distinctly lower than the temperature in the first vessel. In addition, you have a vessel filled with water at a temperature $$t+t_1$$. When now, you hold your fingers in the two outer vessels you will note by your sensations the heat conditions in these vessels. You can then plunge your fingers which have been in the outer vessels into the central vessel and you will see that to the finger which has been in the cold water the water in the central vessel will feel warm, while to the finger which has been in the warm water, the water in the central vessel will feel cold. The same temperature therefore is experienced differently according to the temperature to which one has previously been exposed. Everyone knows that when he goes into a cellar, it may feel different in winter from the way it feels in summer. Even though the thermometer stands at the same point circumstances may be such that the cellar feels warm in the winter and cool in the summer. Indeed, the subjective experience of heat is not uniform and it is necessary to set an objective standard by which to measure the heat condition of any object or location. Now, I need not here go into the elementary phenomena or take up the elementary instruments for measuring heat. It must be assumed that you are acquainted with them. I will simply say that when the temperature condition is measured with a thermometer, there is a feeling that since we measure the degree above or below zero, we are getting an objective temperature measurement. In our thinking we consider that there is a fundamental difference between this objective determination in which we have no part and the subjective determination, where our own organization enters into the experience.
For all that the 19th century has striven to attain it may be said that this view on the matter was, from a certain point of view, fruitful and justified by its results. Now, however, we are in a time when people must pay attention to certain other things if they are to advance their way of thinking and their way of life. From science itself must come certain questions simply overlooked in such conclusions as those I have given. One question is this: Is there a difference, a real objective difference, between the determination of temperature by my organism and by a thermometer, or do I deceive myself for the sake of getting useful practical results when I bring such a difference into my ideas and concepts? This whole course will be designed to show why today such questions must be asked. From the principal questions it will be my object to proceed to those important considerations which have been overlooked owing to exclusive attention to the practical life. How they have been lost for us on account of the attention to technology you will see. I would like to impress you with the fact that we have completely lost our feeling for the real being of heat under the influence of certain ideas to be described presently. And, along with this loss, has gone the possibility of bringing this being of heat into relation with the human organism itself, a relation which must be all means be established in certain aspects of our life. To indicate to you in a merely preliminary way the bearing of these things on the human organism, I may call your attention to the fact that in many cases we are obliged today to measure the temperature of this organism, as for instance, when it is in a feverish condition. This will show you that the relation of the unknown being of heat to the human organism has considerable importance. Those extreme conditions as met with in chemical and technical processes will be dealt with subsequently. A proper attitude toward the relation of the unknown being of heat to the human organism has considerable importance. Those extreme conditions as met with in chemical and technical processes will be dealt with subsequently. A proper attitude toward the relation of the heat-being to the human organism cannot, however, be attained on the basis of a mechanical view of heat. The reason is, that in so doing, one neglects the fact that the various organs are quite different in their sensitiveness to this heat-being, that the heart, the liver, the lungs differ greatly in their capacity to react to the being of heat. Through the purely physical view of heat no foundation is laid for the real study of certain symptoms of disease, since the varying capacity to react to heat of the several organs of the body escapes attention. Today we are in no position to apply to the organic world the physical views built up in the course of the 19th century on the nature of heat. This is obvious to anyone who has an eye to see the harm done by modern physical research, so-called, in dealing with what might be designated the higher branches of knowledge of the living being. Certain questions must be asked, questions that call above everything for clear, lucid ideas. In the so-called “exact science,” nothing has done more harm than the introduction of confused ideas.
What then does it really mean when I say, if I put my fingers in the right and left hand vessels and then into a vessel with a liquid of an intermediate temperature, I get different sensations? Is there really something in the conceptual realm that is different from the so-called objective determination with the thermometer? Consider now, suppose you put thermometers in these two vessels in place of your fingers. You will then get different readings depending on whether you observe the thermometer in the one vessel or the other. If then you place the two thermometers instead of your fingers into the middle vessel, the mercury will act differently on the two. In the one it will rise; in the other it will fall. You see the thermometer does not behave differently from your sensations. For the setting up of a view of the phenomenon, there is no distinction between the two thermometers and the sensation from your finger. In both cases exactly the same thing occurs, namely a difference is shown from the immediately preceding conditions. And the thing our sensation depends on is that we do not within ourselves have any zero or reference point. If we had such a reference point then we would establish not merely the immediate sensation but would have apparatus to relate the temperature subjectively perceived, to such a reference point. We would then attach to the phenomenon just as we do with the thermometers something which really is not inherent in it, namely the variation from the reference point. You see, for the construction of our concept of the process there is no difference.
It is such questions as these that must be raised today if we are to clarify our ideas, or all the present ideas on these things are really confused. Do not imagine for a moment that this is of no consequence. Our whole life process is bound up with this fact that we have in us no temperature reference point. If we could establish such a reference point within ourselves, it would necessitate an entirely different state of consciousness, a different soul life. It is precisely because the reference point is hidden for us that we lead the kind of life we do.
You see, many things in life, in human life and in the animal organism, too, depend on the fact that we do not perceive certain processes. Think what you would have to do if you were obliged to experience subjectively everything that goes on in your organism. Suppose you had to be aware of all the details of the digestive process. A great deal pertaining to our condition of life rests on this fact that we do not bring into our consciousness certain things that take place in our organism. Among these things is that we do not carry within us a temperature reference point—we are not thermometers. A subjective-objective distinction such as is usually made is not therefore adequate for a comprehensive grasp of the physical.
It is this which has been the uncertain point in human thinking since the time of ancient Greeks. It had to be so, but it cannot remain so in the future. For the old Grecian philosophers, Zeno in particular, had already orientated human thinking about certain processes in a manner strikingly opposed to outer reality. I must call your attention to these things even at the risk of seeming pedantic. Let me recall to you the problem of Achilles and the tortoise, a problem I have often spoken about.
Let us assume we have the distance traveled by Achilles in a certain time $$a$$. This represents the rate at which he can travel. And here we have the tortoise $$s$$, who has a start on Achilles. Let us take the moment when Achilles gets to the point marked $$1$$. The tortoise is ahead of him. Since the problem stated that Achilles has to cover every point covered by the tortoise, the tortoise will always be a little ahead and Achilles can never catch up. But, the way people would consider it is this. You would say, yes, I understand the problem all right, but Achilles would soon catch the tortoise. The whole thing is absurd. But if we reason that Achilles must cover the same path as the tortoise and the tortoise is ahead, he will never catch the tortoise. Although people would say this is absurd, nevertheless the conclusion is absolutely necessary and nothing can be urged against it. It is not foolish to come to this conclusion but on the other hand, it is remarkably clever considering only the logic of the matter. It is a necessary conclusion and cannot be avoided. Now what does all this depend on? It depends on this: that as long as you think, you cannot think otherwise than the premise requires. As a matter of fact, you do not depend on thinking strictly, but instead you look at the reality and you realize that it is obvious that Achilles will soon catch the tortoise. And in doing this you uproot thinking by means of reality and abandon the pure thought process. There is no point in admitting the premises and then saying, “Anyone who thinks this way is stupid.” Through thinking alone we can get nothing out of the proposition but that Achilles will never catch the tortoise. And why not? Because when we apply our thinking absolutely to reality, then our conclusions are not in accord with the facts. They cannot be. When we turn our rationalistic thought on reality it does not help us at all that we establish so-called truths which turn out not to be true. For we must conclude if Achilles follows the tortoise that he passes through each point that the tortoise passes through. Ideally this is so; in reality he does nothing of the kind. His stride is greater than that of the tortoise. He does not pass through each point of the path of the tortoise. We must, therefore, consider what Achilles really does, and not simply limit ourselves to mere thinking. Then we come to a different result. People do not bother their heads about these things but in reality they are extraordinarily important. Today especially, in our present scientific development, they are extremely important. For only when we understand that much of our thinking misses the phenomena of nature if we go from observation to so-called explanation, only in this case will we get the proper attitude toward these things.
The observable, however, is something which only needs to be described. That I can do the following for instance, calls simply for a description: here I have a ball which will pass through this opening. We will now warm the ball slightly. Now you see it does not go through. It will only go through when it has cooled sufficiently. As soon as I cool it by pouring this cold water on it, the ball goes through again. This is the observation, and it is this observation that I need only describe. Let us suppose, however, that I begin to theorize. I will do so in a sketchy way with the object merely of introducing the matter. Here is the ball; it consists of a certain number of small parts—molecules, atoms, if you like. This is not observation, but something added to observation in theory. At this moment, I have left the observed and in doing so I assume an extremely tragic role. Only those who are in a position to have insight into these things can realize this tragedy. For you see, if you investigate whether Achilles can catch the tortoise, you may indeed begin by thinking “Achilles must pass over every point covered by the tortoise and can never catch it.” This may be strictly demonstrated. Then you can make an experiment. You place the tortoise ahead and Achilles or some other who does not run even so fast as Achilles, in the rear. And at any time you can show that observation furnishes the opposite of what you conclude from reasoning. The tortoise is soon caught.
When, however, you theorize about the sphere, as to how its atoms and molecules are arranged, and when you abandon the possibility of observation, you cannot in such a case look into the matter and investigate it—you can only theorize. And in this realm you will do no better than you did when you applied your thinking to the course of Achilles. That is to say, you carry the whole incompleteness of your logic into your thinking about something which cannot be made the object of observation. This is the tragedy. We build explanation upon explanation while at the same time we abandon observation, and think we have explained things simply because we have erected hypotheses and theories. And the consequence of this course of forced reliance on our mere thinking is that this same thinking fails us the moment we are able to observe. It no longer agrees with the observation.
You will remember I already pointed out this distinction in the previous course when I indicated the boundary between kinematics and mechanics. Kinematics describes mere motion phenomena or phenomena as expressed by equations, but it is restricted to verifying the data of observation.
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# Find a non subgroup out of subgroups cosets
• Sep 13th 2008, 06:02 AM
Find a non subgroup out of subgroups cosets
Find a group G with subgroups A and B such that AB is not a subgroup.
I think I may be overthinking this, let G be the set of integers, then let A be {1} and B = {0}, then both A and B are subgroup, but there would be no inverse for 0 in the set AB.
Is this right?
Thank you.
• Sep 13th 2008, 11:09 AM
NonCommAlg
Quote:
Find a group G with subgroups A and B such that AB is not a subgroup.
I think I may be overthinking this, let G be the set of integers, then let A be {1} and B = {0}, then both A and B are subgroup, but there would be no inverse for 0 in the set AB.
Is this right? no! the set of integers is an additive group and so {1} is not a subgroup.
you know that if one of A and B is normal, then AB will be a subgroup. so the group G that you're looking for has to be non-abelian. there are many examples. here are two of them:
Example 1 G finite: let $G=S_3, \ A=\{(1),(1,2)\}, \ B=\{(1),(1,3)\}.$ then $|AB|=4,$ which doesn't divide $|S_3|=6.$ so AB is not a subgroup.
Example 2 G infinite: let $G=GL(2,\mathbb{Q}).$ let $A=\{\begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}: \ n \in \mathbb{Z} \}, \ \ B=\{\begin{pmatrix} 1 & 0 \\ n & 1 \end{pmatrix}: \ n \in \mathbb{Z} \}.$ then $AB=\{\begin{pmatrix} nm+1 & n \\ m & 1 \end{pmatrix}: \ n,m \in \mathbb{Z} \}.$ now $x=\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \in AB, \ \ y=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \in AB,$
but $xy=\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} \notin AB.$ so AB is not a subgroup.
• Sep 13th 2008, 04:47 PM
ThePerfectHacker
If you are looking for an infinite case counterexample it is easier just to consider $\mathbb{Z}\times S_3$.
Now choose $A = \mathbb{Z}\times \left< (12) \right>$ and $B = \mathbb{Z}\times \left< (13) \right>$.
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# ME345_Lecture_15 - frequency inferred from the acquired...
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M E 345 Spring 2008 Professor John M. Cimbala Lecture 15 02/18/2008 Today, we will : Review the pdf module: Digital Data Acquisition Do some example problems – digital data acquisition Example Given : The integer 78 (base 10). To do : Write this integer in 8-bit binary format. Solution :
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Example Given : A voltage signal ranges from -1.2 to 1.4 volts. A digital data acquisition system is to be chosen – four choices are available: (a) 12-bit A/D range = -10 to 10 V (b) 8-bit A/D range = -2 to 2 V (c) 10-bit A/D range = -5 to 5 V (d) 14-bit A/D range = -1 to 1 V To do : Determine which system is the best choice for this application, assuming that cost is irrelevant (all four are available in the lab). Solution :
Example Given : Consider the following scenarios. To do : In each case, calculate the perceived
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Unformatted text preview: frequency inferred from the acquired digital data. ( a ) The actual signal is f = 600 Hz. The data are sampled at f s = 2000 Hz. Solution : ( b ) The actual signal is f = 600 Hz. The data are sampled at f s = 900 Hz. Solution : ( c ) The actual signal is f = 600 Hz. The data are sampled at f s = 600 Hz. Solution : ( d ) The actual signal is f = 600 Hz. The data are sampled at f s = 500 Hz. Solution : Example Given : A voltage signal contains a sine wave at 600 Hz. We sample the data digitally at a sampling frequency of 320 Hz. To do : What is the perceived frequency obtained from the digital data acquisition system? Solution : 0 0.2 0.4 0.6 0.8 1.0 f a / f folding f / f folding 2.0 2.2 2.4 2.6 2.8 3.0 4.0 1.2 1.4 1.6 1.8 5.0 4.4 4.6 4.2 4.8 3.4 3.6 3.2 3.8 f / f folding...
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# physics
posted by .
A 1100 kg car rolling on a horizontal surface has speed v = 70 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring stiffness constant of the spring?
• physics -
The car comes to rest (temporarily)at compression distance X, at which point the initial kiteci energy (1/2) M V^2 is converted to spring potential energt, (1/2)kX^2.
Therefore
kX^2 = mV^2
k = m (V/X)^2
Make sure you convert V to meters per second before using the formula. k will be in Newtons/meter
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A 1000 kg car rolling on a horizontal surface has speed v = 55 km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 2.2 m. What is the spring stiffness constant of the spring?
10. ### physics
A 1100-kg car moving on a horizontal surface has speed v = 60km/h when it strikes a horizontal coiled spring and is brought to rest in a distance of 3.1m .What is the spring stiffness constant of the spring?
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https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/567/2/f/g/
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# Properties
Label 567.2.f.g Level $567$ Weight $2$ Character orbit 567.f Analytic conductor $4.528$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$
# Related objects
Show commands: Magma / PariGP / SageMath
## Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [567,2,Mod(190,567)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(567, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("567.190");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.f (of order $$3$$, degree $$2$$, not minimal)
## Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$
## $q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.
$$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + 3 q^{8}+O(q^{10})$$ q + (-z + 1) * q^2 + z * q^4 + 2*z * q^5 + (-z + 1) * q^7 + 3 * q^8 $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + 3 q^{8} + 2 q^{10} + (4 \zeta_{6} - 4) q^{11} + 2 \zeta_{6} q^{13} - \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} - 6 q^{17} + 4 q^{19} + (2 \zeta_{6} - 2) q^{20} + 4 \zeta_{6} q^{22} + ( - \zeta_{6} + 1) q^{25} + 2 q^{26} + q^{28} + ( - 2 \zeta_{6} + 2) q^{29} + 5 \zeta_{6} q^{32} + (6 \zeta_{6} - 6) q^{34} + 2 q^{35} + 6 q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + 6 \zeta_{6} q^{40} - 2 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 4 q^{44} - \zeta_{6} q^{49} - \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} + 6 q^{53} - 8 q^{55} + ( - 3 \zeta_{6} + 3) q^{56} - 2 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + 7 q^{64} + (4 \zeta_{6} - 4) q^{65} - 4 \zeta_{6} q^{67} - 6 \zeta_{6} q^{68} + ( - 2 \zeta_{6} + 2) q^{70} - 6 q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + 4 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + ( - 16 \zeta_{6} + 16) q^{79} + 2 q^{80} - 2 q^{82} + ( - 12 \zeta_{6} + 12) q^{83} - 12 \zeta_{6} q^{85} - 4 \zeta_{6} q^{86} + (12 \zeta_{6} - 12) q^{88} - 14 q^{89} + 2 q^{91} + 8 \zeta_{6} q^{95} + (18 \zeta_{6} - 18) q^{97} - q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 + z * q^4 + 2*z * q^5 + (-z + 1) * q^7 + 3 * q^8 + 2 * q^10 + (4*z - 4) * q^11 + 2*z * q^13 - z * q^14 + (-z + 1) * q^16 - 6 * q^17 + 4 * q^19 + (2*z - 2) * q^20 + 4*z * q^22 + (-z + 1) * q^25 + 2 * q^26 + q^28 + (-2*z + 2) * q^29 + 5*z * q^32 + (6*z - 6) * q^34 + 2 * q^35 + 6 * q^37 + (-4*z + 4) * q^38 + 6*z * q^40 - 2*z * q^41 + (-4*z + 4) * q^43 - 4 * q^44 - z * q^49 - z * q^50 + (2*z - 2) * q^52 + 6 * q^53 - 8 * q^55 + (-3*z + 3) * q^56 - 2*z * q^58 - 12*z * q^59 + (-2*z + 2) * q^61 + 7 * q^64 + (4*z - 4) * q^65 - 4*z * q^67 - 6*z * q^68 + (-2*z + 2) * q^70 - 6 * q^73 + (-6*z + 6) * q^74 + 4*z * q^76 + 4*z * q^77 + (-16*z + 16) * q^79 + 2 * q^80 - 2 * q^82 + (-12*z + 12) * q^83 - 12*z * q^85 - 4*z * q^86 + (12*z - 12) * q^88 - 14 * q^89 + 2 * q^91 + 8*z * q^95 + (18*z - 18) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8 $$2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8} + 4 q^{10} - 4 q^{11} + 2 q^{13} - q^{14} + q^{16} - 12 q^{17} + 8 q^{19} - 2 q^{20} + 4 q^{22} + q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{29} + 5 q^{32} - 6 q^{34} + 4 q^{35} + 12 q^{37} + 4 q^{38} + 6 q^{40} - 2 q^{41} + 4 q^{43} - 8 q^{44} - q^{49} - q^{50} - 2 q^{52} + 12 q^{53} - 16 q^{55} + 3 q^{56} - 2 q^{58} - 12 q^{59} + 2 q^{61} + 14 q^{64} - 4 q^{65} - 4 q^{67} - 6 q^{68} + 2 q^{70} - 12 q^{73} + 6 q^{74} + 4 q^{76} + 4 q^{77} + 16 q^{79} + 4 q^{80} - 4 q^{82} + 12 q^{83} - 12 q^{85} - 4 q^{86} - 12 q^{88} - 28 q^{89} + 4 q^{91} + 8 q^{95} - 18 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8 + 4 * q^10 - 4 * q^11 + 2 * q^13 - q^14 + q^16 - 12 * q^17 + 8 * q^19 - 2 * q^20 + 4 * q^22 + q^25 + 4 * q^26 + 2 * q^28 + 2 * q^29 + 5 * q^32 - 6 * q^34 + 4 * q^35 + 12 * q^37 + 4 * q^38 + 6 * q^40 - 2 * q^41 + 4 * q^43 - 8 * q^44 - q^49 - q^50 - 2 * q^52 + 12 * q^53 - 16 * q^55 + 3 * q^56 - 2 * q^58 - 12 * q^59 + 2 * q^61 + 14 * q^64 - 4 * q^65 - 4 * q^67 - 6 * q^68 + 2 * q^70 - 12 * q^73 + 6 * q^74 + 4 * q^76 + 4 * q^77 + 16 * q^79 + 4 * q^80 - 4 * q^82 + 12 * q^83 - 12 * q^85 - 4 * q^86 - 12 * q^88 - 28 * q^89 + 4 * q^91 + 8 * q^95 - 18 * q^97 - 2 * q^98
## Character values
We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/567\mathbb{Z}\right)^\times$$.
