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The project uses the Pi to directly solve the Rubik’s cube. The BrickPi3 takes the unsolved Rubik’s cube and the Raspberry Pi takes a picture of each side of the Rubik’s cube with the Raspberry Pi Camera. The Pi creates a text map of the color squares that shows where they are located on the cube. When it has fully mapped the cube, the Pi uses the “kociemba” python library to map out the moves needed to solve the Rubik’s cube. This information is taken by the Pi and BrickPi3 to solve the Rubik’s cube using the LEGO motors. The result: a solved Rubik’s cube. Rubiks Build It Solve It
When you get round to building the Rubik’s Cube, you will find it is not as hard as it appears. The instructions are quite easy to follow and it will probably take you about fifteen minutes. When you get round to placing the colored tiles, pay attention to where they are supposed to go. Because once you snap them into place. you will not be able to remove them. Having said that. you can still use the Rubik’s Cube. What you will not be able to do is follow the instruction guide and solve the puzzle. Rubiks Build It Solve It
The standard Rubik's cube has sides of about 2.2 in (5.7 cm) per square. Various other sizes have also been produced such as a 1.5 in (3.8 cm) mini cube, a 0.8 in (2 cm) key chain micro cube, and a 3.5 in (9 cm) giant cube. While the standard cube is a 3 × 3 × 3 segmentation other types have also been introduced. Some of the more interesting ones include the 2 × 2 × 2 cube, the 4 × 4 × 4 cube (called Rubik's Revenge) and the 5 × 5 × 5 cube. The shape has also been varied and puzzles in the form of a tetrahedral, a pyramid, and an octahedral are among types that were produced. The Rubik's cube also led to the development of game derivatives like the Rubik's cube puzzle and the Rub it cube eraser.
Rubik’s cubes are 3-D combination puzzles. The 3x3x3 Rubik’s cubes have nine faces on each side of the square cube and each face has one of six solid colors. A traditional way to solve the Rubik’s cube is by returning the blocks so that each face of the cube has only one color.[1] However, since the cube’s creation in 1974, there have been many other ways found to “solve” the Rubik’s cube. Each of them create some sort of repetitive design over the faces of the cube.
An important thing to note is that this task is not a light one. It may require several hours of attempts. If you'd rather just solve the cube in your hand and forget about it rather than being able to solve any cube you're given, there are plenty of solvers available on the web. However, the satisfaction of holding a completed Rubik's Cube in your hand and thinking “I did that, and I can do it again” is greater than most, mainly due to the fact that the puzzle has been present in all our lives at some point. By the mid 1980's, an estimated fifth of the world's population had attempted to solve the cube. If you want to stand out and say that you can defeat the puzzle, time and determination is a large factor.
Okay, we’re going to be honest with you – you know how aggravating the traditional Rubik’s Cube is, correct? If you’ve ever tried to solve one, you probably know exactly what we’re talking about here. Yes, this kit offers a behind-the-scenes look and even comes with a 10-page instruction manual, but this doesn’t necessarily mean you will be solving the puzzle like a professional within seconds. Rubix Building Solutions
Since the center pieces cannot be moved relatively to each other it's important to solve the edge pieces correctly in relation to each other. For example, when solving the white in our case- the green center piece is to the left of the red center piece, therefore the green-white edge piece should to be solved to the left of the red-white edge piece (see image).
Constructing the Cube is a superb way to exercise those fine motor skills, visual and spatial comprehension and cognitive thinking from children. When the block is placed together, it is going to challenge the small ones to use their spatial and visual understanding as they know to spin the tiles. The block also helps kids learn about colours and fitting them. Rubix Building Products
If you're reading this, you're probably holding a cube in your hand and already feeling bad about yourself for needing to look up the solution. But don't worry! In fact, most of the “super-human-intelligence beings” (a common misconception) who have solved the cube thousands of times in their lifetimes were sitting as you are now. Whether you want to learn it to impress a girl, because your friends bet you couldn't, or just to close the book on the biggest time waste of your childhood by finally defeating it, this guide will take you through the simplest way to conquer the puzzle.
When you get round to building the Rubik’s Cube, you will find it is not as hard as it appears. The instructions are quite easy to follow and it will probably take you about fifteen minutes. When you get round to placing the colored tiles, pay attention to where they are supposed to go. Because once you snap them into place. you will not be able to remove them. Having said that. you can still use the Rubik’s Cube. What you will not be able to do is follow the instruction guide and solve the puzzle. Rubiks Build It Solve It
If you're still reading, congratulations on not being put off by the time requirements! The first thing you are going to need to know about solving the cube is how the turns you make can be represented by letters. Later on in this guide, you're going to need a few algorithms. These are combinations of moves that rotate pieces or just move them around to get them where you want them. These algorithms are written using this notation, so you can always come back to this section if you've forgotten by the time we need them. Rubiks Build It Solve It Instructions
If it comes to constructing the Rubik’s Cube, it’s not as difficult as it seems. In reality, it is going to take approximately fifteen minutes and the directions are simple to follow. If it comes to putting the coloured tiles, be sure to look closely at where you’re supposed to put them since in the event that you snap them in the incorrect location, you won’t have the ability to eliminate them. Yes, you will continue to have the ability to use this Rubik’s Cube, however you won’t be able to follow along with the documentation manual on solving the mystery.
```Repeat the process. Turn back to your blue side and repeat the turns on opposite sides. Then, return once more to the red side and turn the opposite sides in opposite directions. And last, return once more to the blue side and turn the opposite sides in opposite directions. When you finish, you should have a staircase-like zig-zag across four sides of your Rubik’s cube.[4]
```
1 When production is initiated, the plastic pellets are transformed into Rubik's cube parts through injection molding. In this process, the pellets are put into the hopper of an injection molding machine. They are melted when they are passed through a hydraulically controlled screw. As the screw turns, the melted plastic is shuttled through a nozzle and physically forced, or injected, into the mold. Just prior to the arrival of the molten plastic, the two halves of the mold are brought together to create a cavity that has the identical shape of the Rubik's cube part. This could be an edge, a corner, or the center piece. Inside the mold, the plastic is held under pressure for a specific amount of time and then allowed to cool. While cooling, the plastic hardens inside the mold. After enough time passes, the mold halves are opened and the cube pieces are ejected. The mold then closes again and the process begins again. Each time the machine moulds a set of parts is one cycle of the machine. The Rubik's cube cycle time is around 20 seconds. Rubix Building Products
4 Next, the Rubik's cube faces need to be labeled. The labels are made from sheet polypropylene material that is printed with the colors. The printed sheet PP is then laminated with a clear PP protective covering. The material is then die cut with the labels wound onto rolls. The labels are made with all nine squares of each face exactly aligned. This way the labels can be perfectly aligned when they are applied to the cube.
When you get round to building the Rubik’s Cube, you will find it is not as hard as it appears. The instructions are quite easy to follow and it will probably take you about fifteen minutes. When you get round to placing the colored tiles, pay attention to where they are supposed to go. Because once you snap them into place. you will not be able to remove them. Having said that. you can still use the Rubik’s Cube. What you will not be able to do is follow the instruction guide and solve the puzzle.
In the publication, during the very first measure, you’re advised you’ll have to practice and trial by mistake. We believe that this block will be enjoyable for kids and adults who love puzzles and don’t mind the complications supporting a Rubik’s Cube. You should have patience in regards to practicing and building. However, as soon as you’re able to resolve it, you’re likely to be quite proud of yourself and the people around you’re likely to be more amazed since it actually requires a whole lot to resolve those cubes.
hottoysheadquarters.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to amazon.com, or endless.com, MYHABIT.com, SmallParts.com, or AmazonWireless.com and any other site that may be affiliated with Amazon Service LLC associates program.
Finally, we add a camera arm. In the original design by MindCubr, this held the EV3 color sensor over the Rubik’s cube. In our modified design, it holds a Raspberry Pi Camera over the Rubik’s cube. We use two LEGO Mindstorms motors to manipulate the cube: the first sits below the cradle to rotate the cube, and the second moves the shuffler arm to spin the cube on an opposite axis.
Simply put the 1x1x3 is a pseudo puzzle, It fills a gap in the collection but its not exactly complicated to solve. The way this puzzle was made was by using two centres and a core of a QiYi Sail. As these parts already spin like a 1x1x3 should all I had to do was make these parts into cubies by adding some apoxie sculpt and sanding them smooth. This puzzle was made in an afternoon and stickered the following morning while I was also building my 'Mefferts bandage cube'.
For decorative purposes, a colorant is typically added to the plastic. The pieces of a Rubik's cube are typically black. During production, colored stickers are put on the outside of the cube to denote the color of a side. The plastics that are used during production are supplied to the manufacturer in a pellet form complete with the filler and colorants. These pellets can then be loaded into the molding machines directly.
Even in the manual it states that you will need to practice using trial and error. Personally, we have a feeling that people who like doing puzzles and working things out will enjoy this more. It is important that you have patience when it comes to building the Rubik’s Build It Solve It cube. However, once you get the hang of this you are going to feel pretty smart. And your friends will be really impressed because it takes a bit of effort to solve one of these cubes. If you would like to read our review on the d-fantix cyclone 3×3 – please click on the link.
```Rubik’s Build It, Solve It is similar to the conventional Rubik’s block, but with a twist. This block includes each the tools, bits and directions kids need to be able to construct a Rubik’s Cube of the own. After this block was assembled together, there’s an education booklet (it’s’s 10-pages in duration) which will direct you through the procedure for solving the Rubik’s Cube (eventually). Here, you’ll find everything from identifying the areas of the block to solving fundamental puzzles. With this toy, kids will be provided a slow and continuous introduction about the best way best to use the block and progress to harder struggles.
```
Since the center pieces cannot be moved relatively to each other it's important to solve the edge pieces correctly in relation to each other. For example, when solving the white in our case- the green center piece is to the left of the red center piece, therefore the green-white edge piece should to be solved to the left of the red-white edge piece (see image).
```Do you remember those complicated little Rubik’s block that we would sit there trying to figure out for what seems like hours? Did any of you guys/girls ever solve them? Maybe all it takes for us to solve the “cube” would be for us to see what it’s all about. While there are many mesmerizing toys that are about to emerge into our world, today, we would like to take a close look in this Rubik’s Build It Solve It Review, because we believe this is the one-way ticket to finally solving the cube!
```
The Rubik's cube appears to be made up of 26 smaller cubes. In its solved state, it has six faces, each made up of nine small square faces of the same color. While it appears that all of the small faces can be moved, only the corners and edges can actually move. The center cubes are each fixed and only rotate in place. When the cube is taken apart it can be seen that the center cubes are each connected by axles to an inner core. The corners and edges are not fixed to anything. This allows them to move around the center cubes. The cube maintains its shape because the corners and edges hold each other in place and are retained by the center cubes. Each piece has an internal tab that is retained by the center cubes and trapped by the surrounding pieces. These tabs are shaped to fit along a curved track that is created by the backs of the other pieces. The central cubes are fixed with a spring and rivet and retain all the surrounding pieces. The spring exerts just the right pressure to hold all the pieces in place while giving enough flexibility for a smooth and forgiving function.
Whether you complete all 6 stages or 1, be sure to tell your teacher about this program so all your classmates can solve with you! Teachers from all over the country use our program, at no cost, to teach their classes not only to solve, but math, art, science, and more. Hundreds of schools compete at solving cubes as a group and classes create really cool mosaic designs too. We even have ongoing mosaic contests each year. So check out our site and learn how you can do even more with a Rubik's® Cube!
Puzzle makers have been creating problems for people to solve for centuries. Some of the earliest puzzles date back to the time of the ancient Greeks and Romans. The Chinese have a ring puzzle that is thought to have been developed during the second century A.D. This was first described by Italian mathematician Girolamo Carolano (Cardan) in 1550. When the printing press was invented, complete books of mathematical and mechanical problems designed specifically for recreation were circulated.
5 After all the labeling is completed, the cubes are put in their final packaging. This can be a small box that has an instruction booklet included or a plastic blister pack with a cardboard backing. The package serves the dual purpose of protecting the Rubik's cube from damage caused by shipping and advertising the product. The Rubik's cube packages are put into cases and moved to a pallet. The pallets are then loaded on trucks and the products are shipped all over the world. Rubiks Build It Solve It Review
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# McKinsey off-cycle
Author Message
Intern
Joined: 03 Nov 2011
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03 Nov 2011, 19:46
I just applied online for a full-time associate position at McKinsey. I know I'm applying off-cycle and that it's an uphill battle. I'm a current student at a top business school - don't ask me why I didn't apply sooner. How long should I wait to hear back before concluding that it's a ding? Thank you.
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07 Nov 2011, 16:52
The deadlines differ from country to country. Try to reach out to the HR at McKinsey. They are very friendly. Alternatively, if you know anyone at McKinsey, it might be better to contact them and help you in getting an interview. Have you written the test?
http://www.interviewbay.com
MBA | Consulting | Finance
ConsultingBound wrote:
I just applied online for a full-time associate position at McKinsey. I know I'm applying off-cycle and that it's an uphill battle. I'm a current student at a top business school - don't ask me why I didn't apply sooner. How long should I wait to hear back before concluding that it's a ding? Thank you.
Manager
Status: Taking heavily leveraged but calculated risks at all times
Joined: 04 Apr 2010
Posts: 183
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Schools: HBS '15, Stanford '15
GMAT Date: 01-31-2012
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Kudos [?]: 72 [0], given: 12
### Show Tags
02 Dec 2011, 07:01
1. When you say "top", how Top is that? Trust me this question is very relevant.
2. Off cycle hiring? You are better off networking your way in than waiting for a response in that case. So if you are in a top business school, then am sure you wouldn't have a problem finding classmates employed there.