$$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$
## Embeddings
For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
190.1
0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 1.73205i 0 0.500000 + 0.866025i 3.00000 0 2.00000
379.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 + 1.73205i 0 0.500000 0.866025i 3.00000 0 2.00000
$$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles
## Inner twists
Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
## Twists
By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.f.g 2
3.b odd 2 1 567.2.f.b 2
9.c even 3 1 21.2.a.a 1
9.c even 3 1 inner 567.2.f.g 2
9.d odd 6 1 63.2.a.a 1
9.d odd 6 1 567.2.f.b 2
36.f odd 6 1 336.2.a.a 1
36.h even 6 1 1008.2.a.l 1
45.h odd 6 1 1575.2.a.c 1
45.j even 6 1 525.2.a.d 1
45.k odd 12 2 525.2.d.a 2
45.l even 12 2 1575.2.d.a 2
63.g even 3 1 147.2.e.b 2
63.h even 3 1 147.2.e.b 2
63.i even 6 1 441.2.e.b 2
63.j odd 6 1 441.2.e.a 2
63.k odd 6 1 147.2.e.c 2
63.l odd 6 1 147.2.a.a 1
63.n odd 6 1 441.2.e.a 2
63.o even 6 1 441.2.a.f 1
63.s even 6 1 441.2.e.b 2
63.t odd 6 1 147.2.e.c 2
72.j odd 6 1 4032.2.a.h 1
72.l even 6 1 4032.2.a.k 1
72.n even 6 1 1344.2.a.g 1
72.p odd 6 1 1344.2.a.s 1
99.g even 6 1 7623.2.a.g 1
99.h odd 6 1 2541.2.a.j 1
117.t even 6 1 3549.2.a.c 1
144.v odd 12 2 5376.2.c.l 2
144.x even 12 2 5376.2.c.r 2
153.h even 6 1 6069.2.a.b 1
171.o odd 6 1 7581.2.a.d 1
180.p odd 6 1 8400.2.a.bn 1
252.n even 6 1 2352.2.q.e 2
252.s odd 6 1 7056.2.a.p 1
252.u odd 6 1 2352.2.q.x 2
252.bi even 6 1 2352.2.a.v 1
252.bj even 6 1 2352.2.q.e 2
252.bl odd 6 1 2352.2.q.x 2
315.bg odd 6 1 3675.2.a.n 1
504.be even 6 1 9408.2.a.m 1
504.bn odd 6 1 9408.2.a.bv 1
By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 9.c even 3 1
63.2.a.a 1 9.d odd 6 1
147.2.a.a 1 63.l odd 6 1
147.2.e.b 2 63.g even 3 1
147.2.e.b 2 63.h even 3 1
147.2.e.c 2 63.k odd 6 1
147.2.e.c 2 63.t odd 6 1
336.2.a.a 1 36.f odd 6 1
441.2.a.f 1 63.o even 6 1
441.2.e.a 2 63.j odd 6 1
441.2.e.a 2 63.n odd 6 1
441.2.e.b 2 63.i even 6 1
441.2.e.b 2 63.s even 6 1
525.2.a.d 1 45.j even 6 1
525.2.d.a 2 45.k odd 12 2
567.2.f.b 2 3.b odd 2 1
567.2.f.b 2 9.d odd 6 1
567.2.f.g 2 1.a even 1 1 trivial
567.2.f.g 2 9.c even 3 1 inner
1008.2.a.l 1 36.h even 6 1
1344.2.a.g 1 72.n even 6 1
1344.2.a.s 1 72.p odd 6 1
1575.2.a.c 1 45.h odd 6 1
1575.2.d.a 2 45.l even 12 2
2352.2.a.v 1 252.bi even 6 1
2352.2.q.e 2 252.n even 6 1
2352.2.q.e 2 252.bj even 6 1
2352.2.q.x 2 252.u odd 6 1
2352.2.q.x 2 252.bl odd 6 1
2541.2.a.j 1 99.h odd 6 1
3549.2.a.c 1 117.t even 6 1
3675.2.a.n 1 315.bg odd 6 1
4032.2.a.h 1 72.j odd 6 1
4032.2.a.k 1 72.l even 6 1
5376.2.c.l 2 144.v odd 12 2
5376.2.c.r 2 144.x even 12 2
6069.2.a.b 1 153.h even 6 1
7056.2.a.p 1 252.s odd 6 1
7581.2.a.d 1 171.o odd 6 1
7623.2.a.g 1 99.g even 6 1
8400.2.a.bn 1 180.p odd 6 1
9408.2.a.m 1 504.be even 6 1
9408.2.a.bv 1 504.bn odd 6 1
## Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(567, [\chi])$$:
$$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4
## Hecke characteristic polynomials
$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 2T + 4$$
$31$ $$T^{2}$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 2T + 4$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2}$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 18T + 324$$
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Cbc 2.9.9
ClpCholeskyDense.hpp
Go to the documentation of this file.
1 /* \$Id: ClpCholeskyDense.hpp 1910 2013-01-27 02:00:13Z stefan \$ */
2 /*
5
7 */
8 #ifndef ClpCholeskyDense_H
9 #define ClpCholeskyDense_H
10
11 #include "ClpCholeskyBase.hpp"
12 class ClpMatrixBase;
13
15
16 public:
21 virtual int order(ClpInterior * model) ;
26 virtual int symbolic();
29 virtual int factorize(const CoinWorkDouble * diagonal, int * rowsDropped) ;
31 virtual void solve (CoinWorkDouble * region) ;
39 int reserveSpace(const ClpCholeskyBase * factor, int numberRows) ;
41 CoinBigIndex space( int numberRows) const;
43 void factorizePart2(int * rowsDropped) ;
45 void factorizePart3(int * rowsDropped) ;
47 void solveF1(longDouble * a, int n, CoinWorkDouble * region);
48 void solveF2(longDouble * a, int n, CoinWorkDouble * region, CoinWorkDouble * region2);
50 void solveB1(longDouble * a, int n, CoinWorkDouble * region);
51 void solveB2(longDouble * a, int n, CoinWorkDouble * region, CoinWorkDouble * region2);
52 int bNumber(const longDouble * array, int &, int&);
54 inline longDouble * aMatrix() const {
55 return sparseFactor_;
56 }
58 inline longDouble * diagonal() const {
59 return diagonal_;
60 }
69 virtual ~ClpCholeskyDense();
75 virtual ClpCholeskyBase * clone() const ;
79 private:
85 };
86
87 /* structure for C */
88 typedef struct {
92 int * rowsDropped;
93 double doubleParameters_[1]; /* corresponds to 10 */
94 int integerParameters_[2]; /* corresponds to 34, nThreads */
95 int n;
98
99 extern "C" {
100 void ClpCholeskySpawn(void *);
101 }
103 void
105 longDouble * a, int n, int numberBlocks,
106 longDouble * diagonal, longDouble * work, int * rowsDropped);
107
109 void
111 longDouble * aTri, int nThis,
112 longDouble * aUnder, longDouble * diagonal,
113 longDouble * work,
114 int nLeft, int iBlock, int jBlock,
115 int numberBlocks);
117 void
119 longDouble * aUnder, int nTri, int nDo,
120 int iBlock, int jBlock, longDouble * aTri,
121 longDouble * diagonal, longDouble * work,
122 int numberBlocks);
127 void
129 longDouble * above, int nUnder, int nUnderK,
130 int nDo, longDouble * aUnder, longDouble *aOther,
131 longDouble * work,
132 int iBlock, int jBlock,
133 int numberBlocks);
135 void
137 longDouble * a, int n,
138 longDouble * diagonal, longDouble * work,
139 int * rowsDropped);
141 void
142 ClpCholeskyCtriRecLeaf(/*ClpCholeskyDenseC * thisStruct,*/
143 longDouble * aTri, longDouble * aUnder,
144 longDouble * diagonal, longDouble * work,
145 int nUnder);
147 void
148 ClpCholeskyCrecTriLeaf(/*ClpCholeskyDenseC * thisStruct, */
149 longDouble * aUnder, longDouble * aTri,
150 /*longDouble * diagonal,*/ longDouble * work, int nUnder);
155 void
156 ClpCholeskyCrecRecLeaf(/*ClpCholeskyDenseC * thisStruct, */
157 const longDouble * COIN_RESTRICT above,
158 const longDouble * COIN_RESTRICT aUnder,
159 longDouble * COIN_RESTRICT aOther,
160 const longDouble * COIN_RESTRICT work,
161 int nUnder);
162 #endif
longDouble * diagonal_
ClpCholeskyDense & operator=(const ClpCholeskyDense &)
Assignment.
Abstract base class for Clp Matrices.
This solves LPs using interior point methods.
Definition: ClpInterior.hpp:72
ClpCholeskyDense()
Default constructor.
char * rowsDropped() const
rowsDropped - which rows are gone
longDouble * sparseFactor_
sparseFactor.
virtual int factorize(const CoinWorkDouble *diagonal, int *rowsDropped)
Factorize - filling in rowsDropped and returning number dropped.
void solveF2(longDouble *a, int n, CoinWorkDouble *region, CoinWorkDouble *region2)
void ClpCholeskyCfactorLeaf(ClpCholeskyDenseC *thisStruct, longDouble *a, int n, longDouble *diagonal, longDouble *work, int *rowsDropped)
Leaf recursive factor.
void solveB2(longDouble *a, int n, CoinWorkDouble *region, CoinWorkDouble *region2)
void ClpCholeskyCrecTriLeaf(longDouble *aUnder, longDouble *aTri, longDouble *work, int nUnder)
Leaf recursive rectangle triangle update.
int reserveSpace(const ClpCholeskyBase *factor, int numberRows)
Reserves space.
longDouble * diagonal() const
Diagonal.
Base class for Clp Cholesky factorization Will do better factorization.
#define COIN_RESTRICT
void solveF1(longDouble *a, int n, CoinWorkDouble *region)
Forward part of solve.
void ClpCholeskyCrecRecLeaf(const longDouble *COIN_RESTRICT above, const longDouble *COIN_RESTRICT aUnder, longDouble *COIN_RESTRICT aOther, const longDouble *COIN_RESTRICT work, int nUnder)
Leaf recursive rectangle rectangle update, nUnder is number of rows in iBlock, nUnderK is number of r...
void ClpCholeskyCrecRec(ClpCholeskyDenseC *thisStruct, longDouble *above, int nUnder, int nUnderK, int nDo, longDouble *aUnder, longDouble *aOther, longDouble *work, int iBlock, int jBlock, int numberBlocks)
Non leaf recursive rectangle rectangle update, nUnder is number of rows in iBlock, nUnderK is number of rows in kBlock.
int bNumber(const longDouble *array, int &, int &)
int numberRows() const
Return number of rows.
void ClpCholeskyCrecTri(ClpCholeskyDenseC *thisStruct, longDouble *aUnder, int nTri, int nDo, int iBlock, int jBlock, longDouble *aTri, longDouble *diagonal, longDouble *work, int numberBlocks)
Non leaf recursive rectangle triangle update.
longDouble * diagonal_
Diagonal.
void factorizePart3(int *rowsDropped)
part 2 of Factorize - filling in rowsDropped - blocked
void ClpCholeskyCtriRecLeaf(longDouble *aTri, longDouble *aUnder, longDouble *diagonal, longDouble *work, int nUnder)
Leaf recursive triangle rectangle update.
virtual ~ClpCholeskyDense()
Destructor.
double longDouble
virtual int symbolic()
Does Symbolic factorization given permutation.
void factorizePart2(int *rowsDropped)
part 2 of Factorize - filling in rowsDropped
int CoinBigIndex
virtual void solve(CoinWorkDouble *region)
Uses factorization to solve.
double CoinWorkDouble
Definition: CoinTypes.hpp:53
longDouble * aMatrix() const
A.
virtual ClpCholeskyBase * clone() const
Clone.
CoinBigIndex space(int numberRows) const
Returns space needed.
void ClpCholeskySpawn(void *)
virtual int order(ClpInterior *model)
Orders rows and saves pointer to matrix.and model.
void solveB1(longDouble *a, int n, CoinWorkDouble *region)
Backward part of solve.
void ClpCholeskyCfactor(ClpCholeskyDenseC *thisStruct, longDouble *a, int n, int numberBlocks, longDouble *diagonal, longDouble *work, int *rowsDropped)
Non leaf recursive factor.
bool borrowSpace_
Just borrowing space.
void ClpCholeskyCtriRec(ClpCholeskyDenseC *thisStruct, longDouble *aTri, int nThis, longDouble *aUnder, longDouble *diagonal, longDouble *work, int nLeft, int iBlock, int jBlock, int numberBlocks)
Non leaf recursive triangle rectangle update.
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https://oeis.org/A205058
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The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
A205058 Number of (n+1)X3 0..2 arrays with rows and columns of permanents of all 2X2 subblocks lexicographically nondecreasing 1
491, 7572, 111413, 1383154, 16383311, 175923930, 1825712439, 18040920199, 174834354143, 1658203154833, 15589489658454, 145407627330028, 1352240865751568, 12547362532350341, 116355015072768773 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS Column 2 of A205063 LINKS R. H. Hardin, Table of n, a(n) for n = 1..210 EXAMPLE Some solutions for n=4 ..2..1..1....2..0..1....1..1..1....2..1..0....0..1..1....2..2..2....2..0..2 ..2..0..2....0..0..2....0..2..1....2..0..2....0..0..2....0..1..2....2..2..0 ..1..2..2....1..0..2....2..2..2....1..1..1....0..1..0....2..2..2....0..2..2 ..1..2..2....0..1..0....0..2..1....1..1..1....1..1..0....0..1..2....2..0..2 ..1..2..2....1..2..2....2..2..2....2..0..2....0..1..0....2..2..2....2..2..2 CROSSREFS Sequence in context: A260925 A217118 A205200 * A229520 A226715 A205819 Adjacent sequences: A205055 A205056 A205057 * A205059 A205060 A205061 KEYWORD nonn AUTHOR R. H. Hardin Jan 21 2012 STATUS approved
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Last modified March 28 21:15 EDT 2023. Contains 361596 sequences. (Running on oeis4.)
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https://www.bystudin.com/dice-thrown-what-is-the-probability-of-events-a-1-point-has-fallen-q-2-points-dropped-out/
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# Dice thrown. What is the probability of events: A- 1 point has fallen; Q- 2 points dropped out?
Decision:
The number of all possible results is n = 6 (all faces).
a) The number of faces on which only 1 point m = 1:
P = 1/6
b) the number of faces on which only 2 points m = 1:
P = 1/6
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https://math.stackexchange.com/questions/845263/trans-invariant-set-infinite-lebesgue-measure
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# trans invariant set infinite lebesgue measure
I was wondering about the following result: We are given a subset of the reals $A$ such that $A + r = A$ for some real number $r$ (in my application I have in mind for each $r$ belonging to a dense subset of the reals). Then assuming $A$ has a positive non-zero Lebesgue measure it follows that $A$ has infinite Lebesgue measure.
My approach: From the invariance we get a disjoint countable cover by sets $A_n := A \cap [(n-1) r, nr)$. But I don't see how to argue that for infinitely many $n$ the sets $A_n$ have positive Lebesgue measure which would clinch the deal. Ideas?
The background why I thought this is true (and should be easy to prove): In general there is a Theorem like this which is more difficult to prove without the assumption that $A$ be measurable in the first place. Namely if $A$ is translation invariant wrt a dense subset then $A$ is either of infinite measure or immeasurable (if I remember correctly).
EDIT: Additional question: What happens if instead of Lebesgue measure we consider a measure on the real numbers which is not necessarily translation invariant?
Assume $A \cap [a,b]$ has positive measure then chose $n$ such that $$((A\cap[a,b])+nr)\cap [a,b]=\emptyset$$ so the sets $(A\cap [a,b])+knr$ are all disjoint of equal positive meaasure and subsets of $A$.
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https://www.svtuition.org/2011/01/job-costing-problem-of-accountant-of.html
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222
## Balance Sheet \$type=three\$count=6\$author=hide\$comment=hide\$label=hide\$date=hide\$show=home\$s=0
I work for a demo/roll off company. The books were a mess before I came in. I am trying to figure out a way to assign over head costs to all jobs. The company is commercial roll off/demo company. The jobs normally only last a week. I have been trying allocate the over head by a per minute basis, because of the different types of work that we do. For example we do a switch out which takes a 120 minutes, equipment move takes 180 minutes, a final pull takes 96 minutes, and there are couple of other activities the company does. I basically took all the tickets for one month then multiplied them by how many minutes each activity would take to come up with a total minutes. I then took the P/L and divided each account by 12 to get the cost for one month. I divided that number the total number of minutes to come up with a per minute cost of over head. I only used the accounts that would be considered overhead accounts. The per minute number I am coming up with is too high. What would be a better way to figure this out since they were not job costing.
Solution :
Basically job costing is used in that industry who all work is done on job basis. Suppose, we got the order of making 10 chairs and at that time, we can calculate total job cost and each unit cost in that job by dividing the number of units in that job. In your case, I think, you got specific jobs order for this. So, need not any other method for calculating overhead cost per minute number. You did all right. I can more suggestion by knowing exact per minute number.
Name
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# Sets, Vectors & Functions - PowerPoint PPT Presentation
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Sets, Vectors & Functions. IGCSE Chapter 8. Note 1 : Sets. A ∩ B is green. A. B. ∩ - intersection U – Union - ‘is a subset of’. A. B. ∩. B. A. ∩. A B. Note 1 : Sets. X. a. C - ‘is a member of’ ‘belongs to’ b є X ξ – ‘universal set’ ξ = { a,b,c,d,e }
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## Sets, Vectors & Functions
IGCSE Chapter 8
### Note 1: Sets
A ∩ B is green
A
B
∩ - intersection
U – Union
- ‘is a subset of’
A
B
B
A
A B
### Note 1: Sets
X
a
C - ‘is a member of’
‘belongs to’
bєX
ξ – ‘universal set’
ξ = {a,b,c,d,e}
- ‘complement of’
b
c
ξ
d
a
c
b
e
ξ
B
A ‘
A
A U A’ = ξ
### Note 1: Sets
A
a
n(A) - ’the number of elements
in set A’
A = {x : x is an integer, 2 ≤ x ≤ 9}
Reads: A is the set of elements x such that x is an integer and 2 ≤ x ≤ 9
b
n(A) = 3
c
The set A is {2,3,4,5,6,7,8,9}
Ø or { } - ‘empty set’
Ø A for any set A
### Note 1: Sets
ξ
T
R
e.g. In the venn diagram
8
5
11
ξ = {students in year 10}
9
13
15
R = {Yr 10 students who play Rugby}
12
17
V = {Yr 10 students who play Volleyball}
V
T = {Yr 10 students who play Tennis}
a.) How many students play Rugby?
b.) How many students do not play Volleyball?
c.) How many students play Rugby and Tennis?
d.) How many students in total?
e.) How many play only 1 of these 3 sports?
40
41
14
90
31
IGCSE
Ex 1 Pg 242-243
e.g.
If X = {1,2,3,..10}
ξ
Y = {1,3,5,…19}
Y
X
13
1
4
Z = {x : x is an integer, 5≤x≤11}
15
3
17
Find:
2
5
7
19
10
{1,3,5,7,9}
9
a.) X ∩ Y
b.) Y ∩ Z
c.) X ∩ Z
d.) n(X U Y)
e.) n(Z)
f.) X’ U Y’
11
6
8
{5,7,9,11}
{5,6,7,8,9,10}
Z
15
7
Ø or { }
State whether true or false:
False
a.) 7 εX ∩ Y
b.) {5,7,9,11} Z
c.) Z X U Y
True
IGCSE
Ex 2 Pg 244
True
e.g.
Draw and shade this diagram to show the following sets:
a.) X∩ Y
b.) (X U Y)’
c.) X’ ∩ Y
ξ
X
Y
IGCSE
Ex 3 Pg 245-246
e.g.
Logical Problems
If A = { sheep }
B = { sheep dogs }
C = { ‘intelligent animals’ }
D = { good pet }
Express the following in set language:
a.) No sheep are ‘intelligent’ animals
b.) All sheep dogs make good pets
c.) Some sheep make good pets
A ∩ C = Ø
B D
A ∩ D = Ø
Interpret the following statements
a.) B C
b.) B U C = D
c.) AεD
All sheep dogs are intelligent animals
All sheep dogs and all intelligent animals make good pets
Sheep do not make good pets
e.g.
Logical Problems
Of 27 students in the class, 18 play chess, 15 play piano and 7 do both. How many do neither?
C
P
27 students = 11 + 7 + 8 + X
7
11
8
27 – 11 – 7 – 8 = X
X = 1
1
There is only 1 student who does not play either piano or chess
e.g.
Logical Problems
The math results from the Hockey Team show that all 16 players passed at least 2 subjects, 8 passed at least 3 subjects and 5 students passed 4 subjects or more
ξ
How many passed exactly 2 subjects?
What fraction passed exactly 3 subjects?
3
2
3
8
8
5
4
3/16
IGCSE
Ex 4 Pg 247-249
### Note 2: Vectors
A vector is a quantity that has both magnitude and direction
Vectors can be added using scale drawings.
a
a
b
a + b
Therefore:
a + b = b + a
b
* Notice that this is the same result as b + a
### Note 2: Vectors
A scalar is a quantity that has magnitude but no direction.
e.g. ordinary numbers, & quantities like temperature, mass & volume.