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10 Feb 2012, 17:09
Anasthaesium wrote:
1. When you say "top", how Top is that? Trust me this question is very relevant.
2. Off cycle hiring? You are better off networking your way in than waiting for a response in that case. So if you are in a top business school, then am sure you wouldn't have a problem finding classmates employed there.
Yeah, I agree. the best way would be to actually get in touch with people who currently work their, including consultants and HR. I would also think about a story to tell why you didn't apply earlier. Did you change you industry preference? Why did you decide that consulting is a right thing for you now, given you didn't apply several weeks/months ago.
Re: McKinsey off-cycle [#permalink] 10 Feb 2012, 17:09
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Photons absorbed by electrons, selections rule
I know that if a photon with a certain energy $$E_1$$ is absorbed by an electron, for example we are talking about Hydrogen atom, if this energy $$E_1$$ is equal to the difference in energy between two different levels, the elctron jumps on the excited state $$n$$.
If the photon has an energy inconsistent with the difference in energy between two levels, what will happen?
Will the photon be absorbed?
You are asking whether the photon has to have exactly the same energy as the difference between two energy levels (bang gap), to be absorbed by the electron/atom system. The answer is no, first of all, it is QM, and all probabilities, but more importantly, if the photon has excess energy (exceeding the band gap), the electron might still be excited, and the excess energy will be transformed into the kinetic energy of the electron inside the new band.
No, it is sufficient for the photon energy to exceed the band gap. Any excess energy is transformed into kinetic energy for the electron in the new band.
Is the Energy of an absorbed photon exactly the energy of the band gap?
Now you are correct, that if the photon energy does not meet the minimal required energy to excite the electron, then the electron should not be excited (with a high probability).
It is very important to understand that because of the quantum mechanical (probabilistic) nature of the world, we do not need to talk about exact matches between the band gap and the photon energy, first of all, even if the two would exactly match, even then there is only certain probability that the electron will be or will not be excited. Second, and most importantly, it is not correct (in your case) to talk about specific photon energy levels, since photon energy levels are never exactly defined.
You should also take into account that photon energies are never exactly defined except for monochromatic beams with infinite temporal duration. This is exactly because of the energy-time uncertainty relation: the only way to have a perfectly defined photon frequency, and hence energy, is to observe it for an infinitely long time. Thus, the photon energy is always spread out over a finite bandwidth.
• Ok, thank you for your answer, so if the photon does not meet the minimal required energy it will be scattered for example? May 10, 2020 at 12:53
• @Salmone or there is no interaction at all. Scattering can be elastic (photon keeps its energy) or inelastic (photon gives part of its energy to the electron). May 10, 2020 at 16:36
• Ok thank you very much May 10, 2020 at 19:41
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# Exams
There will be two comprehensive exams, based on the homework but with new problems.
For each exam, you may use one sheet of notes that you wrote yourself. However, you may not use your book or anything else not written by you. You certainly should not talk to other people! Calculators are allowed, although you shouldn't really need them; however, a graphing calculator may help with extra credit problems.
The first exam is on May 1 Tuesday, to prepare for the midterm grades that come out later that week.
• Date taken: May 1 Tuesday.
• Aids allowed: Calculator, one sheet of notes.
• Lectures covered: April 3 through April 26.
• Sections of the official textbook: §2-1, §§3-1&3-2, §3-5, §3-7, §§4-3–4-6, §§5-1–5-6.
• Chapters of Calculus Made Easy: §I, §§III–XI.
• Review questions:
• From Chapter 2 Review (pages 120–124): 5, 88, 89, 90;
• From Chapter 3 Review (pages 205–209): 1, 3, 14–20, 25–30, 31 (ignore the hint), 43, 45–49, 52, 53, 60–67 (but maybe not all of them), 78–82, 88, 90, 91, 92, 95, 96, 97;
• From Chapter 4 Review (pages 264&265): 6, 7, 18, 19, 22, 23, 24, 32, 33, 37 (square both sides first), 38, 42, 44;
• From Chapter 5 Review (pages 345–348): 1, 2, 4–7, 11, 15, 16, 21–28 (using a calculator if you like but only after calculating the window size), 32, 37, 44, 45, 51 (read carefully), 58–62, 68, 70, 72, 73.
• My answers: DjVu format, PDF format.
The other exam is on June 5 Tuesday, so that you can see your tentative grade on the last day of class.
• Date taken: June 5 Tuesday.
• Aids allowed: Calculator, one sheet of notes.
• Lectures covered: April 3 through May 29.
• Sections of the official textbook: §2-1, §§3-1–3-7, §§4-1–4-7, §§5-1–5-6, §§6-1–6-5, §§7-1&7-2.
• Chapters of Calculus Made Easy: §I–XI, §XIV, §§XVII–XXI.
• Review questions:
• From Chapter 2 Review (pages 120–124): 5, 88, 89, 90;
• From Chapter 3 Review (pages 205–209): 1, 3, 14–20, 25–30, 31 (ignore the hint), 43, 45–49, 52, 53, 60–67 (but maybe not all of them), 78–82, 87, 88, 90, 91, 92, 95, 96, 97;
• From Chapter 4 Review (pages 264&265): 2, 3, 6, 7, 8, 11, 12, 13, 15, 16, 18, 19, 22, 23, 24, 29, 30, 32, 33, 36, 37 (square both sides first), 38–44;
• From Chapter 5 Review (pages 345–348): 1, 2, 4–7, 11, 15, 16, 21–28 (using a calculator if you like but only after calculating the window size), 32, 37, 44, 45, 51 (read carefully), 58–62, 68, 70, 72, 73;
• From Chapter 6 Review (pages 407–409): 1–6, 9, 10, 11 (doing both upper and lower sums to get the error bound), 28, 32–42 (but maybe not all of them), 43, 46–51, 62, 63, 64;
• From Chapter 7 Review (pages 447&448): 5, 6, 7, 11–16, 22, 25, 26, 38, 41, 42, 43, 44.A, 48, 50, 51.
If you will miss an exam, then tell me as soon as you know.
Go back to the course homepage.
This web page and the files linked from it were written between 2003 and 2012 by Toby Bartels, last edited on 2012 May 29. Toby reserves no legal rights to them.
The permanent URI of this web page is `http://tobybartels.name/MATH-1400/2012s/exams/`.
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# how is the electron geometry of a molecule determined?
Determine the electron geometry, molecular geometry, and idealized bond angles for each molecule. c. CHCI3 For example, the methane molecule, CH 4, which is the major component of natural gas, has four bonding pairs of electrons around the central carbon atom; the electron-pair geometry is tetrahedral, as is the molecular structure (Figure 4.4. The valence shell electron-pair repulsion theory (abbreviated VSEPR) is commonly used to predict molecular geometry. Lewis structures are very useful in predicting the geometry of a molecule or ion. The two X atoms (in white) are 180° away from one another. The shape of a molecule is the structure of the molecule predicted using the bond electron pair on the central atom. Molecular Geometry ... Valence-Shell Electron-Pair Repulsion Theory. The molar mass of KClO3 is, Harlon currently works as a quality moderator and content writer for Difference Wiki. the electron geometry of a molecule is determined by the number of bonded electrons and lone pair electrons around the central atom. It is very important from the onset that students understand the difference between electronic geometry and molecular geometry. It covers all possible shapes for molecules with up to six electron pairs around the central atom. They are both electronically tetrahedrals with the two lone pairs in two sp3 orbitals and the two Brs bonded to the other two sp3 orbitals. Determine the electron geometry (eg) and molecular geometry (mg) of CO32⁻. Square pyramid. 3- Determine the idealized bond angle for each molecule. The following procedure uses VSEPR theory to determine the electron pair geometries and the molecular structures: Write the Lewis structure of the molecule or polyatomic ion. How to find the CO 2 molecular geometry using VSEPR theory is not very difficult using these three steps.. WARNING. Key Takeaways Key Points. Molecular geometry of the molecule can be determined by electron groups as well lone pair of electrons. Valence Shell Electron Pair Repulsion (VSEPR) theory: Principle: Electron pairs around a central atom arrange themselves so that they can be as far apart as possible from each other. You should note that to determine the shape (molecular geometry) of a molecule you must write the Lewis structure and determine the number of bonding groups of electrons and the number of non-bonding pairs of electrons on the central atom, then use the associated name for that shape. Write the complete mechanism, step by step, of the reaction shown in figure A, in such a way that... A: In the presence of H2SO4, tertiary carbocation is formed which undergoes intramolecular nucleophilic... *Response times vary by subject and question complexity. Electron geometry gives us information about the organization of groups of electrons; on the other hand, molecular geometry gives us information about the organization of only atoms in molecule except for lone pairs. one. Electron geometry contains electron pairs; on the flip side, molecular geometry does not contain electron pairs. 3- Determine the idealized bond angle for each molecule. The main difference between the electron geometry and molecular geometry is that when we discuss the shape of the molecule, it means we are determining the molecular geometry; on the other hand, when we discuss the geometry of electron pairs, it means we are determining the electron geometry. Determine the electron geometry, molecular geometry, and idealized bond angles for each molecule. The HCN molecule is a type of AX2 where A denotes central atom (carbon) and X represents surrounding atoms (hydrogen and nitrogen). According to this theory by counting the total number of electron pairs around the central atom the geometry of any given molecule can be determined. A single, double, or triple bond counts as one region of electron density. BClF2. From an electron-group-geometry perspective, GeF 2 has a trigonal planar shape, but its real shape is dictated by the positions of the atoms. Determine the following for {eq}O_2 {/eq}. Determine the electron geometry, molecular geometry and polarity of N2O (N central). Electron geometry gives us the shape of the molecule that includes both bonding and non-bonding electron pairs; on the other hand, molecular geometry gives us the shape of the molecule that includes only the bonding electron pairs. The shape of BrF 5 molecule is square pyramidal. 35. We calculate the number of total electron pairs in electron geometry and not in molecular geometry. 2- Determine the molecular geometry for each molecule. Choose the compound below that contains at least one polar covalent bond, but is nonpolar. How many electron groups there are. Fundamentally, the VSEPR model theorizes that regions of negative electric charge will repel each other, causing them (and the chemical bonds that they form) to stay as far apart as possible. 1. To see how the model works for a molecule with double bonds, consider carbon dioxide, CO 2. Electron geometry helps give the arrangement of electron pairs; on the opposite side, molecular geometry helps give the arrangement of atoms around the central nuclei. Give the number of lone pairs around the central atom and the molecular geometry of SeF 4. Use the SN and VSEPR theory to determine the electron pair geometry of the molecule. Molecular structure, which refers only to the placement of atoms in a molecule and not the electrons, is equivalent to electron-pair geometry only when there are no lone electron pairs around the central atom. Determine the electron geometry, molecular geometry, and tetrahedral. The geometry of each interior atom is as follows. Lewis structures are very useful in predicting the geometry of a molecule or ion. The following procedure uses VSEPR theory to determine the electron pair geometries and the molecular structures: Write the Lewis structure of the molecule or polyatomic ion. a) PF3 (b) SBr2 (c) CH3Br (d) BCl3 I'd really appreciate your help! Molecular geometry or molecular structure is the three-dimensional arrangement of atoms within a molecule. The main purpose of the electron geometry is to find out the geometry of the molecule by the arrangement of atoms and electrons around the central atom. Tetrahedral- CF4 Trigonal Pyramidal- NF3 Bent- OF2 and H2S. trigonal bipyramidal. By decreasing the repulsion between these electronic regions, electron geometry gives us the shape of the molecules. Molecular geometry can be determined by the number of bonds that a particular molecule has. The example of molecular geometry is the water molecule. That gives you the steric number (SN) — the number of bond pairs and lone pairs around the central atom. An example of electron geometry can be given by using the formula of methane. A. eg = trigonal planar, mg = trigonal planar. A molecule containing a central atom with sp3 hybridization has a(n) _____ electron geometry. In electron geometry, it also considers the region of electrons having different densities. d. CS2, Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes! Electron geometry determines the shape of the molecule that involves the electron pair and bond pair; on the other hand, molecular geometry determines the shape of the molecule that involves only the bond pairs. Molecular geometry does not consider the lone pairs as bonds like in electron domain geometry. Valence Shell Electron Pair Repulsion (VSEPR) is a theory that states that the 3d orientation, also known as the molecular geometry, of a molecule is not dependent on its chemical formula but on the repulsion of valence electrons.In other words, two molecules with the general formulas AB_3 may look completely different in real life: one may be a pyramid whereas the other … XeF4 Molecular Geometry. Please add difference.wiki to your ad blocking whitelist or disable your adblocking software. 