We can multiply a vector by a scalar.
e.g.x multiplied by 2 gives 2x
x
x
x
x
x
x
-3x
2x
e.g.x multiplied by -3 gives -3x
* The negative sign reverses the direction of the vector
### Note 2: Vectors
The result of a – b is the same as a + -b
e.g.Find 3a – b
a
a
a
a
a
a
a
a
b
b
b
b
3a – b
e.g.Find -4a + 2b
-4a + 2b
### Note 2: Vectors
d
Starting from Eeach time, find vectors for the following:
c
e.g. Find:
2c
3c + d
-c + d
-d
c – 4d
-2c – 4d
EG
EA
EC
EF
EL
EK
C
A
B
E
F
D
G
J
H
I
M
L
K
### Note 2: Vectors
d
The result of a – b is the same as a + -b
c
e.g. Find:
DE
DG
HJ
GB
BE
GC
CG
= 2c
C
A
B
= c – 2d
F
E
D
= c– 2d
= 2c + 3d
= -c – d
G
H
= 4c + 3d
= -4c – 3d
J
I
### Note 2: Vectors
Write each vector in terms of a, b and c
e.g.
• AB
DE
DC
HD
FE
BE
EA
DG
A
C
2a
a
= -2a
a
B
D
= -a-c
b
c
c
= -4a
F
= 2a+c
b
= b +a
a
a
G
H
E
= -2b
( c )
= -2b + 2a
= 2b – 2a
### Note 2: Vectors
Write each vector in terms of a, bonly
e.g.
DE
HD
A
C
2a
a
a
B
D
= -3a+ 2b
b
c
c
= 4a– 2b
F
b
a
a
G
H
E
IGCSE
Ex 5 Pg 251-252
### Starter
OA = a and OB = b
e.g. Using the figure, express each of the following vectors in terms of a and/or b
AP = OA
OB = BQ
NP = QN
= a
a.) AP
b.) AB
c.) OQ
d.) PO
e.) PQ
f.) PN
g.) ON
h.) AN
i.) BP
j.) QA
= -a + b
= 2b
= -2a
= -2a + 2b
= ½PQ
= -a + b
Q
= 2a + PN
= a + b
b
= a + PN
= b
B
N
b
= -b + 2a
A
P
O
= -2b + a
a
a
### Note 2: Vectors
OA = a and OB = b
e.g. Using the figure, express each of the following vectors in terms of a and/or b
AP = 3OA
OB = 1/2BQ
NP = QN
= 3a
a.) AP
b.) AB
c.) OQ
d.) PO
e.) PQ
f.) PN
g.) ON
h.) AN
i.) BP
j.) QA
= -a + b
= 3b
= -4a
= -4a + 3b
= ½PQ
= -2a + 3/2b
Q
= 4a + PN
= 2a + 3/2b
b
= 3a + PN
= a + 3/2b
N
b
B
= -b + 4a
b
A
P
O
= -3b + a
a
a
a
a
### Note 2: Vectors
ABCDEF is a regular hexagon with AB representing the vector s and AF representing the vector t. Find the vector representing AC
t
s
BC
= ?
= s + t
C
B
s
= AB + BC
AC
A
D
t
= s + s + t
E
IGCSE
Ex 6 Pg 253-255
F
= 2s + t
### Note 3: Column Vectors
The vector AB can be written as a column vector
( )
2
Horizontal Component (Movement in x direction)
AB =
3
Vertical Component (Movement in y direction)
F
C
B
( )
( )
( )
-2
5
0
-3
3
0
E
A
H
G
D
### Note 3: Column Vectors
We can easily add column vectors.
AB + CD =
AB =
CD =
B
= ( )
( )
( )
( )
( ) +
3
7
3
7
10
C
-3
-3
2
2
-1
A
AB + CD
D
* A vector is described by its length and direction, NOT its position
### Note 3: Column Vectors
Similarly, subtracting column vectors.
AB − CD =
AB =
CD =
C
= ( )
( )
( )
( )
( ) −
6
5
6
5
1
-2
-2
1
1
3
AB – CD
B
D
A
* A vector is described by its length and direction, NOT its position
### Note 3: Column Vectors
Multiplying by a scalar
2AB =
Each component is multiplied by 2
AB =
= ( )
2( )
( )
6
12
6
1
2
1
B
2AB
A
* A vector is described by its length and direction, NOT its position
### Note 3: Column Vectors
Vectors are parallel if they have the same direction
i.e. Both components must be in the same ratio
e.g. Is parallel to
IGCSE
Ex 7 Pg 257-258
e.g. Is parallel to
( )
( )
k( )
( )
( )
( )
-10
12
5
a
6
a
4
2
1
b
-2
b
In general, a vector is parallel if it is a scalar multiple
Is parallel to
### Note 3: Column Vectors
y
If A has the coordinate (1,2) and B has the coordinate (6,4), find the column vector for AB
B
AB =
A
( )
( )
5
-5
2
-2
Find the column vector for BA
x
BA =
### Note 3: Column Vectors
y
Given the following diagram, what would be the coordinate of point A such that ABCD is a parallelogram.
C
D
A = (1,0)
( )
( )
1
4
1
3
B
Notice the column vector for
AB = DC
x
A
### Note 3: Column Vectors
y
y = - x
y = x
Find the image of the vector after reflection in the following lines
a.) y = 0
b.) x = 0
c.) y = x
d.) y = -x
x
( )
( )
( )
( )
( )
-5
3
5
-5
5
5
-3
-3
3
3
IGCSE
Ex 8 Pg 258-259
### Note 4: Modulus of a Vector
The modulus of a vector a is written a .
This represents the magnitude (or length) of the vector
As shown in the diagram:
B
a =
( )
5
3
3
We can find the length using Pythagoras’ Theorem
A
5
|a|= √(52+32)
|a|= √34 units
### Note 4: Modulus of a Vector
e.g. If AB = & BC = , find AC
AB + BC =
C
B
=
( )
( )
( ) +
( )
( )
1
-4
5
-4
5
3
3
1
1
4
We can find the length using Pythagoras’ Theorem
A
IGCSE
Ex 9 Pg 260-261
|AC|= √(12+42)
|AC|= √17units
### Note 5: Vector Geometry
Position vectors give the position of a point relative to the origin, O.
In the diagram we see that
OD = 2OA, OE = 4OB, OA = a and OB = b
b.) Express BA in terms of a and b
a.) Express OD and OE in term of a and b respectively
d.) Given that BC = 3BA, express OC in terms of a and b
c.) Express ED in terms of a and b
C
OD = 2a
OE = 4b
BA = -b + a
ED = -4b + 2a
OC = OB + BC
D
A
a
b
O
E
B
OC = b + -3b +3a
OC = -2b +3a
### Note 5: Vector Geometry
Position vectors give the position of a point relative to the origin, O.
In the diagram we see that
OD = 2OA, OE = 4OB, OA = a and OB = b
IGCSE
Ex 10 Pg 262-264
e.) Express EC in terms of a and b
C
EC = -6b + 3a
EC = -4b – 2b + 3a
EC = EO + OC
EC = -4b + OC
ED = -4b + 2a
From part (d)
D
A
a
e.) Show that E, D and C lie on a straight line
b
O
E
B
From part (c)
OC = -2b +3a
EC and ED are parallel vector that pass through the same point, E
### Starter
Note 6: Functions
Function Notation
f(x) = x2+ 3 orf : xx2 +3
We can read this as “the function f, such that, x is mapped on x2 +3
If f(x) = 7x – 5 and g(x) = 2x2 +5, find:
a.) f(2) b.) f(-4) c.) g(3) d.) g(0)
f(2) = 7(2) - 5
f(-4) = 7(-4) - 5
g(3) = 2(3)2 + 5
g(0) = 0 + 5
f(-4) = -33
g(2) = 23
g(2) = 5
f(2) = 9
Note 6: Functions
If f : x and g : x √[5(x+2)], find:
a.) x if f(x) = 40 b.) x if g(x) = 10
5x2
2
√[5(x+2)] = 10
= 40
[5(x+2)] = 102
5x2 = 2 x 40
5(x+2) = 100
5x2 = 80
(x+2) = 20
x2 = 16
x = 18
x = ± 4
Note 6: Functions (flow diagrams)
We can illustrate functions using flow diagrams.
e.g.The function (7x + 8)2 consists of 3 simpler functions
7x
x
7x+8
(7x+8)2
multiply by 7
square
Note 6: Functions
e.g. The functions h and k are defined as follows:
h: x x2 + 1, k:x ax + b, whereaandbare constants.
If h(0) = k(0) and k(2) = 15, find values of aandb
h(0) = 02 + 1
= 1
k(2) = 15
a(2) + 1 = 15
IGCSE
Ex 11 Pg 265-267
k(0) = 1
a(0) + b = 1
2a + 1= 15
2a = 14
a = 7
b = 1
Note 7: Composite Functions
Consider f(x) = x4and g(x) is 2x + 3
fg means that g converts x to 2x + 3, ….. and then……
f converts 2x + 3 to (2x + 3)4
Note 7: Composite Functions
Consider f(x) = x4and g(x) is 2x + 3
fg means that g converts x to 2x + 3, ….. and then……
f converts 2x + 3 to (2x + 3)4
Notice how this is different from gf
gf means that f converts x to x4 , ….. and then……
g converts x4to 2(x4) + 3
You must get used to reading and applying the composite function from right to left
In general, fg=gf
Note 7: Composite Functions
Other common notations show fg as f(g(x)) and gf as g(f(x))
e.g. Given f(x) = 2x + 1 and g(x) = 3 – 4x, find in simplest form:
a.) fgb.) gf
= f(g(x))
= g(f(x))
=2(3 – 4x) + 1
=3 – 4(2x + 1)
=6 – 8x +1
=3 – 8x – 4
=7 – 8x
=– 8x – 1
Note 7: Composite Functions
e.g. Given f:x x3
g:x 4 + 5x
h:x 2x
Find:
a.) fgb.) ghc.)fgh
= g(h(x))
=fg(h(x))
= f(g(x))
=4 + 5(2x)
=(4 + 5x)3
=f(4 + 5(2x))
=4 + 10x
=f(4 + 10x)
d.) gf(2)
d.) ff(-1)
=(4 + 10x)3
= g(f(x))
=(x3)3
=x9
=4 + 5x3
=(-1)9
=4 + 5(2)3
=44
=-1
Note 7: Composite Functions
e.g. Given f:x 2x – 8
g:x x + 4
h:x 7x2
Find:
a.) x if fg= 0b.) ggg(5)c.) x if f(x) = g(x)
2x – 8 = x + 4
g ((x+4) + 4))
f(g(x)) = 0
= g (x+ 8)
x = 4 + 8
2(x + 4) – 8 = 0
x = 12
= (x+4) + 8
2x + 8 – 8 = 0
= x + 12
2x = 0
= 5 + 12
x = 0
IGCSE
Ex 12 Pg 268-269
#1-8
= 17
Note 8: InverseFunctions
Reversing (undoing) a function
The operations: + and –
× and ÷
x2and √x
are all inverse operations because one undoes the other
If a function f maps a number n onto m then its inverse function f-1 will map m onto n.
f-1
### Recall: Column Vectors
y
y = x
Find the image of the vector after reflection in the following lines
a.) y = 0
b.) x = 0
c.) y = x
d.) y = -x
x
( )
( )
( )
( )
( )
5
-5
5
-5
3
3
5
-3
3
-3
Note 8: InverseFunctions
We can find the inverse of a function using a flow diagram
f(x) =
4x+5
x
x
4x
3x
3x-5
Multiply by 3
Subtract 5
Divide by 4
Now, start on the right hand side and replace each operation with its inverse
Multiply by 4
Divide by 3
Thus, f-1(x) =
### You try!
Find the inverse of f(x) = 4x-2
*4
-2
x
4x-2
(x+2)/4
/4
+2
x
So f -1 (x)=(x+2)/4
IGCSE
Ex 12 Pg 268-269
#9-22
### A neat little trick…
• As always in maths, there is a trick to this…
• Write function as a rule in terms of y and x.
• Swap ‘x’ and ‘y’
• Rearrange to get in terms of y.
f(x) = 5x + 7
y = 5x + 7
x = 5y + 7
x -7 = 5y
y = (x-7)/5
f-1(x) =
### Reflecting..
Inverse functions only exist for one-one functions.
i.e. functions where each input (x) can
only lead to one possible output ( f(x) )
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# Swift Program to Calculate Area of Octagon
This tutorial will discuss how to write swift program to calculate area of octagon.
An octagon is a two-dimensional shape with 8 sides and also consists of eight interior and exterior angles. The total amount of space enclosed inside the octagon is known as the area of the octagon.
## Formula:
Following is the formula of the area of the octagon −
Area = 2 *(side)2*(1+√2)
Below is a demonstration of the same −
Input
Suppose our given input is −
side = 15
Output
The desired output would be −
Area of octagon = 1086.3961030678927
## Algorithm
Following is the algorithm −
Step 1- Create a function with return value.
Step 2- Find the area of the octagon using the following formula:
return 2 * q * q * (1+sqrt(2))
Step 3- Calling the function and pass the side in the circle as a parameter.
Step 4- Print the output.
## Example
The following program shows how to calculate the area of octagon.
import Foundation
import Glibc
// Creating a function to find the area of octagon
func octagonArea(q:Double) -> Double{
return 2 * q * q * (1+sqrt(2))
}
var num = 10.0
print("Length of the side is", num)
print("Area of the octagon:", octagonArea(q:num))
## Output
Length of the side is 10.0
Area of the octagon: 482.84271247461896
Here, in the above program we create a function which return the area of the octagon using the following formula −
return 2 * q * q * (1+sqrt(2))
Here, we use sqrt() function to find the square root of 2.
Updated on: 30-Nov-2022
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## Bernoulli and Binomial Distribution
1 review
by: Dresden Mosch
202
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# Bernoulli and Binomial Distribution STAT 225
Marketplace > Purdue University > STAT 225 > Bernoulli and Binomial Distribution
Dresden Mosch
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GPA 3.1
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Stat 225 Notes 2l11l2015 53 Bernoulli and Binomial Distributions rnc mlngialan can log 0FX quot Zaculan nq Fixobab iy Mm FM I Can amenam ai 01r Imbg billyNJ 5 39 a FW mm m asiis gm 4mm v 39 quot l 0 each pair Tb e m 1 OF I v la1a f0 w 0 X JEMMLMQMZQYif 777a mama 0r agaac 016 x x 1 IA 1 oLn 0 KINVIT 13 0 dr ibMHOH quot 39 oil smibuh on of a KQnaom Vam czlble X r Ke ens l0 5 Silppo l J and me warm sway clijl mbcl om V w the Classi ed one5 wnlch 6W6 u I 39 y v39 x h M 39 Iquot 2 39 y quot 39 gang ex emmam mm rw 39 it Wows 0 0 cH39COI Y 5 39 if 2 Pad 1 1 K9 SMCCQSS l org 0L macmew m mm Xgo 5 0 pxI PW0KPJ a H lip F Kanaom vambne 6 Mpporlt pamme 612 e WMF Efx MM X mm 77 7 MM LH x Ff 506665565 024 77 K OMHC B k OUlf Walls A R V BMW0 2 v 39 I h ammch X 0 I Id Emefcq m 57 59 e DdemL e on Random mm ab 6 3 emun Random Vamblgpse Wm myowmo quotV a 9 Wquot 03 mob e m m p o av L w 539 3 3950 07 100 50 0150 T 115 616155 45 0f amamfs Famed exam1 gt 1 N6 Pick a Jmaem m Random and awed urry wm m we n01 Wy paJJe 16 of aweems Paffed r I 1 1quot W106 0F Katmain Valera91 11 m 5 7Why X39 e fnp075 w BarlowIi J mawlt m y have Imial w 39meq he exam w1 a Wagm Pmifsa 25 mm 3 a farMe v3 PMF EU WK x 0j quot225 quot 0K 39pxlxk 5075 I F X X I 7 015 l 1 O f WXJ 075 Stat 225 Notes 2132015 54 Hypergeometric Distribution av r A m w 7 0 W ccmm I FXLI X quot quotNvrrw tamw i 39 nr umwmuW w 41 mm wmmmvwm a u 7 4 mvu w 1 WWW Replacemcn kut i w v 39quotnrmw v Mquot Wm W i V w 39 39n 39vm M A M 4 vmenqsQ u s V 10194 when 1 me quot mas thenwaMD Wbam 1415vnmmen1 1396 pawne mat 1f qnimlfmm I776 max imam LoJeaIWWW a ondr be mace 774m 1 w mv mmm an r a a I wasp nuk A r amtpw w wt I 4 a n ms a m i A xmaae u i Lag I Hum 44 70X 39 baH4Ci i k 2 IZLAQLIUUWSE mdz mdel H Mb 395 amp 1116 39 W a W13 a w t5 NQQOWIOLI quotK a 0 OleFCC 7V5 7W quot 051mww imycigmb elil lf Q mo an 1 w mah m mm appeoPkam 39 Alt00052 2 5 AM TJO NgtMn LnL 50 056 9 000 0 apffewn Y HQA 52 m M Woo me 039 PY B 330 3903925393 1 39 a 07503 Stat 225 Notes 2182015 55 Poisson Distributions a msz f P0SX o 7 07 sz 1 MW 39 J ltampQLQQK QWE k Nat eg 05 39 JLYJIKQQSW Iquot 1 MEEQi Mai HM 45an Q a Mwh m M M 4Ol30 MW wm W M N
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# Question about Math of Superposition
all.
I am currently reading through a quantum mechanics book, and I was struggling to understand an equation presented in the review of the mathematics. The part where I am discusses one-dimensional wave packets. It first presents the differential equation:
$$-\frac{\partial^2 u(x, t)}{\partial x^2} = \alpha i \frac{\partial u(x, t)}{\partial t}$$
With a solution of
$$u_k(x, t) = Be^{i(kx-\omega t)}$$
Any superposition of the possible solutions to the differential equation can be formed by taking
$$A_1 u_{k_1}(x, t) + A_2 u_{k_2}(x, t) + ...$$
where the A's are constant.
The book then goes on to say that the principle of superposition can be generalized to an integration over a continuum of solutions for various k:
$$f(x, t) = \int {A(k)u_k(x, t)dk}$$
However, I'm not sure how this is true, and the book does not seem to give any explanation. I looked around a little bit online to see if I could find an answer, but what I found did not seem to cover this part (or I possibly did not understand). Could anyone help me through this part?
As a side note, I don't formally study physics (I actually study chemistry at my university), however the physical chemistry course I took really piqued my interest in quantum mechanics. I don't mind difficult-to-understand answers, but be prepared for some possibly dumb questions on my part.
• Does this answer your question? Is the superposition principle universal? Jun 16, 2021 at 22:55
• Not quite. I’m wondering how the integration is equal to the superposition of solutions. Whether the superposition principle is universal or not doesn’t have much to do with my question.
– Andy
Jun 16, 2021 at 23:08
• Are you familiar with the definition of the integral? An integral is the limit of a sum, and you say you're fine with sums. Where is the disconnect exactly? Jun 17, 2021 at 1:38
• @BioPhysicist Yeah! I understand that the integral is actually just a Reimann sum with a limit of n approaching infinity. However, when one writes the discrete sum for the superposition, there is no delta k anywhere. I would assume, which is in agreement with what CW Tan said, that A_i = A(k_i)delta k. If you substitute that into the discrete sum and apply a limit where n approaches infinity, then it's clear that you've just come up with the integral in question. However, I don't understand why A_i would equal A(k_i)delta k. Does this make more sense? I apologize for not being clear earlier.
– Andy
Jun 17, 2021 at 3:38
Any solution of the form $$u_k(x,t) \propto e^{i(kx-\omega t)}$$ satisfies the differential equation, so we can construct a general solution from a linear combination of $$u_k(x,t)$$ solutions where $$k$$ can take various values. For example, one possible solution is
$$u(x,t) = A_1 e^{i(k_1 x- \omega_1 t)} + A_2 e^{i(k_2 x- \omega_2 t)} + A_3 e^{i(k_3 x- \omega_3 t)}$$.
Note how $$A_1$$, $$A_2$$ and $$A_3$$ are distinct and associated with the complex exponentials of specific values of $$k$$, i.e. $$A_1$$ is the scaling constant associated with the exponential with a wavevector of $$k_1$$. Also, I distinguished the respective $$\omega$$ values as there usually is some relation between $$\omega$$ and $$k$$ (called a dispersion relation). You should be able to find it by plugging the complex exponential solution into the differential equation you have.
Anyway, an even more general representation would be to write the solution as an arbitrary sum over various $$k$$ values (which probably have their own corresponding $$\omega$$ values), i.e.
$$u(x,t) = \sum_i A_i e^{i(k_i x - \omega_i t)}$$
Note again how each scaling coefficient $$A_i$$ is indexed in a way that is associated with the $$k$$ values. In theory, this could sum over all possible $$k$$ values and we just have to choose the $$A_i$$ values carefully to reflect a particular solution, e.g. you could represent a plane wave state $$A_0 e^{i(k_0x-\omega_0 t)}$$ as the sum but setting all $$A_i=0$$ except for $$A_0$$ that corresponds to $$k_0$$.
We could go one step further and move from this discrete representation to a continuous one using an integral to represent the sum over possible values of $$k$$, i.e.
$$u(x,t) = \int A(k) e^{i(kx-\omega(k)t)} dk$$
Here the $$A(k)$$ again just represents the value of the scaling constant at a specific $$k$$ value and I've written the frequency explicitly as a function of $$k$$, i.e. $$\omega(k)$$, which is the dispersion relation obtained by a simple substitution of a general $$e^{i(kx-\omega t)}$$ into the differential equation.
Hopefully this answered your question - if you understand why a discrete sum represents a superposition of solutions, then going to an integral is just an additional step where we consider a continuous range of $$k$$ values that can be adopted rather than just discrete values. The idea of the superposition just comes from the fact that any $$\textbf{linear combination}$$ of the complex exponential plane wave solutions can satisfy the differential equation, since any individual plane wave solution satisfies the differential equation. (I can show this explicitly if you want, but I assumed your main query was about the integral representation).
• Okay, I think this answer is going in the right direction! The discrete sum makes perfect sense, however I'm struggling to understand how the integral could possibly equal that discrete sum. It would seem to me that you would have to multiply the sum by delta k to actually get a an integral (although, I would assume this is actually wrong). Does what I'm saying make sense?
– Andy
Jun 17, 2021 at 1:30
• This just restates what the OP says they are confused about. Jun 17, 2021 at 1:36
• @BioPhysicist I wrote the general solution explicitly with the summation (Sigma) notation to make it clearer that the integral was just a continuous representation of it, though I agree that I could write the connection even more explicitly. Also I tried to explain the origin of the A(k) term in the integral, so I think it's unfair to imply that I just restated the question without value-adding. As for Andy's comment, (not sure how mathematically rigorous this is but) I think of the A_i in the discrete sum as A(k_i) Δk, which becomes A(k) dk when Δk is very small. Jun 17, 2021 at 2:19
• @CWTan Yeah, I think you're right about A_i being equal to A(k_i)delta k. As I mentioned above though, I'm not actually sure why this is the case.
– Andy
Jun 17, 2021 at 3:41
• @Andy I think this is a general feature of converting from sums to integrals. E.g. with a continuous probability distribution over a variable x, we'd say that the probability of obtaining a value between x and x+dx is P(x)dx, but in the discrete case we'd just say that the probability of getting x_i is P(x_i). In this case, I think the statement we're making is that the contribution of the term with a k-value between k and k+Δk to the sum is given by A(k) e^{...} Δk, which is slightly different from saying that the contribution of a single term corresponding to a particular k is A_k e^{...}. Jun 17, 2021 at 12:55
Well, A wave packet is a superposition of many waves with different phases, in this case I can see that you have the index "k", I automatically associate it to the energy, then you must think about stationary states and not stationary states, the stationary states are those which have a defined energy, like when you solve the box problem, in this case you will have a free particle, and its wave function will be a linear combination of many waves, with different energies. That's why you can get your solution like an integral, since the energy is continuous, whereas in a box it can just take certain values. Another thing you have to take into account is that not all the wave functions are physical.That's what I understand about it, I'm not really an expert, please correct me if anyone can see a mistake.
If I understand your question, you are asking why you should believe that $$f$$ satisfies the differential equation.
So write the differential equation, substituting $$f$$ for $$u$$. Then move the derivatives inside the integral sign. Now the fact that every $$u_k$$ satisfies the differential equation tells you that the thing you are integrating on the left is equal to the thing you are integrating on the right, making the integrals equal as needed.
To do this, you need to justify moving the derivatives inside the integral sign, which you can't absolutely always do, but there are standard theorems in mathematics that allow you to do it under very general conditions. Physicists are happy to assume that $$f$$ satisfies those conditions.
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## Physics Sapling Problem
Two identical guitar strings are prepared such that they have the same length (2.56 m) and are under the same amount of tension. The first string is plucked at one location, primarily exciting the 1st harmonic. The other string is plucked in a different location, primarily exciting the 4th harmonic. The resulting sounds give rise to a beat frequency of 378 Hz. What is the wave propagation speed on the guitar strings?
• Anonymous commented
That is a HUGE guitar if the strings are 2.56 m long...
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# existence of linear transformation
This is a simple question in linear algebra but I just had a hard time thinking about the logic.
Let V and W be two vector spaces over a field F. For simplicity, we assume they have the same finite dimensions and any linear transformation $$T:V \rightarrow W$$, null space is {$$0$$}. Let {$$v_1$$, $$v_2$$, ..., $$v_n$$} be a basis for V. Then by theorem, any linear transformation $$T:V \rightarrow W$$, {$$T(v_1)$$, $$T(v_2)$$, ..., $$T(v_n)$$} should be a basis for W. However, theorem of existence of linear transformation states that there exists precisely one linear transformation
$$T:V \rightarrow W$$ such that $$T(v_i) = w_i$$ where $$w_i, i = 1,..,n$$ is any vectors in W. If {$$w_1$$, $$w_2$$, ..., $$w_n$$} are linear dependent, then one of two theorms is wrong. Did I omit something?
edit: the first theorem can be derived from $$N(T)$$ = {$$0$$}, then T is injective, then T is an isomorphism given dim(V) = dim(V) < $$\infty$$.
• It is not true that if $V$ and $W$ have the same dimension then any linear map $T\colon V\to W$ has $\{0\}$ null space. Feb 14, 2019 at 22:34
• @egreg Those are the assumption. Feb 14, 2019 at 22:46
• You read them wrongly, I'm afraid. The only vector space $V$ such that every linear map $T\colon V\to V$ is injective is the trivial space. Feb 14, 2019 at 22:52
• @egreg I mean that "$V$ and $W$ having the same dimension" and "map $T: V \to W$ having {$0$} null space" are assumptions. Feb 14, 2019 at 22:57
• I think the question really is that there doesn't exist a linear transformation that maps independent basis to dependent vectors when it is an isomorphism. Feb 14, 2019 at 23:03
The image vectors $$\{T(v_{1}),\ldots, T(v_{n})\}$$ can form a basis only if the kernel of $$T$$ is zero. This is not true anymore if $$\{w_{1},\ldots,w_{n}\}$$ are linearly dependent, because by definition you can find scalars $$\lambda_{i}$$'s not all zero such that $$\lambda_{1}w_{1}+\cdots+\lambda_{n}w_{n}=0$$ Then $$T(\lambda_{1}v_{1}+\cdots +\lambda_{n}v_{n})=0$$, so we have a non-zero element in the kernel (if $$\lambda_{1}v_{1}+\cdots +\lambda_{n}v_{n}=0$$ then all $$\lambda_{i}$$'s would be zero by linear independence of the $$v_{i}$$'s).
So in the first theorem you need to assume that $$T$$ is an isomorphism (or equivalently that it has trivial kernel if both spaces have the same dimension).
• I believe I just stated in that way: "null space is {$0$}"/trivial kernel and both spaces have the same dimensions. Feb 14, 2019 at 22:07
• Yes, after your last edit the statement is more clear. So you see that both theorems are fine, but the point is that if the $w_{i}$ are linearly dependent, then the corresponding $T$ does have some non-trivial kernel, so you cannot apply the first theorem Feb 14, 2019 at 22:12
There's a few things to address here. First, the null space is defined in terms of an operator, and is not an intrinsic property of spaces themselves.
Secondly I am not sure where you are getting this theorem, but I believe you may be misinterpreting it. If $$T:V\to W$$ is an isomorphism then yes, $$\{Tv_i\}_{i=1}^n$$ is a basis for for $$W$$, but if not then no. Just consider the $$0$$ operator.
Thirdly if $$\{w_i\}_{i=1}^n$$ is a basis then they will not be linearly dependent.
• I agree with all your points... But my setting is exactly $T:V\to W$ is an isomorphism, $\{Tv_i\}_{i=1}^n$ is a basis for for $W$ and $\{w_i\}_{i=1}^n$ is not a basis. Feb 14, 2019 at 22:10
• @YellowRiver For a given set $\{w_i\}_{i=1}^n$ yes then there is a unique operator $S:V\to W$ satisfying $Sv_i=w_i$, but there is no reason why this operator has to equal $T$? In fact for your given $T$ we know that this can't be $S$ precisely because it maps a linearly independent set to a linearly independent set. Feb 14, 2019 at 22:16
• Nice! I just realized the importance of your first point. When setting a null space to be {$0$}, choice of operator T is limited. Feb 14, 2019 at 22:25
Suppose that the linear transformation being considered is one that takes every vector in $$V$$ to the $$0$$ of $$W$$. Clearly, it could never result in a basis. The theorem that all linear transformations map bases to bases must be wrong - perhaps what was meant is that all invertible linear transformations map linearly independent sets to other linearly independent sets.
• @Display Name you mean linearly independent sets to other linearly independent sets. Feb 14, 2019 at 22:18
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Applying the DeBroglie Equation
$\lambda=\frac{h}{p}$
Vincent Leong 2B
Posts: 207
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Applying the DeBroglie Equation
When is the DeBroglie Equation applicable? More specifically, what type of particles or substances can we use the DeBroglie equation to solve a problem?
Michelle Chan 1J
Posts: 50
Joined: Thu Jul 25, 2019 12:16 am
Re: Applying the DeBroglie Equation
The De Broglie Equation is used for any particle with momentum and has wavelike properties with a wavelength. Remember to not use EM equations for electrons!
nickianel_4b
Posts: 50
Joined: Thu Jul 25, 2019 12:17 am
Re: Applying the DeBroglie Equation
The De Broglie equation says that any moving particle with momentum p has wavelike properties with wavelength λ, so it's applicable for any moving particle.
Malia Shitabata 1F
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Joined: Sat Aug 17, 2019 12:17 am
Re: Applying the DeBroglie Equation
The DeBroglie equation can't be applied to light
DLee_1L
Posts: 103
Joined: Sat Aug 17, 2019 12:17 am
Re: Applying the DeBroglie Equation
Ironically, the Debroglie equation was derived from equations using light, however the wavelength of light cannot be found using debroglie's equation due to the equation requiring mass.
EthanPham_1G
Posts: 104
Joined: Sat Jul 20, 2019 12:17 am
Re: Applying the DeBroglie Equation
The DeBroglie equation is used to find the wavelength of anything that has momentum (mass in motion). With the DeBroglie equation, we can tell if an object displays wave-like properties, depending on its mass.
Rhea Shah 2F
Posts: 97
Joined: Thu Jul 25, 2019 12:17 am
Re: Applying the DeBroglie Equation
The DeBroglie equation can be used on any particle with momentum, and is used to determine the wavelength of the particles movement. Since momentum is calculated by multiplying mass by velocity, the equation cannot be used for light particles as light does not have mass and thus does not have momentum.
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hwsln-421-s06-05
# hwsln-421-s06-05 - ECE 421 Spring 2006 Solutions to HW...
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ECE 421, Spring 2006 Solutions to HW Assignment #5 Problem #B-5-23 From the given G ( s ) , the closed-loop transfer function and closed-loop characteristic equation are T CL ( s )= C ( s ) R ( s ) = G ( s ) 1+ G ( s ) = K s ( s +1)( s +2)+ K = K s 3 +3 s 2 +2 s + K (1) ( s s 3 s 2 s + K (2) and the Routh array is s 3 : 1 2 s 2 : 3 K s 1 : 6 K 3 s 0 : K For closed-loop stability, there must be no sign changes in the f rst column of the array. Checking the s 0 and s 1 rows (the only ones involving K ) produces the following limits on K for stability. s 0 : K> 0 ,s 1 :6 0 K< 6 (3) 0 <K< 6 (4) Problem #B-5-24 G ( s ) , the closed-loop transfer function and closed-loop characteristic equation are T ( s C ( s ) R ( s ) = G ( s ) G ( s ) = 10 s ( s 1) (2 s +3)+10 = 10 2 s 3 + s 2 3 s +10 (5) ( s )=2 s 3 + s 2 3 s (6) It is obvious by inspection of ( s ) that the closed-loop system is unstable since not all the coe cients have the same sign. Forming the Routh array to check this conclusion gives the following result. s 3 : 2 3 s 2 : 1 10 s 1 : 23 s 0 : 10 Since there are 2 sign changes in the f rst column of the array, there are 2 closed-loop poles in the right-half plane. Therefore, the closed-loop system is unstable. Problem #B-5-25 From the given closed-loop characteristic ( s s 4 s 3 +(4+ K ) s 2 +9 s +25=0 (7) the Routh array is 1
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s 4 : 1 (4 + K ) 25 s 3 : 2 9 s 2 : 2 K 1 2 25 s 1 : 18 K 109 2 K 1 s 0 : 25 For closed-loop stability, there must be no sign changes in the f rst column, so the s 2 and s 1 rows must be checked. s 2 :2 K 1 > 0 K> 0 . 5 (8) s 1 :1 8 K 109 > 0 109 / 18 = 6 . 056 (9) Since the constraint from the s 1 row is more restrictive than the costraint from the s 2 row, the limit on K for closed-loop stability is 6 . 056 .
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# 311.8 stones in stones and pounds
## Result
311.8 stones equals 311 stones and 11.2 pounds
You can also convert 311.8 stones to pounds.
## Converter
Three hundred eleven point eight stones is equal to three hundred eleven stones and eleven point two pounds (311.8st = 311st 11.2lb).
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Pendulums_al
# Pendulums_al - % EGM 3400/3401 Classic pendulum dynamics...
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Sheet1 Page 1 % EGM 3400/3401 Classic pendulum dynamics example problems % Based on Chapter 19 in textbook % Example 1: Particle with 1 DOF % Example 2: Rod with 1 DOF % Example 3: Rod with 2 DOFs Overwrite On Frames N % n1> to the right, n2> up, and n3> = n1> x n2> Points Q Particles P Bodies B Variables theta'',Ten,Fx,Fy,x'' Mass B=m,P=m Inertia B,Ixx,Iyy,Izz,0,0,0 Constants g,l % Rotation matrices Simprot(N,B,3,theta) % Angular velocities w_B_N>=theta'*b3> % Example 1: Particle with 1 DOF % Linear velocities v_No_N>=0> p_No_P>=-l*b2> v_P_N>=v_No_N>+Cross(w_B_N>,p_No_P>) % Tension and gravity forces on P F_P>=Ten*b2>-m*g*n2> F_P>=Express(F_P>,B) M_B_P>=0> M_B_No>=Cross(p_No_P>,F_P>) % Linear momentum principle L_P_N>=m*v_P_N> Zero_Lin>=F_P>-Dt(L_P_N>,N) EqnLin1=Dot(Zero_Lin>,n1>) EqnLin2=Dot(Zero_Lin>,n2>) EqnLin3=Dot(Zero_Lin>,b1>) EqnLin4=Dot(Zero_Lin>,b2>) % Angular momentum principle about mass center P H_B_P_N>=Cross(p_P_P>,L_P_N>) Zero_Ang_P>=M_B_P>-Dt(H_B_P_N>,N) % Just gives 0> = 0>
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## Pendulums_al - % EGM 3400/3401 Classic pendulum dynamics...
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## How To Find Vertical Asymptotes
Sm summary sheets – itute., A circle, e.g. {z : z −()1+i = 2}, centred at z =1+i , radius 2 . im re e.g.{z : also defined a circle. 2 z +1 = z −i} let z =x +yi, 2()x +1 +yi = x +()y −1i. Section 1.10: difference quotients – kkuniyuk., (section 1.10: […]
## How To Find Vertical Asymptotes
Page 1 2 14.1 graphing sine, cosine, tangent functions, Page 1 of 2 14.1 graphing sine, cosine, and tangent functions 833 graphing a cosine function graph y=1 3 cosπx. solution the amplitude is a=1 3 and the period is 2. Relations functions (mathematics), Relations and functions (mathematics) copyright © enoch lau 2003 jan 2003 (re-pub […]
## How To Find Vertical Asymptotes
Chapter 2: limits continuity – kkuniyuk., Chapter 2: limits and continuity 2.1: an introduction to limits 2.2: properties of limits 2.3: limits and infinity i: horizontal asymptotes (has). Grade 11 assessment booklet – maths – maths excellence, Grade 11 – 4 – exemplar assessments 2008 instructions and information read the following instructions carefully before answering […]
## How To Find Vertical Asymptotes
Grade 12 /june exam booklet 2016 – reddam, Study tips before the exam day do not miss any days of school. if you are unwell, come to school to write your tests and then go home. then you will be able to see. Di erential equations – theory applications – version, Di erential equations – […]
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SZptFD-06
# SZptFD-06 - 14 SPRING 2012 6. Mean Value Theorem....
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14 SPRING 2012 6. Mean Value Theorem. Clairaut’s Theorem Recall the Mean Value Theorem from Calculus: if f :[ a, b ] R is continuous on [ a, b ]and di f erentiable on ( a, b ), then there exists c between a and b such that f ( b ) f ( a )= f ° ( c )( b a ) . Disappointingly, we can have no such theorem for functions f : U R n R m with m> 1, even though it makes syntactic sense to write the expression f ( z ) f ( x )= f ° ( c )( z x )for such a function. Example 6.1. For instance, consider f : R R 2 de±ned by f ( t )=(co s t, sin t ), so that f ° ( t )=( sin t, cos t )forall t . There is certainly no c between 0 and π such that f ( π ) f (0) = f ° ( c )( π 0), as this would require ( 1 , 0) (1 , 0) = π ( sin c, cos c ) , and the left-hand vector has length 2, while the right-hand vector has length π , irregardless of the value of c . In order to extend the Mean Value Theorem to functions de±ned on subsets of R n ,w e have to ensure that “between” makes sense. A set S R n is convex if (1 λ ) x + λ y S whenever x , y S and 0 λ 1. Theorem 6.2 (Mean Value Theorem) . Let U R n be open and convex, and let f : U R be di f erentiable. For each x and z in U , there exists c on the line segment from x to z such that f (
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## This document was uploaded on 03/11/2012.
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SZptFD-06 - 14 SPRING 2012 6. Mean Value Theorem....
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1 What is the correct total time 2 What is
1. What is the correct total time?
2. What is the critical path?
3. Create a Gantt chart that shows the WBS.
4. Create a PERT/CPM chart.
At Forest Point Construction, your boss says that he can estimate the total project time based on his personal experience. You are trying to convince him that he should use project management techniques to handle a complex project. To prove your point, you decide to use a simple example of a commercial steel building construction project, with eight steps. You create a hypothetical work breakdown structure, as follows:
• Prepare the site (3 days), and then set the building footers (3 days).
• Finish the foundation (5 days), and then assemble the building (3 days).
• When the building is assembled, start two tasks at once: finish the interior work (5 days) and set up an appointment for the final building inspection (15 days).
• When the interior work is done, start two more tasks at once: landscaping (7 days) and driveway paving (3 days).
• When the landscaping and driveway are done, do the painting (2 days).
• Finally, when the painting is done and the final inspection has occurred, arrange the sale (2 days).
Now you ask your boss to review the tasks, estimate the total time, and write the answer on a piece of paper. You look at the paper and see that his guess is wrong.
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# 6b - equilibrium position of the spring Assume air...
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PHY2053 – Section 6168 Quiz #6 Friday, February 20, 2004 Name_______________________________ UFID___________________ Please read the problem carefully and be sure to show all of your work. Partial credit will be given. A 0.50 kg block is placed on a light vertical spring (k = 7.0 x 10 2 N/m) and pushed downward, compressing the spring by 5 cm. After the block is released, it leaves the spring and continues to travel upward. What height will the block reach, relative to the
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Unformatted text preview: equilibrium position of the spring? Assume air resistance is negligible. Solution: Conservation of energy: Potential energy in the spring gets converted to kinetic energy as the block leaves the spring and then gets converted to gravitational potential energy as the block reaches its maximum height. Note that the equilibrium position of the spring is where the block leaves the spring (x=0) ! ½ kx 2 = mgh Solve for h: h = kx 2 /2mg = 17.8 cm...
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# probability and statistics
Apr 6th, 2015
Anonymous
Category:
Accounting
Price: \$5 USD
Question description
You are told that flight times from Newark,NJ to Greenville,SC are uniformly distrubuted from 2.5 to 3.5 hours with standard deviation 0.289 hours.
1. What proportion of flights take between 2.75 and 3.25 hours?
2. The estimated arrival time for the next flight is 6:00 PM. If the plane left at 3:15PM what is the probability that it will be late?
3. Write the sampling distrubution for the mean flight time of a sample of 16 flights from newark to greenville
4. use part c to estimate. Interpret your answers in the context of the problem. For example, is this an unusual event?
(Top Tutor) wenchaochen0814
School: Carnegie Mellon University
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https://blog.betwise.net/2018/05/05/daily-trainer-strike-rates-in-smartform-using-sql/
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Betwise news, analysis and automatic betting info
# Daily trainer strike rates in Smartform
By Nick Franks on Saturday, May 5th, 2018
There are many ways to skin a cat in Smartform. One of the simplest is to stick with SQL and stay in the database, without resorting to programming or exporting the data.
So here is a breakdown of how to generate one of the most commonly used “derived” statistics in horseracing – trainer strike rates. We hear that trainers whose horses are currently in good form are worth backing – but how can we tell?