2- Determine the molecular geometry for each molecule. Electron geometry can be determined by the help of the VESPR Theory; on the flip side, molecular geometry is defined by the arrangement of atoms around the nuclei of the central atom. One Page Lesson: Determining Electron-Group & Molecular Geometry The repulsive forces between bonding and non-bonding electrons determine the three-dimensional geometry of the “groups” of electrons around a central atom. Determine the electron geometry (eg) and molecular geometry (mg) of XeF4. A dipole moment measures a separation of charge. There are four single bonds in the structure of the molecule. Step 1: Draw the Lewis structure for SO 2 using the “easy” method where you calculate the total valence electrons in the molecule to determine the Lewis structure.. C: 4 O: 6×2=12. In which cases do you expect deviations from the idealized bond angle? The geometry of the molecule actually determines number of electron pairs on the central atom. Apply the VSEPR model to determine the geometry of a molecule that contains no lone pairs of electrons on the central atom. b. SB12 The molecular geometry, or three-dimensional shape of a molecule or polyatomic ion, can be determined using . A single, double, or triple bond counts as one region of electron density. The central atom is carbon, and the number of valence electrons is 4. BIF2 III. Determine the electron geometry, molecular geometry and polarity of SF6. Molecular geometry is determined by possible locations of an electron in a valence shell, not by how many how many pairs of valence electrons are present. \mathrm{NF}_{3} \quad \text { c. } \mathrm{OF}_{2} \quad \text { d. } \mathrm{H}_{2} \mathrm{S} Problem 35. A: Assign the priority to the groups at asymmetric centre based on some rules are called CIP rule or se... Q: How many grams of oxygen can be obtained by the determination of 277 g of reactant in the following ... A: The balanced chemical equation is, }\mathrm{CF}_{4} \quad \text { b. } Electron pairs are considered only in the electron geometry and neglected in the molecular geometry. a. PF3 Should be bent with bond angles of around 107 degrees. The three-dimensional structure of a molecule is determined by its valence electrons, not its nucleus or the other electrons in the atoms. The geometric diagram determines this shape. Experimentally the molecular geometry can be observed using various spectroscopic methods and diffraction methods. The above discussion concludes that both electron geometry and molecular geometry is the geometry that is used for the determination of the shape of the molecule. The repulsion between the bonded atoms also considered being less to find out the main purpose of the geometry. This is a LONG document. Determine the molecular geometry. If the bond energy is higher than the Ionization energy, then it can be said that the compound will have a stable condition. It is determined by the central atom and the surrounding atoms and electron pairs. As molecular geometry is used to determine the shape of the molecule, so we must use the lewis structure when we are discussing the shape of molecules in the molecular geometry by drawing it in the form of a lewis method to determine the number of bonding electrons. We consider both lone electron pairs and bond electron pairs while determining the shape of a molecule in electron geometry. 4- In which cases do you expect deviations from the idealized bond angle? Molecular geometry … VSEPR theory is used to predict the arrangement of electron pairs around non-hydrogen atoms in molecules, especially simple and symmetric molecules, where these key, central atoms participate in bonding to two or more other atoms; the geometry of these key atoms and their non-bonding electron pairs in turn determine the geometry of the larger whole. VESPR stands for valence shell electron pair repulsion. molecules also have a general shape, called the vsepr molecule shape, that considers the repulsive and attractive forces between electron pairs. idealized bond angles for each molecule. How many electron groups there are. Some of the most common shapes tha… eg= octahedral, mg= octahedral, nonpolar. Molecular geometry, on the other hand, depends on not only on the number of electron groups, but also on the number of lone pairs. External Customers, Electron geometry is the geometry that determines the shape of the molecule including both the electron pairs and bonds in the molecule, Molecular geometry is the geometry that determines the shape of the molecule including only the bonds in the molecule, Electron geometry includes electron pairs, Molecular geometry does not include electron pairs, Electron geometry includes both bonding electron pairs and non-bonding electron pairs, Molecular geometry includes only bonding electron pairs, Electron geometry is useful in giving the arrangement of the electron groups, Molecular geometry is useful in giving the arrangement of the atoms in a molecule, Electron geometry can also be determined by VESPR Theory, Molecular geometry can also be determined by the arrangement of the atoms around the nuclei of the central atom. According to this theory, electron pairs locate themselves in such a way so that minimum repulsion is created. electron group geometry. It applies a theory called VESPR for short. The electron geometry can be determined using the VESPR Theory. Third, the electron-pair geometry is determined by creating a Lewis dot structure First, the steric number of the central atom is determined. The geometry of an electron determined for the first time by University of Basel An electron is trapped in a quantum dot, which is formed in a two-dimensional gas in a semiconductor wafer. It is also named the Gillespie-Nyholm theory after its two main developers, Ronald Gillespie and Ronald Nyholm. Molecular geometry for each molecule CF4 NF3 OF2 H2S. The shape of the molecule can be predicted using the VSEPR model (valence shell electron pair repulsion model). FREE Expert Solution. linear This theory basically says that bonding and non-bonding electron pairs of the central atom in a molecule will repel (push away from) each other in three dimensional space and this gives the molecules their shape. In other words, the shape of a molecule is determined excluding the lone electron pairs of the central atom. Comes from our online advertising for one bond, but is nonpolar region of electrons of... Arrangement of atoms in the structure of the central atom determine geometry of geometry! Previous question Next question Get more help from Chegg, Flash, animation, obnoxious sound, or ad. The CO 2 molecular geometry and polarity of N2O shape of a molecule is determined by creating a structure. Its nucleus or the other electrons in C2H2 molecule = 2x4+2 =10 e. the Lewis dot structure First the. By using the formula of methane is tetrahedral and the electron pair ) CF4 ( b ) NF3 c. Banner, Flash, animation, obnoxious sound, or three-dimensional shape of molecule. Bonding and non-bonding electrons trigonal Pyramidal- NF3 Bent- OF2 and H2S nucleus how is the electron geometry of a molecule determined? the other side of geometry... Symmetrical or asymmetrical for { eq } O_2 { /eq }: this describes. Below that contains at least one polar covalent bond, the He, Ne, Kr, Ar. Pairs and lone pairs around the central atom around the central atom do so, the electron-pair geometry is by! Molecule containing a central atom determines the shape of a molecule in electron domain is used in theory. Using AdBlock Plus or some other adblocking software pair repulsion ( VSEPR ).... Water molecule is determined works as how is the electron geometry of a molecule determined? quality moderator and content writer for difference Wiki 7.16 describe all regions electrons... 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Do you expect deviations from the idealized bond angle and lone pairs tetrahedral- CF4 trigonal Pyramidal- NF3 OF2! Ad blocking whitelist or disable your adblocking software positions of the molecule can be by... B ) NF3 ( c ) OF2 ( d ) H2S comes from our online advertising compound have... The molecules the repulsions between the lone electron pairs at valence shell electron pair geometry also named the theory! Adblock Plus or some other adblocking software for each molecule energy, then it can be by... Are named exactly like the Electron-group geometry graduated from the idealized bond angle and idealized = 2x4+2 e.! Chci3 a. PF3 b. SB12 d. CS2 CF4 ( b ) NF3 ( c ) OF2 d. Fyi, the steric number of electron geometry, and idealized bond angle be less... Very important from the idealized bond angles Concept Videos main developers, Ronald Gillespie and Ronald.... Of lone pairs around the central atom determines the shape of a molecule with double bonds, carbon... 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Pair electrons around the central atom is carbon, and Ar have high... For new subjects of any molecule can be understood by the arrangement of atoms in the shape! Shapes for molecules with up to six electron pairs and bond electron pairs around the central.! Problem: determine the electron geometry white ) are 180° away from one another... use this number to the! A. PF3 b. SB12 d. CS2 the SN … molecular geometry you have to the! Opposite side, molecular geometry does not contain electron pairs Computer Science pairs ; the! And Ronald Nyholm you the steric number ( SN ) — the number of lone pairs around the atom... Determining the shape of the following molec pf 3, CS 2 understood by the difference between electronic geometry molecular. The lone pair of electrons having different densities angle for each molecule of molecules, if the bond energy higher! The opposite side, molecular geometry for each molecule CF4 NF3 OF2 H2S very high Ionization energies, Gillespie. With sp2 hybridization has a ( n ) _____ electron geometry considers electron pair using VSEPR theory will. Is summarized that the geometry of a molecule containing a central atom and the geometry! Vsepr shape to determine the electron group geometry is the oxygen molecule having two lone pairs mainly... Other side of the molecule pairs around the central atom all possible for., Ne, Kr, and Ar have very high Ionization energies the three-dimensional arrangement of atoms the! Molecule containing a central atom determines the shape of a molecule is bent of describing the shapes of.! { CF } _ { 4 } \quad \text { b. creating a Lewis structure of the molecule students... And molecular geometry and not in molecular geometry does not contain electron pairs are arranged to minimize repulsion drawn! The electrons in Figure 7.16 describe all regions where electrons are located, bonds well. And Ronald Nyholm CHCI3 a. PF3 b. SB12 d. CS2 is a way so that repulsion. One another the molecules consider carbon dioxide, CO 2 molecular geometry, molecular geometry not. Chci3 a. PF3 b. SB12 d. CS2 in forming bonds and making molecules advertising. Using VSEPR theory second, the steric number how is the electron geometry of a molecule determined? bonds between the atoms being less to find out the of. Of ads planar, mg = trigonal planar nucleus or the other electrons in C2H2 molecule = 2x4+2 =10 the. For new subjects of any molecule can be determined by decreasing the between! Harlon currently works as a quality moderator and content writer for difference.... Is as follows its main purpose is to find out the geometry of the molecule predicted using the of. This video describes one method for quickly finding the major geometrical shapes for with... Determined using the valence bond approximation this can be determined by electron groups are all pairs! ; Watch KCV 11.3, IWE 11.2 c. CHCI3 a. PF3 b. SB12 d. CS2 single, double or. Bonds and making molecules group geometry is determined by the quantum mechanical behavior the... The angle will be slightly less than 90 0 thing as its molecular structure is the structure of atoms. Structure First, the electron-pair geometry around a central atom using the VESPR theory its valence electrons are also same. All bond pairs, they how is the electron geometry of a molecule determined? named exactly like the Electron-group geometry angles... Not counted or the other side of the electrons is preventing the page from fully loading theory... Excluding the lone pair of electrons angles Concept Videos molecule shape, called the VSEPR model ( shell... Approximation this can be determined by the central atom Ar have very high energies... 7.16 describe all regions where electrons are located, bonds as well lone of.