The following single SQL produces Trainer strike rates for all daily Flat races in the database, calculated over a period of 14 days. Copy and paste the code at your MySQL command prompt:
``````Select a.*,
ROUND(Winners/Runners * 100,0) AS WinPct,
ROUND(Placers/Runners * 100,0) AS PlacePct
from(
SELECT hru.trainer_name as Trainer, hru.trainer_id,
COUNT(*) AS Runners,
SUM(CASE WHEN hru.finish_position = 1 THEN 1 ELSE 0 END) AS Winners,
sum(case when hra.num_runners < 8 then case when hru.finish_position in ( 2) then 1 else 0 end
else
case when hra.num_runners < 16 then case when hru.finish_position in ( 2,3) then 1 else 0 end
else
case when hra.handicap = 1 then case when hru.finish_position in (2,3,4) then 1 else 0 end
else
case when hru.finish_position in ( 1,2,3) then 1 else 0 end
end
end
end )as Placers,
ROUND(((SUM(CASE WHEN hru.finish_position = 1 THEN (hru.starting_price_decimal -1) ELSE -1 END))),2) AS WinProfit
FROM historic_runners hru
JOIN historic_races hra USING (race_id)
WHERE hra.meeting_date >= ADDDATE(CURDATE(), INTERVAL -14 DAY)
and hra.race_type_id in( 15, 12)
and hru.in_race_comment <> 'Withdrawn'
and hru.starting_price_decimal IS NOT NULL
GROUP BY trainer_name, trainer_id
) a
Where runners >= 5 or winners >= 3
ORDER BY WinProfit desc
LIMIT 20;
``````
The approach is to generate a query within a query – to allow the use of the counts done in the main query to be used to create the strike rate percentages.
The main part joins the historic_runners and historic_races tables by race_id.
The WHERE clause defines the criteria
Firstly, define the period over which you want the analysis, here we set it to analyze races in the last 14 days
``````hra.meeting_date >= ADDDATE(CURDATE(), INTERVAL -14 DAY)
``````
Secondly we define the race types for all Flat races (12 for Turf and 15 for All Weather)
All of these can be found running the SQL statement:
``````select distinct race_type, race_type_id from historic_races; )
``````
The final criteria helps to eliminate horses who have not actually participated that are in the database, non-runners, withdrawn etc. There are a number of ways to do this and one can find their own criteria, but I use this:
``````and hru.in_race_comment <> 'Withdrawn'
and hru.starting_price_decimal IS NOT NULL
``````
The outer section of the SQL selects all the data from the inner part and displays it and calculates the strike rates. The `)a` afterwards is required by SQL.
Finally in Green are the sorting instructions and some other criteria.
Also note that here I am choosing to only display trainers who have had at least 5 runners or at least 3 winners.
``````Where runners >= 5
or winners >= 3
``````
I am ordering them in order of greatest win profit
``````ORDER BY WinProfit desc
``````
And limiting the number of trainers displayed to 20
``````LIMIT 20
``````
So on the day of the 2000 Guineas, Brian Meehan is currently the most profitable trainer to follow, with £47 retuns, but P W Hiatt has the highest strike rate, with 60% – albeit with only 5 runners in the last few days.
If you’re looking for a trainer with a high strike rate from at leasy double digit runners in the last 14 days, then Mark Usher, with a 33% win strike and 25% place strike, is impressive.
Adapt the SQL above to generate daily trainer strike rates so you can judge for yourself.
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# Inital-value problem
1. Mar 11, 2008
### morbello
Have been looking at this problem .A few times nowI've come up with different answers.Its getting close to needing to know or wether, to just put in what i have.so ive placed a question on here to see if im missing something.
The equations that are said to help are
$$\int\f'(x)/f(x) dx = In (f(x))+c (f(x)>0)$$
dy$$/dx = x^4+1/x^5+5x+6 (x>-1)$$ is the equation to work off.
my work so far is
1$$/5 In (x^5+5)+c$$
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
2. Mar 11, 2008
### rock.freak667
3. Mar 12, 2008
### morbello
so the x^5+5x+6 does not differeniate to become x^5+5 ok
there are a few activitys in my books that are like it.
4. Mar 12, 2008
### rock.freak667
$$\int \frac{x^4+1}{x^5+5x+6} dx$$
Let $t=x^5+5x+6 \Rightarrow \frac{dt}{dx}=5(x^4+1)$
$\frac{dt}{5}=(x^4+1)dx$
$$\int \frac {x^4+1}{x^5+5x+6} dx \equiv \int \frac{1}{5} \frac{1}{t} dt$$
and $\int \frac{1}{x} dx = ln(x)+C$
$$\int \frac{1}{5} \frac{1}{t} dt = \frac{1}{5} lnt + C$$
5. Mar 12, 2008
### morbello
that blew me a little bit there.ive sat and looked at it a little and i think i know what you have done just i would not have thought off it. i think we did one function that was worked out that way.ill have to look it up tonight.
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Search i-want-the-solutions-for-this-two-questions-in-four-hours
# I want the solutions for this two questions in four hours
## Top Questions
cific gravity) of 1.24. What is the molarity of this solution? Using this solution how would 1 L of 2N solution be made. 6 (5 points) H3BO3 is needed to neutralize 20 ML of a 2N solution of NaOH. How much of the acid should I put in 40 ML of the water to exactly neutralize this solution? 7 (6 points) Describe how to make the solutions below : 20% w/v Salt in water. 20% v/v alcohol in water 20% w/w NaCl in water. 8 (three points) I have 0.6 g/dl solution of NaOH. What is M? Whan is N? 9 (six points) There are 3000mL of 3M NaOH. How much of the following do I need to neutralize? (watch your M’s and N’s a) 3M H3PO4 b) 2M H2SO4 c) 1M HCL 10 (20 points) The following solutions of NaOH are mixed together 20ML of 3N, 40mL of 2N, 60mL of 1N, 80 mL of 4N, and 100mL of 5N. a) What is the volume and normality of the final solution? b) How much 4M sulfuric acid would I need to neutralize? c) How much stock solution of sulfuric acid with an assay of 77% and a specific gravity of 1.14 would I need? d) How many grams of HCl would I have to put in a 300mL solution of HCl in water to neutralize? 11 (3 points) How much 5N solution can I make with 98 grams of H3PO4 ? 12 (5 points) How much 5N solution of H3AsO4 can I make with 57 mL of stock solution that is 84% assay and 1.14 specific gravity? 13 (5 points) If we have a 4N solution of HCl that has 0.03645g of HCl in the solution, how many microliters of solution do we have? 14 (10 points) If we have 66mL of a solution of concentrated NaOH that has an assay of 88% and a specific gravity of 1.24, how much 3N H3AsO4 can be neutralized? 15 (ten points) If I have 17 mL of a 20% w/v solution of NaOH and I want to neutralize with H2SO4 that is available in a 4% w/v solution, how much of this solution will be required. 16 (ten points) a) I have an 12mL of Ba(Cl)2 that is 78% assay that contains 8 grams of Ba(Cl)2. What is the specific gravity? b) How many Moles of Ba(Cl)2 are there? c) If I have 80 grams of NaOH in a liter of solution that is of an unknown specific gravity, can I calculate molarity and what is it? d) What is the difference between molarity and normality? e) I have 77 ml of 77% salt in water. How much 11% can I make?
View More
1.AU MAT 120 Systems of Linear Equations and Inequalities Discussion
mathematicsalgebra Physics
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# How are laws made in USA?
## How are laws made in USA?
The bill has to be voted on by both houses of Congress: the House of Representatives and the Senate. If they both vote for the bill to become a law, the bill is sent to the President of the United States. He or she can choose whether or not to sign the bill. If the President signs the bill, it becomes a law.
## Who makes the rules for America?
Federal laws are made by Congress on all kinds of matters, such as speed limits on highways. These laws make sure that all people are kept safe. The United States Congress is the lawmaking body of the Federal Government. Congress has two houses: the House of Representatives and the Senate.
Where are the laws of the country made?
Explanation: The United States Congress is the lawmaking body of the Federal Government. Congress has two houses: the House of Representatives and the Senate. Each state also passes its own laws, which you must follow when you are in that state.
How many laws exist in the US?
Laws vs agency rules and regulations. Table compiled by author. Looking back, there have been 88,899 federal rules and regulations since 1995 through December 2016, as the chart shows; but “only” 4,312 laws.
### How are laws for the country made?
Legislative proposals are brought before either house of the Parliament of India in the form of a bill. A bill is the draft of a legislative proposal, which, when passed by both houses of Parliament and assented to by the President, becomes an act of Parliament.
### Who made the law for the whole country?
Option D is the correct answer because it is clear that Parliament which consists of Lok Sabha, Rajya Sabha and President make laws for the entire country. Note: Any of the Lok Sabha, Rajya Sabha or President alone can not make any law for the country. Three of them altogether make laws for the entire country.
Who make the law of the whole country?
Law for the entire country is made in the parliament….. Explanation: because in parliament It has two houses, the Lok Sabha and the Rajya Sabha. They make laws for the whole country.
Who makes the laws and regulations for the country?
2.1 Parliament, as the national legislature, has legislative authority (the power to make laws) in the national sphere of government. Consequently, Parliament has the power to pass new laws, to amend existing laws, and to repeal old laws.
#### What is the process of making a law?
Making a law is a complicated and a lengthy process and which includes a series of steps starting from drafting a bill until the President signs the bill and the same becomes a law. Any issuance of a legal precedent, an enactment of a statute or the making of a new rule is part of making a law.
#### How United States laws are made?
How Federal Laws Are Made. Congress is the legislative branch of the federal government and makes laws for the nation. Congress has two legislative bodies or chambers: the U.S. Senate and the U.S. House of Representatives. Anyone elected to either body can propose a new law. A bill is a proposal for a new law.
How does a federal law get passed?
At the federal level in the United States, legislation (i.e., “statutes” or “statutory law”) consists exclusively of Acts passed by the Congress of the United States and its predecessor, the Continental Congress , that were either signed into law by the President or passed by Congress after a presidential veto.
What is the law making process in Congress?
The Law making process in Congress What gets into the congress as a bill meets a set of steps set to try it out and validate it as a law. These trials mainly done by the law makers in the Senate or the House, sometimes with inclusion of the public or the lobbyists.
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https://math.libretexts.org/Courses/Lumen_Learning/Book%3A_Beginning_Algebra_(Lumen)/05%3A_Polynomials/5.01%3A_Why_It_Matters-_Polynomials
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Skip to main content
$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
# 5.1: Why It Matters- Polynomials
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
## Why learn about polynomials?
If you have ever watched a Pixar movie, you have seen computer generated images. A very common method for generating graphics with a computer is to use what is called a wire mesh. You can think of a wire mesh as a grid – like the ones we have used to graph lines – that has been bent and stretched to make a shape we want, as in the image of a dolphin below.
The dolphin in the image was created by plotting points in space to create connected triangles. This method of rendering graphics works well and is in wide use, but it takes a lot of computer memory. Recently, researchers have been investigating the use of polynomials for rendering graphics in part because it demands less memory in the process.[1]. In this process, the surfaces that are rendered are made from solutions to algebraic polynomials. The image below shows some images of smooth-surfaced objects that were rendered using polynomials by researchers Charles Loop and Jim Blinn from Microsoft.
Surfaces Rendered Using Polynomials.
In this module, you will learn how to identify a polynomial and how to perform algebraic operations on them. Like the linear equations and inequalities you learned about earlier, polynomials are useful in many applications of mathematics as well as in other disciplines like biology, economics, and even cryptology. Gaining a basic understanding of their qualities and how the rules of algebra we have learned so far apply to them will help you learn how to use polynomials both inside and out of your math class.
In your next math class you will likely learn how to solve certain kinds of polynomials and how to graph them as well.
## Learning Outcomes
• Single Variable Polynomials
• Define and evaluate polynomials
• Operations on Polynomials
• Add polynomials
• Multiply polynomials
• Subtract polynomials
• Multiply binomials
• Divide by a polynomial
• Applications of Polynomials
• Divide by a monomial
• Write polynomials involving perimeter, area, and volume
• Write polynomials involving cost, revenue, and profit
1. Loop, Charles, and Jim Blinn. Real-time GPU Rendering of Piecewise Algebraic Surfaces. ACM SIGGRAPH 2006 Papers on - SIGGRAPH '06 (2006): n. pag. Web. ↵
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# Alternating Current (Ac) Generator Quiz
Approved & Edited by ProProfs Editorial Team
The editorial team at ProProfs Quizzes consists of a select group of subject experts, trivia writers, and quiz masters who have authored over 10,000 quizzes taken by more than 100 million users. This team includes our in-house seasoned quiz moderators and subject matter experts. Our editorial experts, spread across the world, are rigorously trained using our comprehensive guidelines to ensure that you receive the highest quality quizzes.
| By Matthew
M
Matthew
Community Contributor
Quizzes Created: 1 | Total Attempts: 840
Questions: 123 | Attempts: 840
Settings
• 1.
### To Change electrical energy to mechanical energy is the purpose of a
• A.
Generator
• B.
Converter
• C.
Inverter
• D.
Motor
D. Motor
Explanation
A motor is a device that converts electrical energy into mechanical energy. It does this by using the principle of electromagnetism, where the interaction between electric current and magnetic fields produces a rotational motion. Unlike a generator, which converts mechanical energy into electrical energy, a motor operates in the opposite direction. Therefore, a motor is the correct answer for the given question.
Rate this question:
• 2.
### When a motor continues turning after reaching its limit, this is known as
• A.
Inertia
• B.
Impedance
• C.
Inductance
• D.
Eddy currents
A. Inertia
Explanation
Inertia refers to the tendency of a motor to continue turning even after it has reached its limit. This is due to the motor's momentum and resistance to changes in its motion. When the motor is powered off or the force causing its rotation is removed, the inertia causes it to keep spinning for a period of time before eventually coming to a stop.
Rate this question:
• 3.
### Torque is defined as
• A.
Wattage
• B.
Horsepower
• C.
Rotational force
• D.
Opposition to applied voltage
C. Rotational force
Explanation
Torque is defined as rotational force. It is a measure of the force that can cause an object to rotate around an axis. It is commonly used in mechanics and engineering to describe the effectiveness of a force in rotating an object. Torque is dependent on both the magnitude of the force and the distance from the axis of rotation.
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• 4.
### In order to operate, a single-phase induction motor requires
• A.
Brushes
• B.
Slip rings
• C.
A commutator
• D.
A starting system
D. A starting system
Explanation
A single-phase induction motor requires a starting system in order to operate. Unlike a three-phase induction motor, a single-phase induction motor cannot start on its own due to the absence of a rotating magnetic field. The starting system provides the necessary initial torque to the motor, allowing it to overcome inertia and start rotating. This can be achieved through various methods such as using a capacitor, a centrifugal switch, or a starting winding. Without a starting system, the motor would not be able to begin its operation.
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• 5.
### The difference in the rotor's speed of rotation and the stator magnetic field is known as
• A.
Differential
• B.
Torque
• C.
Slip
• D.
Drag
C. Slip
Explanation
Slip refers to the difference in speed between the rotor's rotation and the stator magnetic field in an electric motor. This difference in speed is necessary for the motor to generate torque and effectively convert electrical energy into mechanical energy. Slip is a crucial parameter in motor performance and efficiency, as it directly affects the motor's ability to maintain a stable speed under varying load conditions. Therefore, slip is the correct answer to the question.
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• 6.
### Which induction motor has the highest starting torque?
• A.
Capacitive start
• B.
Squirrel cage
• C.
• D.
Split phase
A. Capacitive start
Explanation
The capacitive start induction motor has the highest starting torque among the given options. This is because the capacitive start motor uses a capacitor in the starting winding to create a phase difference between the main and starting windings. This phase difference results in a higher starting torque compared to other types of induction motors. The squirrel cage, shaded pole, and split phase motors have lower starting torque compared to the capacitive start motor.
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• 7.
### The three-phase induction motor does not require
• A.
Slip
• B.
A rotor
• C.
A stator
• D.
A starting device
D. A starting device
Explanation
The three-phase induction motor does not require a starting device because it is a self-starting motor. This is due to the presence of a rotating magnetic field produced by the stator windings. When the motor is powered, the rotating magnetic field induces currents in the rotor, causing it to rotate and start the motor. Therefore, no external starting device is needed to initiate the motor's operation.
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• 8.
### The stator windings of a synchronous motor are spaced
• A.
20 degrees apart
• B.
30 degrees apart
• C.
90 degrees apart
• D.
120 degrees apart
D. 120 degrees apart
Explanation
The stator windings of a synchronous motor are spaced 120 degrees apart. This is because a synchronous motor operates on the principle of magnetic fields rotating at a constant speed. By spacing the stator windings 120 degrees apart, the resulting magnetic fields produced by each winding will also be 120 degrees apart. This allows for a smooth and efficient rotation of the motor's rotor, ensuring synchronous operation.
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• 9.
### During an inspection of a direct current (DC) motor, check the brushes for
• A.
Pitting
• B.
Seating
• C.
Coloration
• D.
Brittleness
B. Seating
Explanation
During an inspection of a DC motor, it is important to check the brushes for seating. Seating refers to the proper alignment and contact of the brushes with the commutator, which is essential for the efficient operation of the motor. If the brushes are not properly seated, it can lead to poor electrical contact, increased friction, and reduced motor performance. Therefore, checking the seating of the brushes ensures that they are in the correct position and making good contact with the commutator.
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• 10.
### The first thing you should do when troubleshooting a motor is perform a
• A.
Voltage check
• B.
Continuity test
• C.
Visual inspection
• D.
Current measurement
C. Visual inspection
Explanation
A visual inspection is the first thing you should do when troubleshooting a motor because it allows you to visually identify any obvious issues such as loose connections, damaged wires, or burnt components. By inspecting the motor, you can quickly determine if there are any visible signs of damage or malfunction that may be causing the problem. This initial step can help narrow down the possible causes and guide further troubleshooting steps.
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• 11.
### In component removal, if wires are to be soldered, do not
• A.
Resolder them until they have been checked
• B.
Cut them, but remove the wiring completely intact
• C.
Resolder them until all wiring has been replaced with the new wires
• D.
Cut them unless there is sufficient length remaining for resoldering
D. Cut them unless there is sufficient length remaining for resoldering
Explanation
When performing component removal and soldering wires, it is important to ensure that the wiring is intact and not damaged. Therefore, the correct approach is to cut the wires only if there is enough length remaining for resoldering. This ensures that the wires can be properly connected and avoids any potential damage or loss of connectivity.
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• 12.
### Perform an operational check of the unit
• A.
At the last minute
• B.
With the defective part removed
• C.
Before final assembly of the enclosure
• D.
As soon as you replace the defective part
C. Before final assembly of the enclosure
Explanation
Performing an operational check of the unit before final assembly of the enclosure ensures that the unit is functioning properly and all components are in working order. This step allows for any defects or issues to be identified and resolved before the final assembly, reducing the risk of having to disassemble the enclosure again. By removing the defective part before the operational check, it ensures that the check is accurate and any issues are not caused by the defective part.
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• 13.
### What controls the amount of fuel injected when the constant-stroke fuel injection pumps are used?
• A.
Injectors
• B.
Control rack
• C.
Delivery valve
• D.
Revolving disk
B. Control rack
Explanation
The control rack controls the amount of fuel injected when constant-stroke fuel injection pumps are used. The control rack is a mechanical device that regulates the movement of the fuel injectors, determining the amount of fuel that is delivered to the engine. By adjusting the position of the control rack, the operator can control the fuel flow and therefore the engine's power output. This makes the control rack an essential component in managing fuel injection in constant-stroke fuel injection systems.
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• 14.
### The constant-stroke pump plunger determines the amount of fuel delivered by the position of a
• A.
Bypass port
• B.
Vertical slot
• C.
Helical groove
• D.
Delivery valve
C. Helical groove
Explanation
The helical groove in the constant-stroke pump plunger determines the amount of fuel delivered. As the plunger moves up and down, the helical groove allows fuel to flow in and out, controlling the amount of fuel delivered to the engine. The depth and pitch of the groove determine the volume of fuel delivered per stroke. Therefore, the helical groove plays a crucial role in regulating the fuel delivery process in the constant-stroke pump.
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• 15.
### What provides the pressure to operate a pressure injector?
• A.
Fuel from the fuel system
• B.
Air from the induction system
• C.
Water from the cooling system
• D.
Oil from the lubricating system
A. Fuel from the fuel system
Explanation
The pressure to operate a pressure injector is provided by the fuel from the fuel system. Pressure injectors are used in various engines, such as gasoline and diesel engines, to deliver fuel into the combustion chamber at high pressure. This high pressure is necessary for efficient fuel atomization and combustion, resulting in optimal engine performance. Therefore, fuel from the fuel system is responsible for generating the required pressure to operate a pressure injector.
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• 16.
### Aside from idle speed, the limiting speed governor through the use of flyweights and spring tension controls
• A.
• B.
Minimum speed
• C.
Maximum speed
• D.
Intermediate speed
C. Maximum speed
Explanation
The limiting speed governor, using flyweights and spring tension controls, regulates the maximum speed of a system. It ensures that the system does not exceed a certain speed limit, preventing any potential damage or accidents that could occur at higher speeds. This governor is responsible for maintaining the system's stability and safety by keeping the speed within a predetermined maximum range.
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• 17.
### What is also reduced when a lubricant reduces friction?
• A.
Dirt and metal particles
• B.
Heat, wear, and resistance
• C.
Blow-by and loss of power
• D.
Dilution and increases cohesion
B. Heat, wear, and resistance
Explanation
When a lubricant reduces friction, it creates a barrier between two surfaces, which reduces the heat generated due to friction. Additionally, the lubricant helps to minimize the wear and tear on the surfaces, reducing the amount of wear. Lastly, by reducing friction, the lubricant also decreases the resistance encountered by the moving parts, allowing for smoother operation.
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• 18.
### Which function of a lubricant compensates for microscopic irregularities in the cylinder wall?
• A.
Cooling
• B.
Sealing
• C.
Cleaning
• D.
Reducing friction
B. Sealing
Explanation
The function of a lubricant that compensates for microscopic irregularities in the cylinder wall is sealing. When a lubricant is applied to the cylinder wall, it forms a thin film that fills in the microscopic gaps and imperfections on the surface. This helps to create a smooth and even surface, reducing friction and preventing metal-to-metal contact. By sealing these irregularities, the lubricant ensures proper lubrication and reduces wear and tear on the engine components.
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• 19.
### What component holds a large amount of coolant in close contact with a large volume of air?
• A.
• B.
Thermostat
• C.
Water pump
• D.
Explanation
The radiator holds a large amount of coolant in close contact with a large volume of air. As coolant flows through the radiator, it is cooled down by the air passing through the radiator fins. This process helps to dissipate heat from the coolant, preventing the engine from overheating. The radiator is designed to maximize the surface area in contact with the air, allowing for efficient heat transfer. Therefore, the radiator is the component that fulfills the given criteria.
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• 20.
### Which liquid cooling system component maintains positive pressure within the cooling system, causes the boiling point of the coolant to be higher, and permits the engine to operate at higher temperature without the coolant boiling?
• A.
Thermostat
• B.
Water pump
• C.
• D.
Vacuum valve
Explanation
The radiator cap is the correct answer because it is designed to maintain positive pressure within the cooling system. This pressure increase raises the boiling point of the coolant, allowing the engine to operate at higher temperatures without the coolant boiling. The radiator cap also acts as a seal to prevent coolant from escaping and allows excess pressure to be released through a pressure relief valve.
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• 21.
### The purpose of the cooling system thermostat is to maintain correct engine
• A.
Operating temperature
• B.
Coolant temperature
• C.
System pressure
• D.
Coolant flow
A. Operating temperature
Explanation
The purpose of the cooling system thermostat is to maintain the correct operating temperature of the engine. This is important because the engine operates most efficiently and effectively within a specific temperature range. The thermostat helps regulate the flow of coolant through the engine, allowing it to warm up quickly and then maintain a consistent temperature. This prevents the engine from overheating or running too cold, which can cause damage and reduce performance.
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• 22.
### What must be accomplished to supercharge a diesel engine?
• A.
Change the overlap and timing so that the intake and exhaust valves are not open at the same time
• B.
Change the overlap and timing so that the intake and exhaust valves are open at the same time
• C.
• D.
Retard intake valve opening time
B. Change the overlap and timing so that the intake and exhaust valves are open at the same time
Explanation
To supercharge a diesel engine, the overlap and timing of the intake and exhaust valves need to be changed so that they are open at the same time. This allows for better air intake and exhaust flow, increasing the efficiency and power of the engine. By having the intake and exhaust valves open simultaneously, more air can enter the combustion chamber, resulting in improved combustion and increased performance.
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• 23.
### The advantage of using a turbocharger on a diesel is the fuel consumption is
• A.
Decreased because of the increased internal operating temperature of the engine
• B.
Increased because of the better mixing of the fuel with the air charge
• C.
Used more efficiently and horsepower is increased
• D.
Used more efficiently and horsepower is decreased
C. Used more efficiently and horsepower is increased
Explanation
A turbocharger on a diesel engine increases the efficiency of fuel usage by compressing the air that enters the engine, allowing more fuel to be burned and producing more power. This increased efficiency leads to improved horsepower output, making the engine more powerful. Therefore, the correct answer is that the fuel is used more efficiently and horsepower is increased.
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• 24.
### The maintenance procedure that is considered to be the cheapest and most effective means of reducing injection equipment problems and maintenance costs is
• A.
Correct defective components only
• B.
Rigid maintenance schedules
• C.
Periodic overhaul schedule
• D.
Preventative maintenance
D. Preventative maintenance
Explanation
Preventative maintenance is the cheapest and most effective means of reducing injection equipment problems and maintenance costs because it involves regularly inspecting and servicing equipment to prevent breakdowns and identify potential issues before they become major problems. This proactive approach helps to avoid costly repairs and downtime, as well as extends the lifespan of the equipment. By regularly performing tasks such as cleaning, lubricating, and replacing worn parts, preventative maintenance ensures that the equipment operates at its optimal level and reduces the likelihood of unexpected failures.
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• 25.
### The filtration step protects the diesel engine fuel system from
• A.
Blockage of airflow
• B.
Lube system malfunctions
• C.
Slipping belts on the blower
• D.
Abrasion by foreign particles
D. Abrasion by foreign particles
Explanation
The filtration step in a diesel engine fuel system is designed to prevent the fuel system from being damaged by foreign particles. These particles can cause abrasion, which can lead to wear and tear on the engine components. By filtering out these particles, the fuel system is protected from potential damage and ensures the engine operates smoothly and efficiently.
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• 26.
### The first thing you should do when operating a diesel engine is to
• A.
Retorque the engine-mount bolts
• B.
Drain moisture from fuel system
• C.
Perform a preoperational check
• D.
Perform a periodic inspection
C. Perform a preoperational check
Explanation
Performing a preoperational check is the first thing you should do when operating a diesel engine. This check ensures that all components of the engine are in proper working condition before starting it. It includes inspecting the fuel system, checking for any leaks or damage, examining the engine oil level and quality, inspecting the cooling system, and verifying that all safety devices are functioning correctly. By performing this check, you can identify any potential issues or malfunctions and address them before starting the engine, ensuring safe and efficient operation.
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• 27.
### To locate a misfiring cylinder in the diesel engine, use a screw driver to
• A.
Hold the injector follower up or open
• B.
Hold the injector follower down or closed
• C.
Isolate the cylinder from the manifold
• D.
Remove the valve cover and align the piston ring slots
B. Hold the injector follower down or closed
Explanation
To locate a misfiring cylinder in a diesel engine, holding the injector follower down or closed with a screwdriver can be effective. By doing so, the fuel supply to that particular cylinder is cut off, causing the engine to run on the remaining cylinders. If the misfiring stops when the injector follower is held down, it indicates that the issue is likely with the fuel injector or its related components in that specific cylinder. This method helps isolate the problematic cylinder and assists in diagnosing and fixing the misfire.
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• 28.
### The component that reduces the turbine engine speed fro accessories is the
• A.
Gear case
• B.
Starter motor
• C.
Tachometer generator
• D.
Multiple centrifugal switch assembly
A. Gear case
Explanation
The gear case is the component that reduces the turbine engine speed for accessories. It is responsible for transmitting power from the engine to various accessories, such as generators, pumps, and fans, at the appropriate speed. By using gears and other mechanical components, the gear case ensures that the accessories operate at the correct speed for optimal performance and efficiency.
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• 29.
### During starting, the turbine engine is rotated up to 20 percent by the
• A.
Alternating current (AC) generator
• B.
Fuel pump and control unit
• C.
Tachometer generator
• D.
Starter motor
D. Starter motor
Explanation
The starter motor is responsible for rotating the turbine engine during starting. It provides the initial power needed to start the engine and get it running. The other options mentioned, such as the AC generator, fuel pump and control unit, and tachometer generator, are not directly involved in the starting process of the turbine engine. Therefore, the correct answer is the starter motor.
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• 30.
### If the drive shaft for the oil pump assembly in a gas turbine engine becomes sheared, what other component is rendered inoperative?
• A.
Alternating current (AC) generator
• B.
Tachometer generator
• C.
Starter motor
• D.
Cooling fan
B. Tachometer generator
Explanation
If the drive shaft for the oil pump assembly in a gas turbine engine becomes sheared, the tachometer generator is rendered inoperative. The tachometer generator is connected to the drive shaft and relies on its rotation to generate electrical signals that are used to measure the engine's speed. Therefore, if the drive shaft is sheared and no longer rotates, the tachometer generator cannot generate the necessary signals and becomes inoperative.
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• 31.
### During gas turbine engine operation, the air is accelerated by the
• A.
Diffusers
• B.
Exducers
• C.
Impellers
• D.
Deswirl ring
C. Impellers
Explanation
Impellers are rotating components in a gas turbine engine that accelerate the air. They consist of a series of blades or vanes that increase the velocity of the air as it passes through them. This increased velocity helps to generate the necessary thrust for the engine to operate efficiently. Diffusers, exducers, and deswirl rings are not directly involved in accelerating the air, making them incorrect options.
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• 32.
### The component of the gas turbine engine that causes the air velocity to be reduced, resulting in an increase in pressure, is the
• A.
Nozzle ring
• B.
Turbine wheel
• C.
Torus assembly
• D.
Divergent diffuser
D. Divergent diffuser
Explanation
The correct answer is divergent diffuser. A divergent diffuser is a component of a gas turbine engine that is responsible for reducing the air velocity and increasing the pressure. It is designed to gradually expand the flow area, allowing the air to slow down and convert its kinetic energy into pressure energy. This process helps to improve the efficiency of the engine by maximizing the pressure difference across the turbine.
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• 33.
### During operation of a turbine engine, fuel and air are mixed and ignited in the
• A.
Fuel atomizer
• B.
Combustor can
• C.
Mixer assembly
• D.
Plenum chamber
B. Combustor can
Explanation
During the operation of a turbine engine, fuel and air are mixed and ignited in the combustor can. The combustor can is specifically designed to contain the combustion process and allow for efficient fuel burning. It is responsible for providing the necessary conditions for the fuel and air mixture to ignite and produce high-temperature gases, which then flow through the turbine and generate thrust. The combustor can is a critical component in the engine, as it ensures the proper combustion of fuel and air, leading to the efficient operation of the turbine engine.
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• 34.
### The boost pump on the gas turbine engine provides fuel to the
• A.
Fuel atomizer
• B.
Main fuel pump
• C.
Combustion chamber
• D.
Fuel air mixture valve
B. Main fuel pump
Explanation
The boost pump on the gas turbine engine provides fuel to the main fuel pump. The main fuel pump is responsible for delivering fuel to the combustion chamber. Therefore, the boost pump plays a crucial role in ensuring that the main fuel pump receives an adequate supply of fuel for combustion.
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• 35.
### The two pressure controls in the fuel pump and control unit of the turbine engine are the
• A.
Governor and pneumatic control device
• B.
Acceleration limiter valve and governor
• C.
Fuel solenoid valve and pneumatic control device
• D.
Acceleration limiter valve and fuel solenoid valve
B. Acceleration limiter valve and governor
Explanation
The acceleration limiter valve and governor are the two pressure controls in the fuel pump and control unit of the turbine engine. The acceleration limiter valve regulates the rate at which fuel is delivered to the engine, limiting the acceleration to prevent sudden surges or drops in power. The governor, on the other hand, maintains a constant engine speed by adjusting the fuel flow according to the load demands. Together, these two controls ensure smooth and efficient operation of the turbine engine.
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• 36.
### Crack pressure is defined as the amount of
• A.
Air pressure that is required to overcome fuel pressure
• B.
Air pressure that is required overcome spring tension
• C.
Fuel pressure that is required to overcome spring tension
• D.
Fuel pressure that is required to overcome air and spring pressure
C. Fuel pressure that is required to overcome spring tension
Explanation
Crack pressure refers to the amount of pressure needed to overcome the spring tension in a system. In this case, the correct answer is "fuel pressure that is required to overcome spring tension." This means that the fuel pressure needs to be sufficient to overcome the tension of the spring in order for the system to function properly.
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• 37.
### Which turbine engine fuel system component is normally closed and is energized open when the engine builds up sufficient oil?
• A.
Governor
• B.
Oil shutoff valve
• C.
Boost pump valve
• D.
Fuel shutoff solenoid valve
D. Fuel shutoff solenoid valve
Explanation
The fuel shutoff solenoid valve is normally closed and is energized open when the engine builds up sufficient oil. This valve is responsible for controlling the flow of fuel to the engine. When the engine has enough oil pressure, the solenoid valve is opened, allowing fuel to flow into the engine. This ensures that the engine only receives fuel when it is ready to start and prevents any potential damage or accidents.
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• 38.
### What types of inputs are used to control the fuel solenoid for the delivery of fuel to the atomizer within the turbine engine fuel system?(418) What types of inputs are used to control the fuel solenoid for the delivery of fuel to the atomizer within the turbine engine fuel system?
• A.
Oil pressure and electrical
• B.
Oil pressure and cooling
• C.
Air and electrical
• D.
Air and cooling
D. Air and cooling
Explanation
The fuel solenoid in a turbine engine fuel system is controlled by inputs of air and cooling. These inputs are necessary to regulate the delivery of fuel to the atomizer. Air is required to mix with the fuel for combustion, while cooling is needed to prevent overheating of the fuel system components. Therefore, air and cooling are the types of inputs used to control the fuel solenoid in the turbine engine fuel system.
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• 39.
### The turbine engine component that prevents fuel from accumulating in the plenum is the
• A.
Atomizer
• B.
Drain valve
• C.
Pneumatic control device
• D.
Acceleration limiter valve
B. Drain valve
Explanation
The drain valve is the turbine engine component that prevents fuel from accumulating in the plenum. The plenum is a chamber where air and fuel mix before entering the combustion chamber. If fuel were to accumulate in the plenum, it could cause a rich mixture and potentially lead to engine damage or failure. The drain valve is responsible for removing any excess fuel or other fluids from the plenum, ensuring proper operation and preventing potential hazards.
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• 40.
### Lubrication system pressure in the turbine engine is regulated through a
• A.
Flow check valve
• B.
Filter bypass valve
• C.
Pressure relief valve
• D.
Temperature regulator
B. Filter bypass valve
Explanation
The correct answer is the Filter bypass valve. In a turbine engine, the lubrication system pressure needs to be regulated to ensure proper lubrication of the engine components. The filter bypass valve is responsible for regulating the pressure by bypassing the oil filter when the pressure exceeds a certain limit. This prevents damage to the engine and ensures continuous lubrication even if the filter becomes clogged or blocked.
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• 41.
### Gas turbine compressor (GTC) operation with insufficient oil pressure is prevented by the
• A.
Oil pressure solenoid
• B.
Oil pressure switch
• C.
Oil drain switch
• D.
Ignition coil
B. Oil pressure switch
Explanation
The oil pressure switch is responsible for preventing gas turbine compressor (GTC) operation with insufficient oil pressure. This switch acts as a safety mechanism by monitoring the oil pressure and shutting down the GTC if the pressure falls below the required level. By doing so, it ensures that the GTC is not operated without sufficient lubrication, which could lead to damage or failure of the compressor.
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• 42.
### The load section of the load control valve consists of chambers
• A.
1 and 2
• B.
1 and 4
• C.
2 and 3
• D.
3 and 4
A. 1 and 2
Explanation
The load section of the load control valve consists of chambers 1 and 2.
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• 43.
### The time required to open the load control valve on the turbine engine is controlled by the
• A.
• B.
Actuator diaphragm
• C.
• D.
Actuator return spring
Explanation
The rate adjustment screw is responsible for controlling the time required to open the load control valve on the turbine engine. This screw allows for fine-tuning of the opening rate, ensuring that the valve opens at the desired speed. By adjusting the rate, the engine can effectively manage the flow of air or fuel, optimizing performance and efficiency. The mechanical linkage, actuator diaphragm, and actuator return spring may play a role in the overall operation of the load control valve, but they do not directly control the time required to open it.
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• 44.
### If the gas turbine compressor (GTC) load control butterfly valve opens to slowly or too quickly, what is the most probable cause of the malfunction?
• A.
Excessive exhaust gas temperature
• B.
• C.
Ruptured actuator diaphragm
• D.
Defective relief valve
Explanation
If the gas turbine compressor (GTC) load control butterfly valve opens too slowly or too quickly, the most probable cause of the malfunction is a maladjusted rate adjustment screw. The rate adjustment screw is responsible for controlling the speed at which the butterfly valve opens and closes. If it is not properly adjusted, it can result in incorrect or inconsistent valve movement, leading to the mentioned malfunction.
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• 45.
### In the turbine engine, what prevents rupture of the rate diaphragm?
• A.
• B.
Actuator regulator
• C.
Rate metering valve
• D.
Rate diaphragm return spring
Explanation
The adjustment screw in the turbine engine prevents rupture of the rate diaphragm. It is responsible for controlling the flow of fuel, ensuring that it is at the correct rate. By adjusting the screw, the operator can regulate the fuel flow and prevent excessive pressure or stress on the rate diaphragm, thus avoiding rupture or damage.
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• 46.
### During a preoperational inspection of a gas turbine engine, you check the intake and exhaust for
• A.
Leakage
• B.
Excess heat
• C.
Hairline cracks
• D.
Foreign objects
D. Foreign objects
Explanation
During a preoperational inspection of a gas turbine engine, checking for foreign objects is important because any debris or objects that enter the engine can cause damage to the internal components. Foreign objects can obstruct the airflow, leading to reduced engine performance or even complete engine failure. Therefore, it is crucial to ensure that the intake and exhaust are free from any foreign objects to maintain the engine's efficiency and prevent potential damage.
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• 47.
### Normal acceleration characteristics of the gas turbine compressor (GTC) include
• A.
Sudden rumbling noises after 35 percent
• B.
A decrease in vibration after 35 percent
• C.
Smooth and quiet acceleration
• D.
Burping
B. A decrease in vibration after 35 percent
Explanation
The correct answer is a decrease in vibration after 35 percent. This means that the gas turbine compressor (GTC) experiences a decrease in vibrations as it accelerates, specifically after it reaches 35 percent of its maximum speed. This characteristic indicates that the GTC is operating smoothly and quietly during acceleration.
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• 48.
### Common or frequently occurring troubles in a unit, their cause, and remedies may be found in the
• A.
Historical record
• B.
Maintenance record
• C.
Troubleshooting chart
• D.
Illustrated parts breakdown
C. Troubleshooting chart
Explanation
A troubleshooting chart is a resource that provides information on common or frequently occurring troubles in a unit, their cause, and remedies. It is a useful tool for identifying and resolving issues in a systematic manner. The historical record may provide information on past issues, but it may not have the specific cause and remedy details needed. The maintenance record is focused on documenting maintenance activities rather than troubleshooting. The illustrated parts breakdown is a visual representation of the unit's components and may not provide troubleshooting information. Therefore, the best option for finding information on common troubles, their cause, and remedies is the troubleshooting chart.
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• 49.
### If no oil pressure is available from the main oil pump assembly on the gas turbine compressor (GTC), what should you replace?
• A.
Oil pump
• B.
Entire assembly
• C.
Flow check valve
• D.
Filter bypass relief valve
B. Entire assembly
Explanation
If no oil pressure is available from the main oil pump assembly on the gas turbine compressor (GTC), the most appropriate course of action would be to replace the entire assembly. This is because if there is no oil pressure, it indicates a failure or malfunction in the pump assembly as a whole, rather than a specific component such as the flow check valve or filter bypass relief valve. By replacing the entire assembly, it ensures that all components are functioning properly and can restore the necessary oil pressure for the GTC.
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• 50.
### Which causes gray or white smoke to pour from the exhaust stack on the gas turbine compressor (GTC)?
• A.
Excessive engine revolutions per minute (rpm)
• B.
Excessive engine exhaust temperature
• C.
Oil that enters the airflow system
• D.
Fuel mixing with the airflow
C. Oil that enters the airflow system
Explanation
When oil enters the airflow system of a gas turbine compressor (GTC), it causes gray or white smoke to pour from the exhaust stack. This occurs because the oil mixes with the airflow and is burned during the combustion process. As a result, the smoke is emitted through the exhaust, indicating the presence of oil in the system. Excessive engine revolutions per minute (rpm) and engine exhaust temperature may cause other issues, but they do not directly cause smoke to pour from the exhaust stack.