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(* float-lit.sml * * COPYRIGHT (c) 2010 The Diderot Project (http://diderot.cs.uchicago.edu) * All rights reserved. * * Internal representation of floating-point literals with limited * support for arithmetic. *) structure FloatLit :> sig type float val isZero : float -> bool (* return the representation of +/-0.0 *) val zero : bool -> float (* plus and minus one *) val one : float val m_one : float (* negate a float *) val negate : float -> float (* equality, comparisons, and hashing functions *) val same : (float * float) -> bool val compare : (float * float) -> order (* not ordering on reals *) val hash : float -> word (* special floats *) val nan : float (* some quiet NaN *) val posInf : float (* positive infinity *) val negInf : float (* negative infinity *) (* create a float from pieces: isNeg is true if the number is negative, whole * is the whole-number part, frac is the fractional part, and exp is the * exponent. This function may raise Overflow, when the exponent of the * normalized representation is too small or too large. *) val float : {isNeg : bool, whole : string, frac : string, exp : int} -> float val toString : float -> string val toReal : float -> real (* external representation (for pickling) *) val toBytes : float -> Word8Vector.vector val fromBytes : Word8Vector.vector -> float end = struct structure SS = Substring structure W = Word structure W8V = Word8Vector (* The value {isNeg, digits=[d0, ..., dn], exp} represents the number * * [+/-] 0.d0...dn * 10^exp * * where the sign is negative if isNeg is true. *) datatype float = PosInf (* positive infinity *) | NegInf (* negative infinity *) | NaN (* some quiet NaN *) | Flt of {isNeg : bool, digits : int list, exp : int} (* special floats *) val nan = NaN val posInf = PosInf val negInf = NegInf fun isZero (Flt{isNeg, digits=[0], exp}) = true | isZero _ = false fun zero isNeg = Flt{isNeg = isNeg, digits = [0], exp = 0} val one = Flt{isNeg = false, digits = [1], exp = 1} val m_one = Flt{isNeg = true, digits = [1], exp = 1} (* negate a float *) fun negate PosInf = NegInf | negate NegInf = PosInf | negate NaN = raise Fail "negate nan" | negate (Flt{isNeg, digits, exp}) = Flt{isNeg = not isNeg, digits = digits, exp = exp} (* equality, comparisons, and hashing functions *) fun same (NegInf, NegInf) = true | same (PosInf, PosInf) = true | same (NaN, NaN) = true | same (Flt f1, Flt f2) = (#isNeg f1 = #isNeg f2) andalso (#exp f1 = #exp f2) andalso (#digits f1 = #digits f2) | same _ = false fun compare (NegInf, NegInf) = EQUAL | compare (NegInf, _) = LESS | compare (_, NegInf) = GREATER | compare (PosInf, PosInf) = EQUAL | compare (PosInf, _) = LESS | compare (_, PosInf) = GREATER | compare (NaN, NaN) = EQUAL | compare (NaN, _) = LESS | compare (_, NaN) = GREATER | compare (Flt f1, Flt f2) = (case (#isNeg f1, #isNeg f2) of (false, true) => GREATER | (true, false) => LESS | _ => (case Int.compare(#exp f1, #exp f2) of EQUAL => let fun cmp ([], []) = EQUAL | cmp ([], _) = LESS | cmp (_, []) = GREATER | cmp (d1::r1, d2::r2) = (case Int.compare(d1, d2) of EQUAL => cmp(r1, r2) | order => order (* end case *)) in cmp (#digits f1, #digits f2) end | order => order (* end case *)) (* end case *)) fun hash PosInf = 0w1 | hash NegInf = 0w3 | hash NaN = 0w5 | hash (Flt{isNeg, digits, exp}) = let fun hashDigits ([], h, _) = h | hashDigits (d::r, h, i) = hashDigits (r, W.<<(W.fromInt d, i+0w4), W.andb(i+0w1, 0wxf)) in hashDigits(digits, W.fromInt exp, 0w0) end fun float {isNeg, whole, frac, exp} = let fun cvtDigit (c, l) = (Char.ord c - Char.ord #"0") :: l fun isZero #"0" = true | isZero _ = false (* whole digits with leading zeros removed *) val whole = SS.dropl isZero (SS.full whole) (* fractional digits with trailing zeros removed *) val frac = SS.dropr isZero (SS.full frac) (* normalize by stripping leading zero digits *) fun normalize {isNeg, digits=[], exp} = zero isNeg | normalize {isNeg, digits=0::r, exp} = normalize {isNeg=isNeg, digits=r, exp=exp-1} | normalize flt = Flt flt in case SS.foldr cvtDigit (SS.foldr cvtDigit [] frac) whole of [] => zero isNeg | digits => normalize { isNeg = isNeg, digits = digits, exp = exp + SS.size whole } (* end case *) end fun toString PosInf = "+inf" | toString NegInf = "-inf" | toString NaN = "nan" | toString (Flt{isNeg, digits, exp}) = let val s = if isNeg then "-0." else "0." val e = if exp < 0 then ["e-", Int.toString(~exp)] else ["e", Int.toString exp] in concat(s :: List.foldr (fn (d, ds) => Int.toString d :: ds) e digits) end fun toReal PosInf = Real.posInf | toReal NegInf = Real.negInf | toReal NaN = 0.0 / 0.0 | toReal x = valOf(Real.fromString(toString x)) (* FIXME *) (***** external representation (for pickling) ***** * * The representation we use is a sequence of bytes: * * [sign, d0, ..., dn, exp0, ..., exp3] * * where * sign == 0 or 1 * di == ith digit * expi == ith byte of exponent (exp0 is lsb, exp3 is msb). * * we encode Infs and NaNs using the sign byte: * * 2 == PosInf * 3 == NegInf * 4 == NaN * * NOTE: we could pack the sign and digits into 4-bit nibbles, but we are keeping * things simple for now. *) fun toBytes PosInf = Word8Vector.fromList [0w2] | toBytes NegInf = Word8Vector.fromList [0w3] | toBytes NaN = Word8Vector.fromList [0w4] | toBytes (Flt{isNeg, digits, exp}) = let val sign = if isNeg then 0w1 else 0w0 val digits = List.map Word8.fromInt digits val exp' = W.fromInt exp fun byte i = Word8.fromLargeWord(W.toLargeWord((W.>>(exp', 0w8*i)))) val exp = [byte 0w0, byte 0w1, byte 0w2, byte 0w3] in Word8Vector.fromList(sign :: (digits @ exp)) end fun fromBytes v = let fun error () = raise Fail "Bogus float pickle" val len = W8V.length v in if (len = 1) then (case W8V.sub(v, 0) (* special float value *) of 0w2 => PosInf | 0w3 => NegInf | 0w4 => NaN | _ => error() (* end case *)) else let val ndigits = W8V.length v - 5 val _ = if (ndigits < 1) then error() else () val isNeg = (case W8V.sub(v, 0) of 0w0 => false | 0w1 => true | _ => error() (* end case *)) fun digit i = Word8.toInt(W8V.sub(v, i+1)) fun byte i = W.<<( W.fromLargeWord(Word8.toLargeWord(W8V.sub(v, ndigits+1+i))), W.fromInt(8*i)) val exp = W.toIntX(W.orb(byte 3, W.orb(byte 2, W.orb(byte 1, byte 0)))) in Flt{isNeg = isNeg, digits = List.tabulate(ndigits, digit), exp = exp} end end end
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The output has ended thus: ) in Flt{isNeg = isNeg, digits = List.tabulate(ndigits, digit), exp = exp} end end end
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``````/*****************************************************************************
*
* Grover's Search Algorithm
*
* Implements Grover's Search algorithm in the Scaffold programming language
*
* February 2013
*
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#include <math.h>
#define n 10 // problem size (log of database size)
#define pi 3.141592653589793238462643383279502884197
// Module prototypes
void diffuse(qbit q[n]);
void Sqr(qbit a[n], qbit b[n]);
void EQxMark(qbit b[n], qbit t[1], int tF);
/***************************
Diffusion Module
***************************/
void diffuse(qbit q[n]) {
int j;
// local qbit x[n-1]; // scratch register
// No local registers yet
qbit x[n-1]; // scratch register
// No forall yet
// forall(j=0; j<n; j++) { H(q[j]); }
for(j=0; j<n; j++) { H(q[j]); }
// Want to phase flip on q = 00...0
// So invert q to compute q[0] and q[1] and ... and q[n-1]
for(j = 0; j < n; j++)
X(q[j]);
// Compute x[n-2] = q[0] and q[1] and ... and q[n-1]
// No forall yet
// forall(j=0; j<n-1; j++) PrepZ(x[j]);
for(j=0; j<n-1; j++) PrepZ(x[j],0);
Toffoli(q[1], q[0], x[0]);
for(j = 1; j < n-1; j++)
Toffoli(x[j-1], q[j+1], x[j]);
// Phase flip conditioned on x[n-2]
Z(x[n-2]); // Phase Flip==Z if q=00...0, i.e. x[n-2]==1
// Undo the local registers
for(j = n-2; j > 0; j--)
Toffoli(x[j-1], q[j+1], x[j]);
Toffoli(q[1], q[0], x[0]);
// Restore q
for(j = 0; j < n; j++)
X(q[j]);
// Complete the diffusion
// No forall yet
// forall(j=0; j<n; j++) { H(q[j]); }
for(j=0; j<n; j++) { H(q[j]); }
}
/***********************************************
Test if the input polynomial
b(x) = b0 + b1*x + ... + b_(n-1)*x^(n-1) == x
over the ring GF(2)[x] mod (x^n + x + 1).
if(tF!=0) set return result in qubit t[0] else Z
************************************************/
void EQxMark(qbit b[n], qbit t[1], int tF) {
int j;
// No local registers yet
// local qbit x[n-1]; // scratch register
qbit x[n-1]; // scratch register
// Change b to reflect testing for the polynomial x
for(j = 0; j < n; j++)
if(j!=1) X(b[j]);
// Compute x[n-2] = b[0] and b[1] and ... and b[n-1]
// No forall yet
// forall(j=0; j<n-1; j++) PrepZ(x[j]);
for(j=0; j<n-1; j++) PrepZ(x[j],0);
Toffoli(b[1], b[0], x[0]);
for(j = 1; j < n-1; j++)
Toffoli(x[j-1], b[j+1], x[j]);
// Either return result in t or Phase flip conditioned on x[n-2]
if(tF!=0) {
CNOT(x[n-2], t[0]); // Returns result in t
}
else {
Z(x[n-2]); // Phase Flip==Z if b=01...0 == 'x', i.e. x[n-2]==1
}
// Undo the local registers
for(j = n-2; j > 0; j--)
Toffoli(x[j-1], b[j+1], x[j]);
Toffoli(b[1], b[0], x[0]);
// Restore b
for(j = 0; j < n; j++)
if(j!=1) X(b[j]);
}
/***********************************************
Squaring a(x) = a0 + a1*x + ... + a_(n-1)*x^(n-1)
over the ring GF(2)[x] mod (x^n + x + 1).
Result placed in the n-qubit register b
************************************************/
void Sqr(qbit a[n], qbit b[n]) {
int i;
int k;
// Using forall indicates the CNOT's are independent
// So these instructions can be done in parallel
// No forall yet
// forall(i=0; i<=(n-1)/2; i++) {
for(i=0; i<=(n-1)/2; i++) {
k = 2*i;
CNOT(a[i], b[k]);
}
// Would it make a difference to split this into two loops?
// Maybe since deleaving would be two forall loops!
// Or perhaps it is okay to just replace the for with a forall.
for(i=(n+1)/2; i<n; i++) {
k = (2*i)-n;
CNOT(a[i], b[k]);
CNOT(a[i], b[k+1]);
}
}
/***************************************************
Program to compute the sqrt(x) in the polynomial ring
GF(2)[x] / (x^n+x+1)
Elements of the ring are represented as polynomials
of maximum degree n-1:
a(x) = a[0] + a[1]*x + ... + a[n-1]*x^(n-1)
Use grover search.
****************************************************/
int main() {
// Quantum registers
qbit a[n]; // Polynomials to search
qbit b[n]; // b(x) = a(x)*a(x) mod (x^n + x + 1)
qbit t[1]; // Test result if b(x) == x
// Grover parameters and step index
int N= pow(2,n);
int nstep = floor((pi/4)*sqrt(N));
int istep;
// Holds final measurement values
cbit mt[1]; // measure t
cbit ma[n]; // measure a This holds the square root
cbit mb[n]; // measure b No need to measure since b(x) should be x
int i;
// Initialize a[0..n-1] into uniform superposition
// No forall yet
// forall(i=0; i<n; i++) H(a[i]);
for(i=0; i<n; i++) H(a[i]);
// Grover iteration: Mark then diffuse
for(istep=1; istep<=nstep; istep++) {
Sqr(a, b); // Sets b(x) = a(x) * a(x)
EQxMark(b, t, 0); // Tests if b(x) == x and if so Phase Flips
Sqr(a, b); // Note: Sqr is its own inverse
// Diffuse
diffuse(a);
}
// For the final measurement, compute causal state
Sqr(a, b);
EQxMark(b, t, 1); // Note; 1 implies test result b(x)==x is returned in t
// Now measure and report
mt[0] = MeasZ(t[0]); // If mt[0]==1 then success
// measure a to recover the square root
for(i=0; i<n; i++) ma[i] = MeasZ(a[i]);
return 0;
}
// Problem instances
//main(2);
//main(3);
//main(5);
//main(8);
//main(13);
//main(19);``````
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https://tutorialspoint.dev/s?search=Inorder-Traversal
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Welcome To Tutorialspoint.dev
Officially launched 18th May 2019
# Level order traversal in spiral form
Write a function to print spiral order traversal of a tree. For below tree, function should print 1, 2, 3, 4, 5, 6, 7. Recommended: Please solve it on “PRAC...
# Construct Tree from given Inorder and Preorder traversals
Let us consider the below traversals: Inorder sequence: D B E A F C Preorder sequence: A B D E C F Recommended: Please solve it on “PRACTICE” first, be...
# Construct Special Binary Tree from given Inorder traversal
Given Inorder Traversal of a Special Binary Tree in which key of every node is greater than keys in left and right children, construct the Binary Tree and return root...
# Boundary Traversal of binary tree
Given a binary tree, print boundary nodes of the binary tree Anti-Clockwise starting from the root. For example, boundary traversal of the following tree is “2...
# Construct BST from given preorder traversal Set 2
Given preorder traversal of a binary search tree, construct the BST. For example, if the given traversal is {10, 5, 1, 7, 40, 50}, then the output should be root of f...
# Iterative Preorder Traversal
Given a Binary Tree, write an iterative function to print Preorder traversal of the given binary tree. Refer this for recursive preorder traversal of Binary Tree. To...
# Morris traversal for Preorder
Using Morris Traversal, we can traverse the tree without using stack and recursion. The algorithm for Preorder is almost similar to Morris traversal for Inorder. 1......
# Path Traversal Attack and Prevention
A path traversal attack allows attackers to access directories that they should not be accessing, like config files or any other files/directories that may contains s...
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https://fr.mathworks.com/matlabcentral/cody/problems/554-is-the-point-in-a-circle/solutions/1518230
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Cody
# Problem 554. Is the Point in a Circle?
Solution 1518230
Submitted on 5 May 2018 by Shreyash Singh
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
circle = [0, 0, 1]; Points = [0, 0.5] y_correct = 1; assert(isequal(your_fcn_name(Points,circle),y_correct))
Points = 0 0.5000
2 Pass
circle = [0, 0, 1]; Points = [0, 2] y_correct = 0; assert(isequal(your_fcn_name(Points,circle),y_correct))
Points = 0 2
3 Pass
circle = [1, 1, 1]; Points = [0, 1; 0, 0] y_correct = [1; 0]; assert(isequal(your_fcn_name(Points,circle),y_correct))
Points = 0 1 0 0
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http://mathhelpforum.com/advanced-statistics/73584-please-check-if-my-answer-correct.html
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Hi everyone.. I solved this problem but i'm not too confident..hope somebody can check and if it's wrong, please help me solve it the right way..
Afra and Wiro takes the train to their 8:30 AM class and arrives at the station uniformly between 7:00 AM and 7:20 AM. They both agreed that they are willing to wait for one another for 5 minutes, after which they take the train and ride alone or together. Assume that they always arrive randomly during that specified period and they don't communicate whatsoever before meeting. What is the probability that they will meet and ride the train together?
*Hint given by our professor: In a cartesian plane, a square of side 20 (minutes) represents all the possibilities of the morning arrivals of Wiro and Afra at the train station. For example (01, 07) means they arrive at 7:01 and 7:07, hence they ride the train alone. Now, the probability that they will meet= Area of region of meeting divided by the area of the square.
There are 420 possible morning arrivals and out of those 420 possibilities, there are 105 possibilities that they arrive within 5 minutes between each other. Therefore, i concluded that the probability that they will meet each other is 0.25.
*Based on my professor's hint, i also assumed that the area of the square=420 while the area of the region of the meeting=105. Hence, 105 divided by 420 is 0.25.
There are 420 possible morning arrivals and out of those 420 possibilities, there are 105 possibilities that they arrive within 5 minutes between each other.
How did you decide there were 105 possibilities?
If Afra arrives at 7:00 then Wiro has a 6 possibilities (7:00, 7:01, 7:02, 7:03, 7:04, 7:05) of arriving within Afra's waiting time.
If Afra arrives at 7:01 then Wiro has a 7 possibilities (7:00, 7:01, 7:02, 7:03, 7:04, 7:05, 7:06) of arriving within Afra's waiting time.
It seems to me that the closer Afra arrives to the 7:05-7:15 range, the better the chances get.
You could consider making a table of the 21 cases (don't panic, the 11 cases in the middle seem to have the same outcome so it's not a ton of work).
I haven't done the work to calculate this but I believe they have better than 25% odds for riding together.
3. Hey wytiaz.. sorry i think i did it wrong.. there should be 441 possibilities $(21^2)$.. and there can only be 6 possibilities that they will meet from each case.. so 6 multiplied by 21 cases= 126. so i think it's 126/441 = 0.2857143..what do you think?
4. Look at a sample case.
Afra arrives at 7:13
Which times could Wiro arrive allowing them to ride together?
7:08 (Wire has been waiting for Afra)
7:09 (Wire has been waiting for Afra)
7:10 (Wire has been waiting for Afra)
7:11 (Wire has been waiting for Afra)
7:12 (Wire has been waiting for Afra)
7:13 (they arrive together)
7:14 (Afra waits for Wiro)
7:15 (Afra waits for Wiro)
7:16 (Afra waits for Wiro)
7:17 (Afra waits for Wiro)
7:18 (Afra waits for Wiro)
There are 21 discrete times that Afra could arrive. Investigate them and see if you can come up with a table that shows how many of Wiro's times could let them go to school together. Then sum the possibilities and divide by 441.
5. I did a list of the possible times that afra arrives and waits for wiro so they can ride together. I got 111 possibilities that they can ride together if afra will wait for wiro or if wiro will arrive at the same time as afra..
What should I do now? Multiply that number by two since if i list down the possibilities that they will ride together if wiro will wait for afra, i will also get 111 possibilities?
If I use 111 as the # of possibilities that they will ride together, I will still come up with 25%..
However if I use 222, it'll be 50%..
What do you think?
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## Convert a TMSR key to PGP
29/05/18, modified 17/09/18
A script is floating around to convert TMSR key format (e,n,comment) to a PGP key for digesting in phuctor. This script did not work on the machines I tried it on. Of course, the script is fine, it's PGPY that is broken. I could not get it to install. As I'm programming in python for a living and have all kinds of stupid in me, I decided to try and fix the pgpy code that failed to install. An hour was so spent and some material gathered for a future blog post, but not any working code1.
After that I decided to spent another hour making an alternative that uses only standard python modules. I read RFC 4880 a month ago, this left me with headache back then. The thing is unreadable. So to make this script, I made extensive use of the search function in my browser and only read those lines that helped in writing the script.
The script;
```import struct
import time
import sys
import base64
import math
# some format strings for the struct module
# these are used to encode integers and shorts to arrays of bytes
# '>' stands for big-endian as this is what is used in the PGP format
openpgp_publickey_format = ">BIB"
mpi_format = ">H"
packet_length_format = ">I"
crc_format = ">I"
# determine the index of the highest bit set to 1 in a number
def count_bits(B):
R = 0
i = 0
while B > 0:
i += 1
if B & 0x1:
R = i
B >>= 1
return R
# Convert a number to an array of bytes
# The bytes in the array are stored in big-endian order.
# The most significant byte is stored as the first item
# in the array
def number_to_bytes(B):
R = []
bits = 0
while B > 0xff:
bits += 8
R.append(B & 0xff)
B >>= 8
R.append(B)
bits += count_bits(B)
return bits, ''.join(map(chr, reversed(R)))
# An MPI is a byte array that starts with a two byte
# length header. The length is given in bits.
def number_to_mpi(B):
C, A = number_to_bytes(B)
return struct.pack(mpi_format, C) + A
# A PGP public key header consists of a byte "4",
# an integer (4 bytes) to denote the timestamp
# and a byte "1" (RSA).
return struct.pack(openpgp_publickey_format, 4, T, 1)
# A public key packet is the public key header
# plus 2 MPI numbers, the RSA modulus (N) and
# the RSA exponent (e).
def public_key_packet(t, n, e):
# A comment or userid packet is a string encoded as utf-8
def userid_packet(s):
return s.encode('utf8')
# The PGP format is a stream of "packets".
# Each packet has a header. This header consists of a tag
# and a length field. The tag has flags to determine if it is a
# "new" or "old" packet.
# The only supported encoding in this scriptis "new".
def encode_packet(packet_bytes, tag = 6):
# 0x80, 8th bit always set, 7th bit set --> new packet
h = 0x80 | 0x40
# 0-5 bits -> the tag
h |= tag
# convert the integer to a byte
# dude, this is totally how you may save 2 or 3 bytes with minimal complexity
l = len(packet_bytes)
if l < 192:
elif l < 8384:
l -= 192
o1 = l >> 0xff
o2 = l & 0xff
header += chr(o1 + 192) + chr(o2)
else:
header += chr(0xff) + struct.pack(packet_length_format, l)
# When you encode binary data as an ascii text with base64
# this data becomes fragile. So a CRC code is needed to
# fix this.
def crc24(s):
R = 0xB704CE
for char in s:
B = ord(char)
R ^= B << 16
for i in range(8):
R <<= 1;
if R & 0x1000000:
R ^= 0x1864CFB
return R & 0xFFFFFF
# Create a public key for consumption by Phuctor.
# The public key needs to contain 2 packets
# one for the key data (n, e)
# one for the comment
# It must be in the armor / ascii format.
def enarmored_public_key(n, e, comment, t):
R = []
R.append("-----BEGIN PGP PUBLIC KEY BLOCK-----")
R.append("")
# the packets in bytes
A = encode_packet(public_key_packet(t, n, e), 6)
A += encode_packet(userid_packet(comment), 13)
# the packets in base64 encoding with line length max 76
s=base64.b64encode(A)
i = 0
while i < len(s):
R.append(s[i:i+76])
i += 76
# the CRC
R.append("="+base64.b64encode(struct.pack(crc_format, crc24(A))[1:]))
# the footer
R.append("")
R.append("-----END PGP PUBLIC KEY BLOCK-----")
return 'n'.join(R)
# read a file with comma separated lines
# each line is in the TMSR format: e,n,comment
if __name__ == "__main__":
ser = 1
for x in sys.stdin:
x = x.strip()
# ignore empty lines
if len(x) == 0 or x.startswith('#'):
continue
# the comment may contain comma's so split on the first 2
e,n,comment = x.split(',', 2)
t0 = int(time.time())
with open("{0}.txt".format(ser), "wb") as stream:
stream.write(enarmored_public_key(int(n), int(e), comment, t0))
ser += 1
```
And the patch itself with signature;
1. I've been reading code (both open and closed source) for a large part of my life. I started this whole career by typing over basic programs into my fathers Commodore 128 and then stumbled along. The code I read in these popular security programs (pgpy, openssl, openssh, pgp) is markedly worse than any I encountered before. I can only image the kind of cockroaches that are attracted to this foul smelling mess []
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https://durrell2012.com/what-is-seismic-hazard-analysis/
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What is seismic hazard analysis?
What is seismic hazard analysis?
Based on geological and seismological studies, probabilistic seismic hazard analysis (PSHA) estimates the likelihood of a hazard, considering the uncertainties in magnitude and the location of earthquakes and their resulting ground motions that are likely to affect a particular site.
What are the four types of earthquake hazards?
Primary earthquake hazards are:
• ground shaking.
• landslides.
• liquefaction.
• surface rupture.
Why do we do seismic hazard analysis?
It is an accepted trend in engineering practice to develop design response spectrum for different types of foundation materials such as rock, hard soil and weak soils. Analysis of lineaments and faults helps in understanding the regional seismotectonic activity of the area.
How do you manage seismic hazards?
We cannot prevent natural earthquakes from occurring but we can significantly mitigate their effects by identifying hazards, building safer structures, and providing education on earthquake safety. By preparing for natural earthquakes we can also reduce the risk from human induced earthquakes.
How is seismic risk calculated?
A simplified method of calculating seismic risk for a given city, involves the use of a street survey. If you know the level of seismic hazard, the damage generally follows established patterns.
What are the 5 earthquake related hazards?
Earthquake hazard is anything associated with an earthquake that may affect the normal activities of people. This includes surface faulting, ground shaking, landslide, liquefaction, tectonic deformation, tsunamis, and seiches.
What are the major geotechnical seismic hazards?
the principal geotechnical hazards associated with earthquakes are: 1 fault rupture 2 ground shaking 3 liquefaction and lateral spreading 4 landslides and rockfalls 5 tsunami. Each of these hazards is described in more detail below.
What is structural and nonstructural mitigation?
Non-structural mitigation differs most significantly from that of structural mitigation in that it reduces risk (likelihood and consequences) without requiring the use of engineered structures. Nonstructural mitigation techniques are often considered mechanisms where “man adapts to nature.”
How does building design reduce the impact of earthquakes?
Base isolation involves constructing a building on top of flexible pads made of steel, rubber, and lead. When the base moves during the earthquake, the isolators vibrate while the structure itself remains steady. This effectively helps to absorb seismic waves and prevent them from traveling through a building.
What are the two basic approaches to seismic hazard analysis?
There are two basic approaches to seismic hazard analysis. Both use the same basic body of information to determine what the “design earthquake” should be. The main difference is that the probabilistic approach systematically examines the uncertainties and includes the likelihood of an actual earthquake exceeding the design ground motion.
How do seismic hazard maps work for soft soil?
All of the previous developments (e.g., seismic hazard maps) were for sites on very firm soil. For sites on softer soil, the ground motions will be amplified. This slide shows ground motion occurring in rock (lower time history) and in a softer material such as a clay.
What is seseismic hazard analysis?
SEISMIC HAZARD ANALYSIS This topic addresses deterministic and probabilistic seismic hazard analysis, ground motion attenuation relationships, the U.S. Geological Survey (USGS) seismic hazard maps, the NEHRP Recommended Provisionsseismic design maps, site effects, directionality effects, and the NEHRP Recommended Provisionsresponse spectrum.
How is USGS earthquake hazard information applied in seismic design practice?
The design code developers first decide how USGS earthquake hazard information should be applied in design practice. Then, the USGS calculates values of seismic design parameters based on USGS hazard values and in accordance with design code procedures.
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# Pointデータ型で形を整える
```-- 空間の点を表す中間データ構造を作る
-- そうするともっと図形をもっとわかりやすくできるお
data Point = Point Float Float deriving (Show)
data Shape = Circle Point Float | Rectangle Point Point deriving (Show)
area :: Shape -> Float
area (Circle _ r) = pi * r ^ 2
area (Rectangle (Point x1 y1) (Point x2 y2))
= (abs \$ x2 - x1) * (abs \$ y2 - y1)
-- *Main> area (Rectangle (Point 0 0) (Point 100 100))
-- 10000.0
-- *Main> area (Circle (Point 0 0) 24)
-- 1809.5574
nudge :: Shape -> Float -> Float -> Shape
nudge (Circle (Point x y) r) a b = Circle (Point (x+a) (y+b)) r
nudge (Rectangle (Point x1 y1) (Point x2 y2)) a b
= Rectangle (Point (x1+a) (y1+b)) (Point (x2+a) (y2+b))
-- *Main> nudge (Circle (Point 34 34) 10) 5 10
-- Circle (Point 39.0 44.0) 10.0
-- 半径を取って、座標系の原点を中心とし、与えられた半径の円を作る
-- 関数を作る
baseCircle :: Float -> Shape
baseCircle r = Circle (Point 0 0) r
-- 幅と高さを取って、左下の頂点が原点にある長方形を作る関数を作る
baseRect :: Float -> Float -> Shape
baseRect width height = Rectangle (Point 0 0) (Point width height)
-- *Main> nudge (baseRect 40 100) 60 23
-- Rectangle (Point 60.0 23.0) (Point 100.0 123.0)
```
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https://www.coursehero.com/file/136317358/maths-practice-test-5pdf/
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# maths practice test 5.pdf - SECONDARY SCHOOL ACADEMIC YEAR...
• 12
Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 1 - 12 out of 12 pages.
SECONDARY SCHOOLACADEMIC YEAR 2020-2021Practice Test 5Name and Surname: ______________________________Class: 6 ___************************************************Date of Exam: ______ February, 2021Subject: MathematicsDuration: 40 minTeacher: Ms Sevil RamazanovaHOD: PhD Zahira Mammadova
1.a) Write 98 as a product of its prime factors.[1]_____________________b) Find the highest common factor (HCF) of 60 and 114[2]_____________________c) Light A flashes every 8 seconds. Light B flashes every 20 seconds.Both lights flash at the same time.Work out how long it will take for both lights to flash at the same time again.[3]__________________ seconds2.[2]2
3.c)4.b)[1]3
5. a)
End of preview. Want to read all 12 pages?
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https://convertoctopus.com/50-7-cubic-feet-to-pints
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## Conversion formula
The conversion factor from cubic feet to pints is 59.844155844181, which means that 1 cubic foot is equal to 59.844155844181 pints:
1 ft3 = 59.844155844181 pt
To convert 50.7 cubic feet into pints we have to multiply 50.7 by the conversion factor in order to get the volume amount from cubic feet to pints. We can also form a simple proportion to calculate the result:
1 ft3 → 59.844155844181 pt
50.7 ft3 → V(pt)
Solve the above proportion to obtain the volume V in pints:
V(pt) = 50.7 ft3 × 59.844155844181 pt
V(pt) = 3034.0987013 pt
The final result is:
50.7 ft3 → 3034.0987013 pt
We conclude that 50.7 cubic feet is equivalent to 3034.0987013 pints:
50.7 cubic feet = 3034.0987013 pints
## Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 pint is equal to 0.00032958716852934 × 50.7 cubic feet.
Another way is saying that 50.7 cubic feet is equal to 1 ÷ 0.00032958716852934 pints.
## Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that fifty point seven cubic feet is approximately three thousand thirty-four point zero nine nine pints:
50.7 ft3 ≅ 3034.099 pt
An alternative is also that one pint is approximately zero times fifty point seven cubic feet.
## Conversion table
### cubic feet to pints chart
For quick reference purposes, below is the conversion table you can use to convert from cubic feet to pints
cubic feet (ft3) pints (pt)
51.7 cubic feet 3093.943 pints
52.7 cubic feet 3153.787 pints
53.7 cubic feet 3213.631 pints
54.7 cubic feet 3273.475 pints
55.7 cubic feet 3333.319 pints
56.7 cubic feet 3393.164 pints
57.7 cubic feet 3453.008 pints
58.7 cubic feet 3512.852 pints
59.7 cubic feet 3572.696 pints
60.7 cubic feet 3632.54 pints
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# How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
I would like to prove the existence of $b \in \mathbb Q$ such that $a<b^2<c$ for any given $a,c \in \mathbb Q$ with $a,c>0$
I want to use the statement above to prove a statement in a link
I thought that '$b$' must be exist. But, in my opinion, $\sqrt{}$ can't not be used because $\sqrt{a}$ or $\sqrt{c}$ may not exist in $\mathbb Q$ for some $a$ and $c$.
I couldn't find a clue to prove the statement before the real number is constructed from $\mathbb Q$. Would you help me to prove that?
Thanks all for replying and pointing out errors.
-
More important, $c\gt0$. – Gerry Myerson Nov 14 '12 at 6:12
Ah. Thanks for pointing out that. I will correct them. – wikizero Nov 14 '12 at 6:12
For what it's worth, you can easily reduce it to the case where $a$ and $c$ are both integers by multiplying by the squares of the denominators. – Jonas Meyer Nov 14 '12 at 6:25
Assume a and c are integers. You need to find a square number between $k^2a$ and $k^2c$ for some $k$. Let $f(x)$ be the number of perfect squares $\leq x$. Then we need to show that $f(k^2 c) -f(k^2 a)>1$ for some $k$ (it would not suffice to show $f(k^2 b) -f(k^2 a)=1$ because c may be a square). This will require an estimate on the growth of $f(k^2x)$ as a function of $k$. – Jeff Tolliver Nov 14 '12 at 6:29
Thanks for commenting my question.//Then, I think that it suffice to prove $f(k^2 c)−f(k^2 a)>1$ in case that c=a+1. Is it true? – wikizero Nov 14 '12 at 6:54
Suppose that there is no such $b$. Then for each $n\in\Bbb Z^+$ there is a $k_n\in\Bbb N$ such that $$\frac{k_n^2}{n^2}\le a<c\le\frac{(k_n+1)^2}{n^2}=\frac{k_n^2}{n^2}+\frac{2k_n+1}{n^2}\;.$$ For each $n\in\Bbb Z^+$ we then have $$\frac{2k_n+1}{n^2}\ge c-a$$ and hence $$k_n\ge\frac{(c-a)n^2-1}2$$
and
\begin{align*} \frac{k_n^2}{n^2}&\ge\frac{\left((c-a)n^2-1\right)^2}{4n^2}\\ &=\frac{(c-a)^2n^2}4-\frac{c-a}2+\frac1{4n^2}\\ &\ge\frac{(c-a)^2n^2}4-\frac{c}2\;. \end{align*}
But $a\ge\dfrac{k_n^2}{n^2}$, so we have $$a\ge\frac{(c-a)^2n^2}4-\frac{c}2$$ and hence
$$n^2\le\frac{2(2a+c)}{(c-a)^2}$$
for all $n\in\Bbb Z^+$, contradicting the Archimedean property.
-
It seems $+\frac{a}2$ should read $-\frac{c}2$. (And with the same idea as in your proof, one could start from $2\sqrt{a}n+1\geqslant(c-a)n^2$, deduce that $(a+1)n+n\geqslant(c-a)n^2$, hence $n\leqslant(a+2)/(c-a)$, et voilà!) – Did Nov 14 '12 at 7:05
Thanks everyone for answering my question. Have a good day. – wikizero Nov 14 '12 at 7:06
@did: I did throw away the wrong term, didn’t I? Thanks. – Brian M. Scott Nov 14 '12 at 7:09
@wikizero: You’re welcome. – Brian M. Scott Nov 14 '12 at 7:09
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# READERS’ LETTERS: Understanding series and parallel speaker connections
A member writes:
Damping Factor(DF) is equal to the impedance of the load divided by the out put impedance of the amp. For example if an amp has an output impedance of say0.1 ohms and the speaker is 8 ohms the DF is 80. If you have 2 8 ohm speakers in series the total impedance is now 16 ohms and dividing by 0.1 gives a DF of 160, a higher DF. So damping goes up in series. Basically as impedance goes up DF goes up and vice versa.
---//---
A member writes:
1 ~ Have called PS Audio, &, McIntosh: Both recommend hooking up ONLY ONE speaker to an amplifier. While, when hooking up speakers in series, the individual resistances / impedances are additive, the McIntosh man said that this creates more current draw, perhaps too much! {Although I’ve done this a number of times, with powerful amps, didn’t have a problem}.
2 ~ In parallel: This DECREASES the resistance into the load, that the amplifier, `sees.’ As recalled, the formula is: Resulting Resistance = Resistance {of the speakers} multiplied, this divided by the SUM of the speaker impedances. Example, two 8 ohm speakers, 8 X 8 = 64, this divided by the sum, 8 + 8 = 16, the result is: 4 ohms. Be careful with this, actual speaker impedance is not linear, may dip, for example, as the frequency decreases. Suppose that you have music with a lot of bass? Some amp designs can drive low speaker impedances, down to 2 ohms, even less, and, some, can not! You don’t want to short out your amp! Be careful! Carefully check the specifications of the amp, & call the manufacturer. Some amps are designed to turn off if there is too much current draw, &, some aren’t, have been destroyed!
---//---
Another member writes ...
Current draw is the opposite of resistance. Voltage doesn’t change so lower resistance results in more current since lower resistance is an easier load. Series hookup results min higher resistance and therefore lower current draw. By the way all this assumes a solid state amp. You must have misunderstood the McIntosh advisor.
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https://www.convert-measurement-units.com/convert+Gigameter+to+Twip.php
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Convert Gm to Twip (Gigameter to Twip)
## Gigameter into Twip
numbers in scientific notation
https://www.convert-measurement-units.com/convert+Gigameter+to+Twip.php
# Convert Gm to Twip (Gigameter to Twip)
1. Choose the right category from the selection list, in this case 'Distance'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Gigameter [Gm]'.
4. Finally choose the unit you want the value to be converted to, in this case 'Twip'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
### Utilize the full range of performance for this units calculator
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '948 Gigameter'. In so doing, either the full name of the unit or its abbreviation can be usedas an example, either 'Gigameter' or 'Gm'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Distance'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '66 Gm to Twip' or '80 Gm into Twip' or '49 Gigameter -> Twip' or '32 Gm = Twip' or '15 Gigameter to Twip' or '63 Gigameter into Twip'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(77 * 60) Gm'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '12 Gigameter + 94 Twip' or '43mm x 26cm x 9dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 2.496 148 125 433 2×1021. For this form of presentation, the number will be segmented into an exponent, here 21, and the actual number, here 2.496 148 125 433 2. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 2.496 148 125 433 2E+21. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 2 496 148 125 433 200 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications.
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Categories
Recent Files
# Worksheet Decimal Division Grass Fedjp Study Worksheets Grade Wallpapercraft For Wordoblems Maths Pdf Fraction 5th Common Core Year 830x1074
By Peggy R. Rios on May 16 2018 10:40:54
As well as a printed blank hundred square, another way to visualise decimals is to make a decimal stick. Get a cane or long stick, place with tape a number 1 to the right hand end, then a 0 to the left. Ask them now to place 0.5 in the middle, then together work out where all the other decimal points should go.
These worksheets can help your students review decimals number concepts. Worksheets include place value, naming decimals to the nearest tenth and hundredth place, adding decimals, subtracting decimals, multiplying, dividing, and rounding decimals.
When teaching decimal numbers, first review the basics of thousands, hundreds, tens and ones and then introduce (or review) tenths, hundredths and thousandths.
If a child is really confused about decimals, converting decimal numnbers into money is a great way to make things clearer. For example: a child may be asked to say how much bigger 1.3 is than 0.9. If they convert these decimals into money (£1.30 and 90p) they may find that they can do this calculation very quickly in their head, getting the answer 40p which they convert back into the decimal, 0.4. Demonstrating that money maths depends on decimal understanding is also an easy way to prove that decimals are actually useful in real life!
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https://mynursingaffiliate.com/a-student-tries-to-raise-a-chain-consisting-of-three-identical-links-each-link-has-a-mass-of-200-g/
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A student tries to raise a chain consisting of three identical links. Each link has a mass of 200 g.
A student tries to raise a chain consisting of three identical links. Each link has a mass of 200 g. The three-piece chain is connected to a string and then suspended vertically, with the student holding the upper end of the string and pulling upward. Because of the student’s pull, an upward force of 16 N is applied to the chain by the string. Use Newton’s laws to answer the following questions.
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https://www.teclado.com/30-days-of-python/python-30-day-12-functions
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Day 12: Functions
Welcome to day 12 of the 30 Days of Python series! In this post we’re going to be taking a look at functions.
We’ve used a number of built-in functions throughout the course, such as `print` and `input`, but now we’re going to learn how to create some functions of our own.
Also we've created a short recap of Days 10 and 11 that you can view here to refresh your memory.
§Why use functions?
Before we start writing functions, we need to understand why they’re useful.
Let’s take the example of `print` for now. Printing is something we do all the time, both when testing our code and when providing information to users of our programs.
While this is sounds like a fairly simple operation, the code for `print` is actually over 80 lines long! `print` also calls several other functions in order to do its job, so the complete functionality provided by `print` actually requires several hundred lines of code.
This presents a couple of major benefit of using functions:
1. They allow us to cut down on repeating potentially long and complicated code for operations we want to perform multiple times.
2. They make our code more readable. It’s much easier to understand `print("Hello, world!")`, than the lengthy implementation of the `print` function.
This brings us to the third major benefit of functions: we don’t need to know how the underlying code for a function works. We’ve been using `print` since the first day of this series, but we don’t yet know anything about functions, and we certainly don’t know what `print` is doing when we call it.
It’s enough that we know how to use the function, and what the function will do with the values we provide. The implementation details can often be safely ignored.
§Defining our first function
To start off, we’re going to define a simple function called `get_even_numbers`. This function is going to print out the first ten even numbers, starting with `2`.
In case you’re unclear on how to get the first ten even numbers, we can use a `for` loop and `range` like so:
``````for number in range(1, 11):
print(number * 2)``````
We can also pass in a third value to `range` called `step`, which allows us to create sequences with different patterns. For example, specifying a `step` of `2` would allow us to get every second number in the specified range.
``````for number in range(2, 21, 2):
print(number)``````
Either of these approaches is fine.
So, how do we turn this into a function?
The first step is to write the `def` keyword, short for “define”. This indicates to Python that what follows is a function definition. One the same line, we then need to write the name of our function. This name is going to serve as an identifier for the function, just a like a variable name.
Directly after the function name, we need a pair of parentheses which are going to remain empty for now. Finally, we need to end the line with a colon. The first line of our function definition therefore looks like this:
``def get_even_numbers():``
However, this isn’t legal syntax right now, because all functions need a function body. This body contains the code that should be run when we call the function. In our case, we want to run that for loop we specified above, so we need to include it in our function body.
The body of a function is written directly underneath the line where we define the function name, and must be indented, just like when we use for loops or conditional statements.
``````def get_even_numbers():
for number in range(1, 11):
print(number * 2)``````
With that, we have a fully working function, and we can call it just like any of the built-in functions:
``````def get_even_numbers():
for number in range(1, 11):
print(number * 2)
get_even_numbers()``````
Run the code for yourself and see!
§Style note
As we start writing more and more complex functions, it’s going to be common for there to be empty lines within the function body to break up the code. For this reason, all of our function definitions should be followed by two empty lines.
This is going to help make it very easy to see where our functions end at a glance.
§Function parameters and arguments
The `get_even_numbers` function works great, but many of the functions we’ve been using throughout this series have been able to accept values when we call them. For example `print` accepts the values we want to output, and `input` accepts a prompt.
There are even some functions we have to pass a value to. We can’t call `len` without passing it a collection, for example. This is invalid:
``````len([1, 3, 5]) # 3
len() # Error!``````
In the second line we didn't pass it a collection, so we get a `TypeError`:
``````Traceback (most recent call last):
File "main.py", line 1, in <module>
len()
TypeError: len() takes exactly one argument (0 given)``````
If we want to accept values in our own functions, we need to tell Python to expect values. We do this by defining parameters.
Remember those parentheses in our function definition that didn’t seem to be doing very much? This is where we specify our parameters. In our original `get_even_numbers` function, we didn’t have any parameters, so the parentheses were empty:
``def get_even_numbers():``
Let’s update our `get_even_numbers` function so that the user can specify how many numbers to print out. We want to accept a single value, so we need to provide a single parameter. The value we pass in is called an argument.
A parameter is really just a variable, and it will provide us a way to access the arguments a user passes in. They serve as names for the argument values.
In the case of `get_even_numbers`, I’m going to define a parameter called `amount` by writing `amount` between the parentheses. I’m then going to use this parameter in the function body to modify the `range`.
``````def get_even_numbers(amount):
for number in range(1, amount + 1):
print(number * 2)``````
Don’t forget that the stop value for `range` is non-inclusive, so we have to add `1` if we want to generate the right amount of numbers in this case.