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# Data Representation / Numbering Conversion / File Permissions
### Main Objectives of this Practice Tutorial
• Understand the importance of how computers store data (i.e. data representation)
• Understand the purpose of decimal, binary, octal and hexadecimal numbers
• Perform various numbering conversions between the decimal, binary, octal and hexadecimal numbering systems
by hand without the use of a computer or calculator
• Identify which numbering system conversion method to use when required to perform a numbering conversion
• Understand directory and regular file permissions
• Learn how to set directory and regular file permissions with the chmod command (symbolic and octal methods)
• Learn how to use the umask command to have permissions for directories and files automatically set upon their creation
### Tutorial Reference Material
Course Notes Numbering Conversion / File Permissions Reference YouTube Videos Course Notes:PDF | PPTX Data Representation Definition Decimal, Binary, Octal, Hexadecimal Numbers Numbering Conversion Binary to Decimal / Decimal to Binary Binary to Octal / Octal to Binary Binary to Hexadecimal / Hexadecimal to Binary Octal to Hexadecimal / Hexadecimal to Octal File Permissions Instructional Videos:x x x x
### Why Study Data Representation?
A series of binary numbers form a byte to represent numbers. Bytes can be used to also represent characters. It is job of a program to know if bytes are used to represent numbers or characters. Learning to convert numbering systems(like Hexadecimal to Binary) can be used to know how a character is represented in binary.
Data (treated as singular, plural, or as a mass noun) is any sequence of one or more symbols given meaning by specific act(s) of interpretation. Digital data is data that is represented using the binary number system of ones (1) and zeros (0), as opposed to analog representation.
Reference: https://en.wikipedia.org/wiki/Data_(computing)
Therefore, computers process and store information in a binary number system consisting of 0s and 1s. For many aspects of programming and networking, the details of data representation must be understood.
Reasons to Understand Data Representation:
• C Programming: Sending information over networks, files
• Web Development: Setting color codes for webpage background or text
• Allowing or Limiting Unix / Linux File Access: Setting permissions for files and directories
In terms of this course, we will learn how a simple decimal number (integer) is stored into the computer system as a binary number. We will also learn other numbering systems (octal and hexadecimal) that can be used as a "short-cut" to represent binary numbers.
### Decimal / Binary / Octal / Hexadecimal Numbering Systems
File:Decimal-number-1.png
The decimal numbering system is a numbering system where each digit can be represented by numbers 0 to 9'. The numbering system is based on sums of the power of 10'Bold text.
The decimal numbering system is a numbering system where each digit can be represented by numbers 0 - 9. The reason for this system may be attributed to the fact that humans were used to counting on their fingers and thumbs.
The numbering system is based on sums of the power of 10.
According to the diagram to the right, each digit moving to the left of units value is the placeholder multiplied to the power of ten. Units are ten to the power of zero (which is 1), tens are ten to the power of one, etc.
x
x
x.
x.
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<
## The Cambridge Positioning System
A cache by mjouk Message this owner
Hidden : 02/26/2014
Difficulty:
Terrain:
Size: (small)
#### Watch
How Geocaching Works
Please note Use of geocaching.com services is subject to the terms and conditions in our disclaimer.
### Geocache Description:
A puzzle inspired by the mathematics used in the GPS.
## Caveat Cacher
The coordinates above are convenient for parking, but there's nothing to see there, except a handy car park.
## Introduction
The GPS is almost miraculous. It allows us to work out where we are by measuring the time it takes for a radio signal to travel from a network of satellites to our receiver. Signals from nearer satellites will arrive more quickly, so if we know where the satellites are we can work out where we are. As a bonus, we can also work out what the time is.
Happily our GPSr does all the maths for us, but to find this cache, you'll have to do a similar calculation yourself.
## The Puzzle
Rather than satellites orbiting the Earth, imagine a cyclist travelling east along an East-West path. When he starts cycling he briefly rings his (very loud) bell; when he reaches the middle of his he rings it again; when he gets to the end he rings it a final time.
Our cyclist takes three minutes to cover the 1.2km track and so rings:
• At 11:58:30.00 when he's at TL 38839 53670;
• at noon when he's at TL 39439 53670;
• at 12:01:30.00 when he's at TL 40039 53670.
These coordinates are close to the One-Mile Radio Telescope at Lord's Bridge whose track is almost perfectly East-West (to within 1cm in 1 mile).
If you were sitting on the cache armed only with a stopwatch you could measure the intervals between the rings: 89.20s and 89.40s respectively.
Your task is to work out where you are, and exactly when you heard the middle ring.
You should:
• Do all your calculations using OSGB36 National Grid coordinates.
• Assume that the speed of sound is 300ms-1.
• Assume that the cache is north of the track.
The puzzle can be solved with nothing more than pen and paper, but you'll probably find a calculator makes it easier.
## Ordnance Survey (OSGB) Coordinates
The Earth's surface is curved which makes it difficult to do calculations. Things are even more complicated if we specify our position with angles of latitude and longitude. Happily the Ordnance Survey invented a set of coordinates which make things easier.
If you use six-figure references for the Easting and Northing, each unit corresponds to 1m on the ground. It might be useful to know that TL eeeee nnnnn corresponds to 5eeeee 2nnnnn.
You can use the normal Cartesian formula to calculate (approximate) distances.
This approximation isn't perfect: for example it means that a line of constant Northings isn't a line of constant latitude. Our cyclist's track is the former; the One-Mile's track the latter.
Converting between OSGB coordinates, and latitude and longitude is fiddly and some online converters give different answers. To avoid such problems I suggest you use the OS's own converter.
## Acknowledgements
Thank you to Cambridge Past, Present & Future for their help in setting up this cache.
## Congratulations!
...to crb11 for being first to find it.
Frr gur trbpurpxre.
Decryption Key
A|B|C|D|E|F|G|H|I|J|K|L|M
-------------------------
N|O|P|Q|R|S|T|U|V|W|X|Y|Z
(letter above equals below, and vice versa)
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QUESTION
# A certain echocardiogram uses ultrasounds with a frequency f = 6.1130 MHz. The speed of sound in blood is 1570 m/s.
A certain echocardiogram uses ultrasounds with a frequency f = 6.1130 MHz. The speed of sound in blood is 1570 m/s. The patient is aligned so that blood is flowing through the heart directly towards the transducer (the transducer produces and detects the ultrasound waves).
1. If blood flows through the artery being measured at a speed of 39.79 cm/s, what frequency signal would be detected by a red blood cell in the artery? Give your answer to four decimal places.
fdetected = _______MHz
2.What is the frequency of the signal that is returned to the transducer after being reflected from the red blood cell? Give your answer to four decimal places.
freflected = _______ MHz
3. What is the wavelength of the ultrasound wave travelling through the blood away from the transducer?
λ = _____ m
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https://stats.stackexchange.com/questions/218467/help-interpreting-formula-for-multi-class-hinge-loss?noredirect=1
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# Help interpreting formula for multi-class hinge loss
As I'm reading from wikipedia, and this Cross Validated question: Gradient for hinge loss multiclass, the gradient value for a training feature set is somewhat straightforward. However if I'm interpreting this correctly, this only gives a gradient value for the weight of the 'true' class. How do I find the gradients for the other weights? That is, if my total weight vector is [W1, W2, ... Wy... Wk], where y is the class of the training sample then what are the gradient/loss values for every weight that isn't Wy?
Since the classifier for the $j$ - th class is given by the row $j$ of $W$ (Which is notated at the answer as ${W}_{j}$) all you need on each iteration is to update ${W}_{j}$ according to ${\nabla}_{{W}_{j}} {L}_{i}$ according to train sample ${x}_{i}$.
Namely, ${W}_{j}^{\left( k + 1 \right)} = {W}_{j}^{\left( k \right)} - \eta {\nabla}_{ {W}_{j} } {L}_{i}$, Where $j$ is the index of the updated row, $k$ is the iteration counter and $\eta$ is the Step Size / Learning Rate..
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# Rank of the given Matrix
• Jan 21st 2010, 11:55 PM
Chandru1
Rank of the given Matrix
Let $\bf{x}$ and $\bf{y} \in \mathbb{R}^{n}$ be 2 non zero column vectors. Let $\bf{y}^{T}$ denote the transpose of $\bf{y}$. Let $A=\bf{x} \bf{y}^{T}$. What is the rank of A?
• Jan 22nd 2010, 01:16 AM
Roam
What do you mean by $y^T$, because we don't know what "y" is! I think maybe you're referring to the case where x=y. And I think in this case $xx^T$ will have the full column rank if $det(x^Tx) \neq 0$, and it will have the full row rank if $det(xx^T) \neq 0$.
Since $x^Tx$ is square, and it is an nxn matrix, it follows that $x^Tx$ is invertible if and only if $x^Tx$ has rank n. However $x^Tx$has the same rank as x itself, so $x^Tx$ is invertible if rank(x)=n (this means if x has full column rank).
Again I'm assuming x=y.
• Jan 22nd 2010, 04:13 AM
HallsofIvy
I see no reason to assume that x= y. The problem makes sense without that. For example, if $x= \begin{bmatrix}a \\ b\end{bmatrix}$ and $y= \begin{bmatrix}c \\ d\end{bmatrix}$, then $y^T= \begin{bmatrix}c & d\end{bmatrix}$ and
$xy^T= \begin{bmatrix}a \\ b\end{bmatrix}\begin{bmatrix}c & d\end{bmatrix}=$ $\begin{bmatrix}ac & ad \\ bc & bd\end{bmatrix}$. The determinant of that is is acbd- adbc= 0 so it does NOT have rank 2. It is not, unless at least one of x and y is the 0 vector, all "0"s so it does not have rank 0. It has rank 1.
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https://www.gradesaver.com/textbooks/math/other-math/basic-college-mathematics-9th-edition/chapter-4-decimals-4-2-rounding-decimal-numbers-4-2-exercises-page-280/43
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## Basic College Mathematics (9th Edition)
Published by Pearson
# Chapter 4 - Decimals - 4.2 Rounding Decimal Numbers - 4.2 Exercises: 43
#### Answer
\$0.0015 rounds to 0 cents. This happens because the number following the hundredths place is less than 5.
#### Work Step by Step
1. Find the decimal place of cents > hundredths place 2. Check if the number following the hundredths place is less than 5 or greater than or equal to 5. Since the number is 1, it is less than 5 and we round the hundredths place down. 3. 0 rounded down is 0 4. Conclude 0 cents
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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# Karma
Karma is a Turing tarpit created by revcompgeek that was designed to have no explicit loops. Instead it has lines that can be jumped between, acting somewhat like functions. It has one stack and one deque as its only data structures. The stack and deque hold bytes and are unbounded (interpreter is limited by memory).
## Information
Karma's commands each consist of a single character. The program ends when the interpreter runs out of characters to execute. Errors are caused whenever an unknown command is encountered or when a command tries to move up or down out of the range of the program.
## Syntax
### Math
For these commands, first is the top of the stack, second is the value underneath
```+ - pop two values, push first + second
- - pop two values, push first - second
* - pop two values, push first * second
/ - pop two values, push first / second (integer division)
% - pop two values, push first % second (modulus)
& - pop two values, push first & second (bitwise AND)
| - pop two values, push first | second (bitwise OR)
^ - pop two values, push first ^ second (bitwise XOR)
~ - bitwise NOT the top of the stack
! - logical NOT the top of the stack
```
### Logic
Each of these pops the top of the stack
```= - push 1 if top of stack equals front of deque, else push 0
> - push 1 if top of stack is greater than front of deque, else push 0
@ - skip the next command if top of stack is not 1
```
### Stack/Deque
```digit - push digit onto stack
} - pop, insert front
{ - remove from front, push
[ - pop, insert back
] - remove from back, push
# - destroy top of stack
\ - clone the top of the stack
```
### IO
```? - input and push char
: - pop and output char
; - pop and output as number
```
### Jumping
```, - move down and begin at beginning of the line
. - move down and continue where it last left off
' - move back up and return where it last left off
< - jump back to the beginning of the current line
```
## Examples
Comments are not allowed, but any character not encountered by the interpreter will be ignored. This means that any line can begin with ,' and it will be effectively ignored anywhere in the program. Also, if the last instruction in a line is ' then you can safely put anything after it (as long as it isn't after a @)
```Example:
, This line will be ignored
12+;
The last line will end and therefore so will the program, this is never executed.
```
Programs can be designed so that lines can be called like functions. The following is a working example of this idea:
```, This program will "call" the numbered function below it
1},5},2},5},4},5},3},
1=!\@,@'{#1;1' (1)
2=!\@,@'{#456**:1' (x)
3=!\@,@'{#81;;1' (18)
4=!\@,@'{#855+*:1' (P)
5=!\@,@'{#55+:1' (newline)
```
In this program, =!\@,@'{# is what separates the function number with the code, and }, will call whatever function is on top of the stack.
## Turing completeness
Proof by emulation of Bitwise Cyclic Tag (actual code untested but principle is sound):
```,Data string in order with 2 and 5 delimiter:
1[1[1[0[0[1[1[0[1[0[2[5,
,Command string in reverse order with 4 delimiter:
40101110101,
,Following is the actual BCT interpreter:
4}\=!@,6,
{#,
,<
0}\>\@,@'{#{#['
#{#1}\>\@,@'[\[[311,1'
#{#1}\>\@,@'{#{<
#{#2}\>\@,@']}1,1'
#{#2}\>\@,@'{#}<
#{#3}\>\@,@'{##{1'
#{#4}\>\@,@']<
#{#5}\>@,[1111111'
1
```
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Main Page | See live article | Alphabetical index
The ohm is the SI unit of electrical resistance. Its symbol is the Greek capital letter omega (Ω). The ohm is named for Georg Ohm, a German physicist who discovered the relation between voltage and current, expressed in Ohm's Law.
By definition in Ohm's Law, 1 ohm equals 1 volt divided by 1 ampere. In other words, a device has a resistance of 1 ohm if a voltage of 1 volt will cause a current of 1 ampere to flow.
A thousand ohms is called a kilohm (not kilo-ohm). A million ohms is called a megohm (not mega-ohm).
A measurement in ohms is the reciprocal of a measurement in siemens (also called mho), the SI unit of electrical conductance. Note that 'siemens' is both singular and plural.
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https://de.mathworks.com/matlabcentral/fileexchange/46872-intuitive-rgb-color-values-from-xkcd
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Intuitive RGB color values from XKCD
Version 1.5.0.0 (53,8 KB) von
RGB triplets for 949 colors, by how they're perceived on a computer monitor and their common names.
Aktualisiert 12. Jun 2017
Lizenz anzeigen
This function returns the RGB triplet for almost any color. And unlike the way colors are somewhat officially defined (http://en.wikipedia.org/wiki/Web_colors), this data set describes returns values for the ways that colors are actually perceived on computer monitors. Color names are intuitive because they are based on a survey of over 200,000 user sessions, wherein respondents were shown colors and asked to fill in their word for that color. You won't find "puke green" on Wikipedia's list of web colors, despite the fact that it's a color name that people use and understand.
Data come from an XKCD survey described here: http://blog.xkcd.com/2010/05/03/color-survey-results/
A chart of available colors and their most common names can be found here: http://xkcd.com/color/rgb/
Syntax
RGB = rgb('Color Name')
RGB = rgb('Color Name 1','Color Name 2',...,'Color Name N')
RGB = rgb({'Color Name 1','Color Name 2',...,'Color Name N'})
Description
RGB = rgb('Color Name') returns the RGB triplet for a color described by 'Color Name'.
RGB = rgb('Color Name 1','Color Name 2',...,'Color Name N') returns an N by 3 matrix containing RGB triplets for each color name.
RGB = rgb({'Color Name 1','Color Name 2',...,'Color Name N'}) accepts list of color names as a character array.
Zitieren als
Chad Greene (2024). Intuitive RGB color values from XKCD (https://www.mathworks.com/matlabcentral/fileexchange/46872-intuitive-rgb-color-values-from-xkcd), MATLAB Central File Exchange. Abgerufen .
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Start Hunting!
XKCD_RGB/html/
Version Veröffentlicht Versionshinweise
1.5.0.0
The rgb data are now included in the zip file.
1.4.0.0
Fixed a bug in the installation script. Installation is now performed fully within the rgb function.
1.3.0.0
Included hex2rgb function and improved error handling.
1.2.0.0
Now supports multiple inputs and offers help for misspelled inputs.
1.1.0.0
Link to hex2rgb, rgb2hex.
1.0.0.0
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How copying an int made my code 11 times faster 168 points by tdurden on Feb 19, 2017 | hide | past | web | favorite | 51 comments
This reminds me of this classic StackOverflow question:Why is it faster to process a sorted array than an unsorted array?
I wrote up a small script to test this out for python (at bottom of this post). The results on my laptop are about 8.0 seconds for unsorted and 7.6 seconds for the sorted version. I'm assuming that the discrepancy for python is much smaller due to the slow nature and high overhead of the language (or at least the way I've used it here), but I would be interested to know: how would one go about finding out what the python interpreter is doing beneath the surface?Edit: After running with a wider range of parameters, it seems that the difference is always roughly the same order of magnitude. To investigate further, I included the sort into the second timing to double check and for 3276800 elements it's still a bit faster overall when you sort the array.`````` #!/usr/bin/env python import time import numpy as np def main(n=32768): arr = np.random.randint(0, 256, n) t0 = time.time() sum1 = do_loop(arr) t1 = time.time() arr = np.sort(arr) t2 = time.time() sum2 = do_loop(arr) t3 = time.time() assert sum1 == sum2 print(" Unsorted execution time: {} seconds".format(t1-t0)) print(" Sorted execution time: {} seconds".format(t3-t2)) def run_many(func): def wrapper(arg): for t in range(1000): func(arg) return func(arg) return wrapper @run_many def do_loop(arr): tot = 0 for i in arr: if i >= 128: tot += i return tot if __name__ == '__main__': main()``````
I tried this on my machine, then tried a pure Python version; I only changed three lines, to:`````` import random ... arr = [random.randint(0, 256) for x in range(n)] ... arr = sorted(arr) `````` Here are my times:`````` \$ time python2.7 hackernews_13682929.py Unsorted execution time: 4.33348608017 seconds Sorted execution time: 4.09405398369 seconds \$ time python3.5 hackernews_13682929.py Unsorted execution time: 4.4200146198272705 seconds Sorted execution time: 4.188237905502319 seconds \$ time python2.7 hackernews_13682929_purepython.py Unsorted execution time: 0.981621026993 seconds Sorted execution time: 0.832424879074 seconds \$ time python3.5 hackernews_13682929_purepython.py Unsorted execution time: 1.3005650043487549 seconds Sorted execution time: 1.157465934753418 seconds \$ time pypy hackernews_13682929_purepython.py Unsorted execution time: 0.239459037781 seconds Sorted execution time: 0.0910339355469 seconds `````` As you can see, the pure Python version is faster than the Numpy version, and also has a larger margin between unsorted and sorted. PyPy is of course faster than both, and also has an even greater margin between unsorted and sorted (2.63x faster).
Good call on going pure python. To take this a bit further I made your changes and used numba with @jit(nopython=True, cache=True), for some interesting results. If I do include the sorting into the timing:`````` Unsorted execution time: 0.2175428867340088 seconds Sorted execution time: 1.133354663848877 seconds `````` And if I don't:`````` Unsorted execution time: 0.21171283721923828 seconds Sorted execution time: 0.08376479148864746 seconds``````
Using python for micro benchmarks usually doesnt work. The mere fact you use python says you dont care about performance, but convenience.Just for fun add "randomvariable = 1"(you know, 1 one cycle machine op) in a tight loop and watch tenths of a second added to your "benchmark".
I'm not quite sure what you mean by "using python for micro benchmarks usually doesn't work". To the extent that "microbenchmarking" is useful at all, then sure it works. There are just different considerations from other languages, and it depends on which Python implementation and libraries you're using.Also, while I'll grant you that using Python implies that convenience, programmer time, and/or development speed is a higher priority than performance, that doesn't at all mean that people who use Python "don't care about performance".
How is this article related to branch prediction? I don't see any connection here.
I read it a while ago, and going on what I remember, I still can't see the connection. Would you mind explaining?
I don't think it's related to branch prediction in particular, just spooky action-at-a-distance where a change makes seemingly unrelated code much slower or faster.
What makes you say this? The author provides an extremely persuasive case, with perf timings. Additionally updated it with compiler enhancements from GCC and Intel that remove the branch mispredictions entirely and do perform as predicted.
I think you misunderstood me. I agree that the StackOverflow post has to do with branch prediction. I just meant that I don't think that's why the earlier poster thinks it's parallel to the situation described in the article.
Ah, my mistake. I thought the OP a bit further up didn't understand the relation of the stack overflow answer to the array question.
This is really surprising. I thought that part of Rust's pitch was that the explicit ownership tracking made optimizations much easier.Is there a bug filed to fix this?
It does make optimization easier but until MIR landed, many of the best optimizations weren't really possible. The problem is that a lot of type information is lost between Rust and LLVM IR, where the compiler does the really serious optimizations. For example, Rust can't tell LLVM about its pointer aliasing guarantees (immutable and mutable borrows can't be done at the same time without unsafe) so a lot of optimizations like storing a highly used value from the heap in a register are passed over because of conservative heuristics.Now that MIR has landed, Rust will eventually get much better optimizations from both rustc and the llvm optimization passes but other things are a much higher priority like non-lexical scoping.