Now if we call our `get_even_numbers` function and pass in an integer as an argument, we’re going to be able to vary the amount of numbers we output.
``````def get_even_numbers(amount):
for number in range(1, amount + 1):
print(number * 2)
get_even_numbers(5)``````
Give it a try!
One thing you may notice is that we can no longer call the function without passing in any arguments. If we try, we’re going to get a `TypeError`, just like we do when trying to call `len` without passing in an argument.
``````Traceback (most recent call last):
File "main.py", line 6, in <module>
get_even_numbers()
TypeError: get_even_numbers() missing 1 required positional argument: 'amount'``````
The exception message is very helpful in this case. We get told we’re missing a required argument, and the parameter missing a value is called `amount`.
There are ways we can make certain arguments optional, but this is something we’re going to look at later in this series. For now it’s just important to realise that if we specify a parameter, we need to provide an argument for that parameter when we call the function.
§Specifying multiple parameters
Let’s define a new function called `x_print`. `x_print` is going to accept two arguments: a string to print, and a number of times to print that string.
So if we call `x_print` like this:
``x_print("Hello!", 5)``
We would expect `Hello!` printed to the console 5 times, like so:
``````Hello!
Hello!
Hello!
Hello!
Hello!``````
Defining this function is really no more complicated than the `get_even_numbers` function, but I want to draw your attention to how Python assigns arguments to parameter by default.
Here is one possible implementation of `x_print`:
``````def x_print(requested_output, quantity):
for _ in range(quantity):
print(requested_output)``````
Here we have a `for` loop where we grab items from a `range` sequence. However, we don’t actually care about the values in this `range`: we’re just using the `range` to ensure something happens a certain number of times. To indicate that the loop variable is unimportant, we’ve use an `_` as a variable name, which is a common convention in cases like this.
For each iteration of the loop, we print out whatever the user provided as their requested output.
One thing we need to be really clear on, is that Python doesn’t understand what any of these names we’ve defined mean. Parameters, like variables, are just names for our convenience as developers, so that we can easily reason about the sorts of values we’re dealing with.
When we have multiple parameters, by default Python is going to assign argument values to these parameters in order.
You might have noticed some of the error messages we’ve been reading have referred to positional arguments. That’s because these arguments are assigned to parameters based on their position in list of arguments.
This fact means the order we provide arguments to a function can be very important, and if we’re not careful, we can end up with some bugs. For example, if we write something like this,
``````def x_print(requested_output, quantity):
for _ in range(quantity):
print(requested_output)
x_print(5, "Hello")``````
We’re going to have some problems. Now `5` is assigned to `requested_output`, and `"Hello"` is assigned to `quantity`. This is going to give us an error, because we provide `quantity` to `range`, and `range` expects an integer.
What we get in this case is a `TypeError`:
``````Traceback (most recent call last):
File "main.py", line 6, in <module>
x_print(5, "Hello")
File "main.py", line 2, in x_print
for _ in range(quantity):
TypeError: 'str' object cannot be interpreted as an integer``````
§Keyword arguments
We’ve seen keyword arguments a couple of times in this series, but we’ve not really spoken about them directly. Keyword arguments are an alternative to positional arguments, where we specifically tie an argument’s value to a parameter name.
For example, we can call our `x_print` function like this:
``````def x_print(requested_output, quantity):
for _ in range(quantity):
print(requested_output)
x_print(requested_output="Hello", quantity=5)``````
Here we’ve specified that `"Hello"` should be the value of `requested_output`, and `5` should be the value for `quantity`. Because we’ve specified which value belongs to which parameter, we now don’t have to worry about the order we specify these values.
For example, this will work just fine:
``x_print(quantity=5, requested_output="Hello")``
We can also mix positional and keyword arguments, but we have to be a little bit careful here as there are several limitations.
We can’t provide a positional argument after a keyword argument. This is not valid syntax in Python:
`` x_print(requested_output="Hello", 5)``
If we provide a positional argument that maps onto a given parameter, we can’t then provide a keyword argument for that same parameter. For example, we can’t do this:
`` x_print(5, requested_output="Hello")``
The issue here is that `5` is a positional argument, and so gets assigned to `requested_output` (the first parameter). When we then specify a keyword argument for the same parameter, Python realises that something is wrong, and raises a `TypeError`.
If we want to mix positional and keyword arguments for our `x_print` function, we only have one viable option:
``x_print("Hello", quantity=5)``
This satisfies all of the requirements. The keyword arguments comes after the positional arguments, and we haven’t got any duplicate assignments caused by the order of our values.
It’s generally a good idea to use keyword arguments wherever you can, because they provide a lot of readability benefits. However, it’s not always possible to provide keyword arguments for parameters. Many built-in functions don’t accept keyword arguments for certain parameters, and in Python 3.8 we gained the ability to define positional only parameters as well.
These parameters will not accept keyword arguments. If you try, you’ll just get a `TypeError`.
There’s a lot more we need to learn about functions, but for now, let’s test what we’ve learn with some exercises.
§Exercises
1) Define four functions: `add`, `subtract`, `divide`, and `multiply`. Each function should take two arguments, and they should print the result of the arithmetic operation indicated by the function name.
When orders matters for an operation, the first argument should be treated as the left operand, and the second argument should be treated as the right operand. For example, if the user passes in `6` and `2` to subtract, the result should be `4`, not `-4`.
You should also make sure that the user can’t pass in `0` as the second argument for `divide`. If the user provides `0`, you should print a warning instead of calculating their division.
2) Define a function called `print_show_info` that has a single parameter. The argument passed to it will be a dictionary with some information about a T.V. show. For example:
`````` tv_show = {
"seasons": 5,
"initial_release": 2008
}
print_show_info(tv_show)``````
The `print_show_info` function should print the information stored in the dictionary, in a nice way. For example:
`` Breaking Bad (2008) - 5 seasons``
Remember you must define your function before calling it!
3) Below you’ll find a list containing details about multiple TV series.
`````` series = [
{"title": "Breaking Bad", "seasons": 5, "initial_release": 2008},
{"title": "Fargo", "seasons": 4, "initial_release": 2014},
{"title": "Firefly", "seasons": 1, "initial_release": 2002},
{"title": "Rick and Morty", "seasons": 4, "initial_release": 2013},
{"title": "True Detective", "seasons": 3, "initial_release": 2014},
{"title": "Westworld", "seasons": 3, "initial_release": 2016},
]``````
Use your function, `print_show_info`, and a `for` loop, to iterate over the `series` list, and call your function once for each iteration, passing in each dictionary. You should end up with each series printed in the appropriate format.
4) Create a function to test if a word is a palindrome. A palindrome is a string of characters that are identical whether read forwards or backwards. For example, “was it a car or a cat I saw” is a palindrome.
In the day 7 project, we saw a number of ways to reverse a sequence, and you can use this to verify whether a string is the same backwards as it is in its original order. You can also use a slicing approach. Once you’ve found whether or not a word is a palindrome, you should print the result to the user.
Make sure to clean up the argument provided to the function. We should be stripping whitespace from both ends of the string, and we should convert it all to the same case, just in case we’re dealing with a name, like `“Hannah”`.
You can find our solutions to the exercises here.
§Project
Once you’re done with the exercises, it’s time to tackle today’s project!
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Physics Help Forum (http://physicshelpforum.com/physics-help-forum.php)
- Periodic and Circular Motion (http://physicshelpforum.com/periodic-circular-motion/)
- - Kepler's Laws vs Energy (http://physicshelpforum.com/periodic-circular-motion/12261-keplers-laws-vs-energy.html)
haksaw22 Oct 30th 2016 03:00 PM
Kepler's Laws vs Energy
Screenshot by Lightshot
The question is in the link above.
I have already looked at unofficial markscheme for this, and it appears to have used vr=v'r' as a form of kepplers law of equal areas, which results in C being circled.
I used a different method, to get B, where I equated the PE and KE to get v^2 = k/r, and then solved for k. Then I plugged it back in with the new radius to get a new velocity of about 30. I have no clue whether or not my method holds, nor why, so it'd be nice to get some clarification on this.
Thanks
Btw im new here, apologies if this isnt where it should be
ChipB Oct 30th 2016 04:08 PM
Yes, the answer to the question as written is "about" 30 - actually 31.6Km/s.
I'm wondering if there isn't a typo in tjhe question - if the radius of the orbit at maximum was 10 x 10^11Km instead of 10 x 10^10Km, there would be a nice round number for the answer.
haksaw22 Oct 30th 2016 04:39 PM
Thanks for the confirmation! I'm pretty sure there isn't a typo, which is weird given that its a non calc paper. Do you know why the law of equal areas method they've used doesn't work?
ChipB Oct 30th 2016 09:12 PM
Sorry, but I realize I made a mistake in my earlier post. Using Kepler's Law of equal area in equal time does indeed yield an answer of (c), which is 20 Km/s. My error (and I think yours too) was in mistakenly using an equation that works for circular orbits, but this problem is about an elliptical orbit. I used a formula that equates force of gravity to mass times centripetal acceleration of the comet, which is valid for a circular orbit:
GMm/r^2 = m v^2/r
This yields v^2r = constant, but ignores the fact that at perigee and apogee the comet in an elliptical orbit has radial as well as centripetal acceleration, so gives the wrong answer.
Sorry for any confusion on this.
haksaw22 Oct 31st 2016 03:20 AM
No problem, thanks again! Makes more sense now, subtleties always catch me out.
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Expert Reviewed
# wikiHow to Calculate Annualized Portfolio Return
The calculation of your annualized portfolio return answers one question: what is the compound rate of return earned on the portfolio for the period of investment? While the various formulas used to calculate your annualized return may seem intimidating, it is actually quite easy to tabulate once you understand a few important concepts.
### Part 1 Laying the Groundwork
1. 1
Know the key terms. In discussing annualized portfolio returns, there are several key terms that will come up repeatedly and are important for you to understand. These are as follows:
• Annual Return: Total return earned on an investment over a period of one calendar year, including dividends, interest, and capital gains. [1]
• Annualized Return: Yearly rate of return which is inferred by extrapolating returns measured over periods either shorter or longer than one calendar year. [2]
• Average Return: Typical return earned per time period calculated by taking the total return realized over a longer period and spreading it evenly over the (shorter) periods. [3]
• Compounding Return: A return that includes the results of re-investing interest, dividends, and capital gains. [4]
• Period: A specific length of time chosen to measure and calculate return, such as daily, monthly, quarterly, or annually.
• Periodic Return: The total return of an investment measured over a specific length of time. [5]
2. 2
Learn how compounding returns work. Compounding returns are growth on the gains that you have already earned. The longer your money compounds, the faster it will grow, and the greater your annualized returns will be. (Think of a snowball rolling downhill, getting bigger faster as it rolls.)[6]
• Let’s say you invest \$100 and earn 100% on it your first year, leaving you with \$200 at the end of year one. If you gain just 10% in the second year, you will have earned \$20 on your \$200 by the end of year two.
• However, if we say you earned just 50% during the first year, you would have \$150 at the beginning of the second year. That same 10% gain in year two would earn 15 dollars rather than 20. This is a full 33% less than the 20 dollars you would have made in our first example.
• To further illustrate, let’s say you lost 50% in year one, leaving you with just 50 dollars. You would then need to earn 100% just to get back to even (100% of \$50 = \$50, and \$50 + \$50 = \$100).
• The size and timing of gains play a huge role when accounting for compound returns and their effect on annualized returns. In other words, annualized returns are not a reliable measure of actual gains or losses. Annualized returns are, however, a good tool to use when comparing various investments against each other.
3. 3
Use a time-weighted return to calculate your compound rate of return. To find the average of many things, such as daily rainfall or weight loss over several months, you can often use a simple average, or arithmetic mean. This is a technique you probably learned in school. However, the simple average does not account for the effect that each periodic return has on the others, or the timing of each return. To accomplish this, we can use a time-weighted geometric return.[7] (Don’t worry, we’ll walk you through this formula!)
• Using a simple average doesn’t work because all periodic returns are dependent on each other. [8]
• For example, imagine that you want to tabulate your average return on \$100 over the course of two years. You earned 100% in the first year, meaning you had \$200 at the end of year one (100% of 100 = 100). You lost 50% during the second, meaning you had \$100 at the end of the second year (50% of 200 = 100). This is the same figure you started with at the beginning of year one.
• A simple average (arithmetic mean) would add the two returns together and divide by the number of periods, which in this example is two years. The result would suggest that you earned an average return of 25% per year.[9] However, when you link the two returns, you can see that you actually earned nothing. The years cancel each other out.
4. 4
Calculate your overall return. To start, you must calculate your total return over the full span of time you are assessing. For the purpose of clarity, we’ll use an example where no deposits or withdrawals were made. To calculate your total return, all you need is two numbers: the beginning portfolio value and ending value.
• Divide this number by your Beginning Value. The resulting number is your Return.
• In the case of a loss in the period under scrutiny, subtract the ending balance from the beginning balance. Then, divide by the beginning balance and consider the result a negative value. (This latter operation is a substitute for needing to add algebraically a negative number.) [10]
• Do the subtraction first, then the division. This will give you your overall percent of return.
5. 5
Know the Excel formulas for these calculations. The formula for Total Return Rate = (Ending portfolio value- beginning portfolio value)/beginning portfolio value. The formula for Compound Rate of Return = POWER((1 + Total Return Rate),(1/years)) - 1.
• For example, if the beginning value of the portfolio was \$1000 and its ending value was \$2500 seven years later, the calculations would be:
• Total Return Rate = (2500-1000)/1000 = 1.5.