> For example, Rust can't tell LLVM about its pointer aliasing guaranteesFalse.
From what I understand, LLVM still doesn't take as much advantage of that information as it could, given Rust input. It's too geared toward the C family of languages. (But as sibling comment says, the problem was partially the Rust compiler's fault.)
C has a pointer aliasing keyword, "restrict".Also: "Originally implemented for C and C++, the language-agnostic design of LLVM has since spawned a wide variety of front ends: languages with compilers that use LLVM include ActionScript, Ada, C#,[4][5][6] Common Lisp, Crystal, D, Delphi, Fortran, OpenGL Shading Language, Halide, Haskell, Java bytecode, Julia, Lua, Objective-C, Pony,[7] Python, R, Ruby, Rust, CUDA, Scala,[8] and Swift."Most of these are nothing like C when it comes to pointers and memory layout.
Again, from what I understand, `restrict` is not enough to convey everything Rust knows.Further, those other languages may be nothing like C, but they're even less like Rust. So because of its heritage, even given the information Rust knows, LLVM simply doesn't have the optimization passes to take full advantage of it.
Ok. Like what ?
The point was that Rust _can_ (more easily) tell LLVM now that MIR has landed.
It's not only a MIR issue, it's also an issue of how many optimizations should be allowed when you also need to be able to make some assumptions about your data in unsafe blocks. For example if you simply added the obvious attributes everywhere, interior mutability (Cell, RefCell, UnsafeCell) would most certainly break. The exact rules around unsafe rules are still being discussed (see https://github.com/nikomatsakis/rust-memory-model), so until that is done some of these optimizations are very unlikely to be implemented because they can make a lot of unsafe code illegal.
Note that the compiler already adds "obvious" attributes to function calls, being careful to not emit them for things that contain an `UnsafeCell`. This is why that type exists, as the building block for interior mutability that the compiler understand and can optimise around.
I guess having taken a compilers class a decade ago has given me a Dunning-Kruger sense of optimism but the explanation sounds kind of like bullshit to me. So what if LLVM doesn't have the type information? There's no type information at all in the machine codeā¦ optimize stuff yourself before you hand it to LLVM?
LLVM is literally designed as an production quality optimisation/code-generation backend, so it's seems perfectly reasonable for projects to rely on LLVM's optimisations without doing their own, especially when there are other things people can work on than the large effort to optimise some programs slightly better (and even fewer programs a lot better, but these are often microbenchmarks, or tight loops like the OP that can be diagnosed with a profiler relatively easily and rewritten) by doing language-specific optimisations. It's definitely a good long-term goal, but it requires a lot of infrastructure, and ends up requiring duplicating a lot of the optimisations and analysis already in LLVM (e.g. inlining is a critical transformation for enabling other optimisations) before one can get great results from the custom optimisations.In any case, compiler optimisations are usually easy to implement on specific styles of internal representations, and this is MIR for Rust, which was only introduced recently, after requiring a very large refactoring of the compiler's internal pipeline.
As I understand it, this is partially the plan. Historically it has just been waiting for a compiler refactor to MIR (which is now complete). I'm not sure why more optimization RFCs haven't been created and prioritized.
I think it's kinda interesting what tiny little tweaks will affect code speed- I recently discovered that in python, if you're only going to use an import for one or two functions, importing it locally shaves off a good bit of time depending on the function, in my case, .2 seconds!
I don't think the function itself matters; what's going on is that if you import it locally, the reference you're using is in the local namespace versus being in a module's namespace, which means it takes less time to get to the function object. I'm guessing the function is being called a fair number of times in a loop or some such similar manner?
You could use `from some_module import some_func` to get the same effect as some_func will be in the local namespace.
That's what the top-level commenter was doing.
Ok. It wasn't clear what "importing it locally" meant. I've worked with some people who would use that phrase to mean "copy paste it into the local file".
Fair point :P. I think I might have seen someone mean it like that as well in the past.
And a common stdlib trick is to bind these via function arguments e.g.`````` def foo(a, b, thing=util.thing): ...``````
I agree, seems likely. gp can use dis.dis() to confirm this...
Python is notable for how it harasses the filesystem. If your python program is running off NFS or parallel filesystems (e.g. gpfs, panasas, lustre, etc) then you may get a lot more performance from avoiding imports. Especially if pkg_resource is being used.
`````` like all generic interfaces in Rust, printing takes arguments by reference, regardless of whether they are Copy or not. The println! macro hides this from you by implicitly borrowing the arguments, `````` What's the reason for this? Seems a bit inconsistent that for functions you must explicitly pass with &, but for print it's automatic.
The reason I can think of are:- it'd be annoying to have to write `println!("{}", &s);`,- this is not an inconsistency one thinks about in practice: ownership is important in Rust, but the compiler does all the annoying checking, so IME one can just let it fade into the background and not think about it until the compiler tells you there's a problem,- it's an "permissive" inconsistency: writing a & will still compile and give the same output,- it's a syntax-level inconsistency created by the println! macro packaging up the normal syntax, not some special rules for special functions: there's still a literal & borrow in there (see the macro expansion),- historical reasons, and no-one was annoyed enough (or even noticed it enough) to change it.
Hmm. If people would think it was annoying to println!("{}", &s) they wouldnt use rust in the first place because you have to do that everywhere else for plain functions.Beeing permissive makes it even worse as there are now 2 inconsistent ways to do the same thing.Maybe I need to understand rust macros better. I guess this all boils down to: macros are a very bad idea, as in other programming languages.
And if print can already have a special treatment, it should surely be the one where the value is copied if can fit the basic type size, that is, at least for int like and float like stuff not have automatic pass by reference but automatic pass by value to it?
The problem with this is print is a macro and that macros are expanded before the type checker. This makes it impossible to know if something can be copied or not, because this is type information.
I believe it could be a solvable problem? Imagine that the macro expands every parameter x to some language construct f( x ) where f( x ) has access to the type of x and the type of its result is different depending of the type of x, returning a value for for int and float-like variables, and the reference for others.
Since LLVM does not have the necessary information to do the optimizations, I wonder whether the same problem occurs in C++ code compiled with clang.
In this particular case, you'd likely be using printf or cout, both of which perform a copy of integer parameters.So you wouldn't see this there.
It may not. It's possible that this may be happening because rustc is failing to mark the reference as both readonly and unaliased, whereas clang might mark the equivalent as both. You'd need to examine the IR from both front ends to really figure out the source of any differences in the generated ASM
Both Rust playgrounds (official[1], my take[2]) allow viewing the LLVM IR of a Rust program.The call to the print is`````` call void @_ZN3std2io5stdio6_print17h690779b3bd8114d5E(%"core::fmt::Arguments"* noalias nocapture nonnull dereferenceable(48) %_3) `````` The entire chunk of `main` preceding that call:`````` %size = alloca i64, align 8 %_3 = alloca %"core::fmt::Arguments", align 8 %_8 = alloca [1 x %"core::fmt::ArgumentV1"], align 8 %0 = bitcast i64* %size to i8* call void @llvm.lifetime.start(i64 8, i8* %0) store i64 33554432, i64* %size, align 8 %1 = bitcast %"core::fmt::Arguments"* %_3 to i8* call void @llvm.lifetime.start(i64 48, i8* %1) %2 = bitcast [1 x %"core::fmt::ArgumentV1"]* %_8 to i8* call void @llvm.lifetime.start(i64 16, i8* %2) %3 = ptrtoint i64* %size to i64 %4 = bitcast [1 x %"core::fmt::ArgumentV1"]* %_8 to i64* store i64 %3, i64* %4, align 8 %5 = getelementptr inbounds [1 x %"core::fmt::ArgumentV1"], [1 x %"core::fmt::ArgumentV1"]* %_8, i64 0, i64 0, i32 1 %6 = bitcast i8 (%"core::fmt::Void"*, %"core::fmt::Formatter"*)** %5 to i64* store i64 ptrtoint (i8 (i64*, %"core::fmt::Formatter"*)* @"_ZN4core3fmt3num54_\$LT\$impl\$u20\$core..fmt..Display\$u20\$for\$u20\$usize\$GT\$3fmt17hb872170870cc06d9E" to i64), i64* %6, align 8 %7 = getelementptr inbounds [1 x %"core::fmt::ArgumentV1"], [1 x %"core::fmt::ArgumentV1"]* %_8, i64 0, i64 0 %8 = getelementptr inbounds %"core::fmt::Arguments", %"core::fmt::Arguments"* %_3, i64 0, i32 0, i32 0 store %str_slice* getelementptr inbounds ([2 x %str_slice], [2 x %str_slice]* @ref.8, i64 0, i64 0), %str_slice** %8, align 8, !alias.scope !1, !noalias !4 %9 = getelementptr inbounds %"core::fmt::Arguments", %"core::fmt::Arguments"* %_3, i64 0, i32 0, i32 1 store i64 2, i64* %9, align 8, !alias.scope !1, !noalias !4 %_6.sroa.0.0..sroa_idx.i = getelementptr inbounds %"core::fmt::Arguments", %"core::fmt::Arguments"* %_3, i64 0, i32 1, i32 0, i32 0 store %"core::fmt::rt::v1::Argument"* null, %"core::fmt::rt::v1::Argument"** %_6.sroa.0.0..sroa_idx.i, align 8, !alias.scope !1, !noalias !4 %10 = getelementptr inbounds %"core::fmt::Arguments", %"core::fmt::Arguments"* %_3, i64 0, i32 2, i32 0 store %"core::fmt::ArgumentV1"* %7, %"core::fmt::ArgumentV1"** %10, align 8, !alias.scope !1, !noalias !4 %11 = getelementptr inbounds %"core::fmt::Arguments", %"core::fmt::Arguments"* %_3, i64 0, i32 2, i32 1 store i64 1, i64* %11, align 8, !alias.scope !1, !noalias !4 call void @_ZN3std2io5stdio6_print17h690779b3bd8114d5E(%"core::fmt::Arguments"* noalias nocapture nonnull dereferenceable(48) %_3) `````` [1]: https://play.rust-lang.org/
Am I reading it right? It looks to me like rustc didn't mark the immutable borrow as readonly, so LLVM went ahead and assumed print could mutate it.
I'm not 100% sure, but I believe LLVM doesn't have a way to understand the (im)mutability of pointers rewritten into memory, like the borrow is here.
If you use printf and cout in the same program then yes you can have dramatic performance problems with cout unless you use `std::ios_base::sync_with_stdio(false);`It's not the same issue but it's the same area of stupid stuff you have to find out about and roll your eyes.
This is awful. Was just reading how Rust is all shiny and special and much better than awful, naughty C++, and now I read how the print method is magic goo because "developers don't want to type & in front of ints". Back in my day, bah, grumble...
One of the reasons that ! is in macros is to let you know that they can do near-arbitrary things inside of their ()s.
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Mathematics
Normal percentile chapter 5 homework
### Question Description
I need support with this Statistics question so I can learn better.
The average beginning annual salary for teachers in California is \$41,181. Assume that this is a normal distribution with a standard deviation of \$725. One quarter of starting teachers in California will make " at least" much money per year?
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mean =41181
sigma =725
z value for 25 percentile or first quartile is -0.67
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# MYP Mathematics 7 (MYP 2) 3rd edition
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### MYP Mathematics 7 (MYP 2) 3rd edition
Product description
Mathematics 7 (MYP 2) third edition has been designed and written for the International Baccalaureate Middle Years Programme (IB MYP) Mathematics framework, providing complete coverage of the content and expectations outlined.
Discussions, Activities, Investigations, and Research exercises are used throughout the chapters to develop conceptual understanding. Material is presented in a clear, easy-to-follow style to aid comprehension and retention, especially for English Language Learners. Each chapter ends with extensive review sets and an online multiple-choice quiz.
The associated digital Snowflake subscription supports the textbook content with interactive and engaging resources for students and educators.
The Global Context projects highlight the use of mathematics in understanding history, culture, science, society, and environment. We have aimed to provide a diversity of topics and styles to create interest for all students and illustrate the real-world application of mathematics.
We have developed this book in consultation with experienced teachers of IB Mathematics internationally but independent of the International Baccalaureate Organisation (IBO). It is not endorsed by the IBO.
We have endeavoured to publish a stimulating and thorough textbook and digital resource to develop and encourage student understanding and nurturing an appreciation of mathematics.
MYP 2 3rd Edition Changes
Multiplying fractions by a whole number has been added to Chapter 6 (Fractions). This gives a more natural lead in to multiplying two fractions.
Rounding decimal numbers to significant figures has been added to Chapter 7 (Decimals).
Formulae has been added to Chapter 8 (Algebra).
Experimental probability has been added to Chapter 17 (Probability).
Enlargements and reductions have been added to Chapter 19 (Transformations).
Material that has been removed:
Finding a percentage change, and profit and loss, have been moved to MYP 3.
Conversion of units of area and volume has been removed.
Time has been moved to MYP 3.
Chapter 17 (Circles) has been removed.
Scale diagrams, speed and density have been moved to MYP 3.
Material that has been restructured:
The material on number strategies, estimation, and order of operations has been moved into a new Chapter 4 (Number strategies and order of operations). This allows estimation to
be studied after the number strategies, so that students can use the strategies in their estimation.
Squares and cubes have been moved to Chapter 2 (Number properties), and incorporates the square root and cube root material that previously existed in this chapter.
The material on prime and composite numbers has been moved between factors and highest common factor, so students can use prime factorization to find highest common factors.
Chapter 5 (Positive and negative numbers) has been reordered. We start by using the number line to illustrate the relationship between positive and negative numbers. This is followed by an exploration of words associated with position and direction, and their relation to positive and negative numbers.
Chapter 8 (Algebra) has been restructured to give a more logical progression of ideas.
In Chapter 10 (Equations) the guess, check and improve method has been moved to an Activity, and replaced by the method of inspection.
More broadly, we have adjusted how we organise the sections in each chapter, so there are less subsections within sections. This will make it easier for teachers and students to see what is in each chapter from the table of contents or the chapter title page.
Each chapter now ends with an online interactive multiple choice test, which is accessed through the digital version of the book.
The book now contains more Activities and Discussions throughout the chapters. These give students the opportunity to have more informal conversations about mathematics, and are also a good source of extra material for students who have finished their work early.
We have updated the Global Context projects, and added some new projects, so there are now nine projects in total. We have also vastly improved the task-specific descriptors, making it much easier for teachers to assess each task against the relevant assessment criterion.
The Self-Tutor scripts have also been updated, with a greater emphasis placed on enhancing student understanding, by providing more detailed explanations than can be given on the pages of a textbook
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Author: Michael Haese
Curriculum: International Baccalaureate Middle Years Programme
Edition: 3rd edition
Format: Paperbak
Language: English
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https://www.r-bloggers.com/2012/02/pca-for-nir-spectra_part-001-plotting-the-loadings/
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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
There are different algorithms to calculate the Principal Components (PCs). Kurt Varmuza & Peter Filzmozer explain them in their book: “Introduction to Multivariate Statistical Analysis in Chemometrics”.
I´m going to apply one of them, to the Yarn spectra.
Previously we have to center the X matrix, let´s call it Xc.
> Xc<-scale(yarn\$NIR, center = TRUE, scale = FALSE)
The algorithm I´m going to apply is “Singular Value Decomposition”.
> Xc_svd<-svd(Xc)
The idea of this post is just to look to the loadings matrix (P). Loadings are spectra. which reconstruct together with the score matrix (T), and an error matrix (E), the original X matrix.
For this case 3 components in enough, because explain almost 99% of the variance, so let´s have a look to the first three loadings:
> P<-Xc_svd\$v
> P3cp<-P[,1:3]
> matplot(wavelengths,P3cp,pch=21,lty=1,
+ xlab=”data_points(nm)”,ylab=”log(1/R)”)
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http://bevinslessor.co.nz/epub/algebraic-geometry-and-analysis-geometry
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# Download PDF by Akira Fujiki, etc., Kazuya Kato, T. Katsura, Y. Kawamata, Y.: Algebraic Geometry and Analysis Geometry
By Akira Fujiki, etc., Kazuya Kato, T. Katsura, Y. Kawamata, Y. Miyaoka
ISBN-10: 0387700862
ISBN-13: 9780387700861
ISBN-10: 4431700862
ISBN-13: 9784431700869
This quantity documents the lawsuits of a world convention held in Tokyo, Japan in August 1990 at the topics of algebraic geometry and analytic geometry.
Similar geometry and topology books
Download e-book for kindle: Geometric Aspects of Functional Analysis by Joram Lindenstrauss, Vitali D. Milman
This is often the 3rd released quantity of the lawsuits of the Israel Seminar on Geometric facets of practical research. the big majority of the papers during this quantity are unique learn papers. there has been final 12 months a robust emphasis on classical finite-dimensional convexity concept and its reference to Banach area concept.
Additional resources for Algebraic Geometry and Analysis Geometry
Sample text
Indeed, set A˜ to be the disjoint union of two isometric sets. Then we can keep one of these sets fixed and put on the top of it the other one, using only translations. The width will change. We can define other invariants though, using the heights and the number of connected components, for example. By using lifts of symplectic flows one can define symplectic invariants. Conversely, with any symplectic capacity comes an invariant of Hom(H(n), vol, Lip). Indeed, let c ˜ = c(A) is an invariant. be a capacity.
Therefore at the limit ε → 0 we get a Lie bracket. Moreover, it is straightforward to see from the definition of [x, y]n that δε is an algebra isomorphism. We conclude that (g, [·, ·]n ) is the Lie algebra of a Carnot group with dilatations δε . , Xp } of D. 2). Moreover the identiˆ i = Xi gives a Lie algebra isomorphism between (g, [·, ·]n ) and n(g, D). fication X Proof. , Xdim g} the Lie bracket on g looks like this: [Xi , Xj ] = Cijk Xk where cijk = 0 if l(Xi ) + l(Xj ) < l(Xk ). From here the first part of the proposition is straightforward.
This is because linear maps are not only group morphisms, but also commute with dilatations. 33 Let N , M be Carnot groups. We write N ≤ M if there is an injective group morphism f : N → M which commutes with dilatations. 34 Let X,Y be Lipschitz manifolds over the Carnot groups M, respectively N. If N ≤ M then there is no bi-Lipschitz embedding of X in Y . Rigidity in the sense of this section manifests in subtler ways. The purpose of Pansu paper [25] was to extend a result of Mostow [22], called Mostow rigidity.
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# What 2D shapes make up a triangular pyramid?
## What two shapes make a prism?
Two squares joined together to form a prism. Two-dimensional shapes make the flat top and bottom sides, or faces, of a prism.
## What 2D shapes make up a triangular pyramid?
The net of a triangular pyramid consists of four triangles. The base of the pyramid is a triangle, and the lateral faces are also triangles. The net of a rectangular pyramid consists of one rectangle and four triangles. The rectangle is the base of the pyramid, and the triangles are the lateral faces.
## What is an example of a prism?
Prism-shaped objects you’ll see in everyday life include ice cubes, barns and candy bars. The regular geometry of the prism makes it useful for designing buildings and simple products. You’ll also find prisms in the natural world, such as mineral crystals.
## What is not a prism?
The prisms are polyhedrons or objects with multiple flat faces. A prism can not have any side which is curved thus objects like cylinder, cone or sphere are not prisms.
## What shapes are needed to construct a triangular pyramid?
A triangular-based pyramid has 4 faces, 4 vertices including the apex and 6 edges. To build a triangular pyramid with construction materials, we would need 4 triangles that join at the edges to make a closed three-dimensional shape or 6 edge pieces and 4 corner pieces to make a frame.
## What are 3 things pyramids and prisms have in common?
Similarities: Common Features Prisms and pyramids are three-dimensional solid shapes that contain sides and faces that are polygons — two-dimensional shapes with straight sides. Both shapes fall under the large category — polyhedrons — because the sides and bases are polygons.
## What is a real life example of a triangular prism?
Examples of Triangular Prism Some of the real-life examples of a triangular prism include triangular roofs, camping tents, Toblerone wrappers, and chocolate candy bars.
## How can you identify a prism?
A prism is a type of three-dimensional (3D) shape with flat sides. It has two ends that are the same shape and size (and look like a 2D shape). It has the same cross-section all along the shape from end to end; that means if you cut through it you would see the same 2D shape as on either end.
## What is the example of prism?
These are all Prisms:
Square Prism:Cross-Section:
Cube:Cross-Section:
(yes, a cube is a prism, because it is a square all along its length) (Also see Rectangular Prisms )
Triangular Prism:Cross-Section:
Pentagonal Prism:Cross-Section:
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## So erstellen Sie kreative Doppelbelichtungsfotos auf dem iPhone
Suchen Sie nach einer Möglichkeit, zwei Fotos auf Ihrem iPhone zusammenzufügen? Ob Sie es glauben oder nicht, es gibt eine Reihe von Anwendungen, mit...
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