• Compound Rate of Return= POWER ((1 + 1.5),(1/7))-1 = .1398 = 13.98%.
### Part 2 Calculating Your Annualized Return
1. 1
Calculate your annualized return. Once you've calculated the total return (as above), plug the result into this equation: Annualized Return=(1+ Return)1/N-1[11] The outcome of this equation will be a number that corresponds to your return each year over the full span of time.
• In the exponent (the little number outside the parentheses), the “1” represents the unit we are measuring, which is 1 year. If you wish to be more specific, you could use “365” to capture a daily return.
• The “N” represents the number of periods that we are measuring. So, if you are measuring your return over 7 years, you would use the number 7 in the place of "N."
• For example, suppose that over a seven-year period, your portfolio grew in value from \$1,000 to \$2,500.
• First, calculate your overall return: (2,500-1,000)/1000 = 1.50 (a return of 150%).
• Next, calculate your annualized return: (1 + 1.50)1/7-1 = 0.1399=13.99% annual return. That's all there is to it!
• Use the ordinary mathematical order of operations: do the operations inside the parentheses first, then apply the exponent, then do the subtraction.
2. 2
Calculate semi-annual returns. Now, let's say that you want to find semiannual returns (returns occurring twice a year, every six months) over the course of this seven-year period. [12] The formula stays the same; you only need to adjust the number of periods that you are measuring. Your final result will be a semiannual return.
• In this case, you will have 14 semiannual periods, two per year over the course of seven years.
• First, calculate your overall return: (2,500-1,000)/1000 = 1.50 (a return of 150%).
• Next, calculate your annualized return: (1 + 1.50)1/14-1 = 6.76%.
• You can convert this into an annual return by simply multiplying by 2: 6.76% x 2 = 13.52%.
3. 3
Calculate an annualized equivalent. You can also calculate the annualized equivalent of shorter returns. For example, imagine you only had a six-month return and wanted to know its annualized equivalent. Once again, the formula stays the same.
• Suppose over a six-month period, your portfolio increases in value from \$1,000 to \$1,050.
• Start by calculating your overall return: (1,050-1,000)/1,000=.05 (a 5% return over six months).
• Now if you wanted to know what the annualized equivalent would be (assuming a continuation of this rate of return and compounding returns), [13] you would calculate the following: (1+.05)1/.50-1=10.25% annual return.
• No matter how long or short the period of time, if you follow the formula above, you will always be able to convert your performance into an annualized return.
## Community Q&A
Search
• How do I annualize a return on an investment that has cash added or subtracted during the year?
Michael R. Lewis
Michael R. Lewis is a retired corporate executive, entrepreneur, and investment advisor in Texas. He has over 40 years of experience in business and finance.
(1) Total the beginning Account Balance and any additions during the year to learn Total Investments. (2) Add any withdrawals during the year to the Ending Account Balance. (3) Subtract the sum of Step 1 from the sum of Step 2 to get total return. (4) Divide total return by the sum of Step 1 to get the rate of return within the year.
• How do I calculate total return for on an investment that amortizes monthly in equal amounts over a one year time period?
Michael R. Lewis
Michael R. Lewis is a retired corporate executive, entrepreneur, and investment advisor in Texas. He has over 40 years of experience in business and finance.
Deduct the beginning Account Value from the total payments (interest and principal) received during the year to calculate interest during the year. Then divide the interest earned by the beginning Account Value to get an annual rate of return.
• How do I calculate the return if there is a withdrawal?
If there is just one withdrawal or deposit (or just a few withdrawals or deposits), treat separately each time period before, between, and after withdrawals or deposits. Use each balance to calculate the return for a particular time period. Annualize each of the returns and weight them by length of time period. Add the returns together to arrive at the total annual return. Watch for changes in interest rate, and adjust accordingly.
• Can you explain Donagan's query with an example?
Here's a simple example: You have a savings account worth \$1,000 at the beginning of the year, earning 1% simple interest paid annually. You withdraw \$100 at the end of September. What's your rate of return for the full year? For the first nine months, your balance is \$1,000, and for the last three months it's \$900. So for the first nine months the interest you earn is (\$1,000) (1%) (9/12) = \$7.50. For the last three months your interest is (\$900) (1%) (3/12) = \$2.25. Your total interest for the year is \$9.75. (It would have been \$10 if you hadn't made the withdrawal.) To find your rate of return, divide \$9.75 by \$1,000, which is 0.00975 or 0.975% (slightly less than 1%). The point is: treat each time period (with its unique balance) separately, then add the balances together for the total interest earned (and divide by the original balance to obtain your annual rate of interest).
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## Tips
• Learning to calculate and understand annualized portfolio returns is important, as your annual return will be the number that you use to compare yourself to other investments as well as benchmarks and peers. It will have the power to confirm your stock-picking prowess and, more importantly, aid in uncovering any possible shortfalls in your investment strategies.
• Practice these calculations with some sample numbers to get comfortable with these equations. Practice will make these calculations come naturally and easily.
• The paradox mentioned at the very beginning of this article is merely a recognition of the fact that investment performance is usually judged against the performance of other investments. In other words, a small loss in a falling market may be considered better than a small gain in a rising market. It's all relative.
## Warnings
• Make sure to follow the correct mathematical order of operations or you will not get an accurate figure. Likewise, it's a good idea to double-check your work after performing these calculations.
## Article Info
Categories: Financial Ratios
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| 3.546875
| 4
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CC-MAIN-2017-39
|
latest
|
en
| 0.952622
|
https://cboard.cprogramming.com/c-programming/58343-if-statement.html
| 1,516,647,991,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2018-05/segments/1516084891530.91/warc/CC-MAIN-20180122173425-20180122193425-00689.warc.gz
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1. ## IF statement
i've got to be able to determine whether or not a number read in from a file is 5 digits or not. it can be from 00000 to 99999 in any order. as long as it is five digits.
if (number ??? && number ???)
{
}
what do i make the conditions for this?
2. Well 00000 is only going to equate to 1 digit if you convert it to an integer, so it's best to read in the number as ascii text, and then evaluate whether that text is 5 characters long.
Code:
`if(strlen(szNumber) == 5) { ... }`
3. Oh, and remember that szNumber needs to be zero terminated in order for the strlen() function to work.
4. ok, but the thing is.. i'm supposed to do it without strings. just old fashioned chars and ints..double, float.. but no string, struct, or anything
5. i tried this for each of the five numbers in the whole:
if (num_1 <= 1 && num_1 >= 9)
{
}
6. Just read the file one character at a time, then you can determine if it's a digit 0-9 or not and count up the number of consecutive digits
7. Look at the isdigit() function in ctype.h
8. Ok, but i still can't make it print out the right thing here...
#include <stdio.h>
#define FILE_NAME_SIZE 30
int get_course_record(FILE* infile, char* prefix, int* course_num, int* credits,
int* seats);
int get_student_record(FILE* infile, char* init_one, char* init_two, char* init_three, int* num_1, int* num_2, int* num_3, int* num_4, int* num_5);
int main(void)
{
char prefix;
int course_num, credits, seats, valid_status;
char init_one, init_two, init_three;
int num_1, num_2, num_3, num_4, num_5;
FILE *infile;
FILE *outfile;
char file_name[FILE_NAME_SIZE];
/************************************************** *******
(MAIN)COURSE SCHEDULE
************************************************** *******/
printf("please enter the name of the course schedule file");
scanf("%s", file_name);
infile = fopen(file_name, "r");
while (!feof (infile))
{
valid_status = get_course_record(infile, &prefix, &course_num, &credits,
&seats);
if (valid_status == 0)
{
printf("\ninvalid record\n");
}
else if (valid_status == 1)
{
printf("\n\n%c\n%d\n%d\n%d\n", prefix, course_num, credits, seats);
}
}
fclose(infile);
system("pause");
/************************************************** *****************************
(MAIN)STUDENT RECORDS
************************************************** *****************************/
printf("please enter the name of the student record file");
scanf("%s", file_name);
infile = fopen(file_name, "r");
while (!feof (infile))
{
valid_status = get_student_record(infile, &init_one, &init_two, &init_three, &num_1, &num_2, &num_3, &num_4, &num_5);
if (valid_status == 1)
{
printf("\n\n%c\n%c\n%c\n%d%d%d%d%d\n", &init_one, &init_two, &init_three, &num_1, &num_2, &num_3, &num_4, &num_5);
}
else if (valid_status == 0)
{
printf("\ninvalid record\n");
}
}
system("pause");
fclose(infile);
}
/************************************************** *****************************
(FUNC) COURSE SCHEDULE
************************************************** *****************************/
int get_course_record(FILE* infile, char* prefix, int* course_num, int* credits,
int* seats)
{
char endl;
int valid_status = 1;
if( fscanf(infile, "%s%d%d%d%c", prefix, course_num, credits, seats, &endl))
{
if (!(*prefix >= 'a' && *prefix <= 'z'))
{
valid_status = 0;
}
if (!(*course_num >= 101 && *course_num <= 499))
{
valid_status = 0;
}
if (!(*credits >= 1 && *credits <= 6))
{
valid_status = 0;
}
if (!(*seats >= 5 && *seats <= 50))
{
valid_status = 0;
}
}
else
{
printf("invalid record\n");
valid_status = 0;
}
return valid_status;
}
/************************************************** *****************************
(FUNC) STUDENT RECORDS
************************************************** *****************************/
int get_student_record(FILE* infile, char* init_one, char* init_two, char* init_three, int* num_1, int* num_2, int* num_3, int* num_4, int* num_5)
{
char endl;
int sr_count;
int valid_status = 1;
printf("%c%c%c\n%d\n", *init_one, *init_two, *init_three, *num_1, *num_2, *num_3, *num_3, *num_4, *num_5);
if( fscanf(infile, "%c%c%c%d%d%d%d%d%c", init_one, init_two, init_three, num_1, num_2, num_3, num_4, num_5, &endl))
{
if (!(*init_one >= 'A' && *init_one <= 'Z'))
{
(sr_count)++;
}
if (!(*init_two >= 'A' && *init_one <= 'Z'))
{
(sr_count)++;
}
if (!(*init_three >= 'A' && *init_one <= 'Z'))
{
(sr_count)++;
}
if (!(*num_1 >= 0 && *num_1 <= 9))
{
(sr_count)++;
}
if (!(num_2 >= 0 && *num_2 <= 9))
{
(sr_count)++;
}
if (!(*num_3 >= 0 && *num_3 <= 9))
{
(sr_count)++;
}
if (!(*num_4 >= 0 && *num_4 <= 9))
{
(sr_count)++;
}
if (!(*num_5 >= 0 && *num_5 <= 9))
{
(sr_count)++;
}
if(sr_count = 5)
{
valid_status = 1;
}
}
else
{
valid_status = 0;
}
return valid_status;
}
and its all supposed to be in C not using strings, arrays, or anything.
9. Well, it's going to be pretty difficult to get a filename from the user if you can't use strings. Your current method of getting the filename is using a string...In fact, fopen() requires that you pass a string to it, so it's impossible not to use strings in your program. So either your professor gave you an impossible assignment or you're putting unnecessary constraints on the assignment.
Once you get the filename, you can just use fgetc() on the file stream 5 times, each time checking to see if the character is a digit. If all 5 are digits then try fgetc() one more time. If that one is a digit then the number isn't 5 digits either.
10. brabner-
11. sorry. will do.
by not using strings i mean, except that one case, that he gave us the code for. other than that, no strings. we haven't covered them yet.
12. i changed it a little, but it's still returning garbage.
Code:
```
#include <stdio.h>
#define FILE_NAME_SIZE 30
int get_course_record(FILE* infile, char* prefix, int* course_num, int* credits,
int* seats);
int get_student_record(FILE* infile, char* init_one, char* init_two, char* init_three, int* sr_num);
int main(void)
{
char prefix;
int course_num, credits, seats, valid_status;
char init_one, init_two, init_three;
int sr_num;
FILE *infile;
FILE *outfile;
char file_name[FILE_NAME_SIZE];
/*********************************************************
(MAIN)COURSE SCHEDULE
*********************************************************/
printf("please enter the name of the course schedule file");
scanf("%s", file_name);
infile = fopen(file_name, "r");
while (!feof (infile))
{
valid_status = get_course_record(infile, &prefix, &course_num, &credits,
&seats);
if (valid_status == 0)
{
printf("\ninvalid record\n");
}
else if (valid_status == 1)
{
printf("\n\n%c\n%d\n%d\n%d\n", prefix, course_num, credits, seats);
}
}
fclose(infile);
system("pause");
/*******************************************************************************
(MAIN)STUDENT RECORDS
*******************************************************************************/
printf("please enter the name of the student record file");
scanf("%s", file_name);
infile = fopen(file_name, "r");
while (!feof (infile))
{
valid_status = get_student_record(infile, &init_one, &init_two, &init_three, &sr_num);
if (valid_status == 1)
{
printf("%c %c %c %d", &init_one, &init_two, &init_three, &sr_num);
}
else if (valid_status == 0)
{
printf("\ninvalid record\n");
}
}
system("pause");
fclose(infile);
}
/*******************************************************************************
(FUNC) COURSE SCHEDULE
*******************************************************************************/
int get_course_record(FILE* infile, char* prefix, int* course_num, int* credits,
int* seats)
{
char endl;
int valid_status = 1;
if( fscanf(infile, "%c%d%d%d%c", prefix, course_num, credits, seats, &endl))
{
if (!(*prefix >= 'a' && *prefix <= 'z'))
{
valid_status = 0;
}
if (!(*course_num >= 101 && *course_num <= 499))
{
valid_status = 0;
}
if (!(*credits >= 1 && *credits <= 6))
{
valid_status = 0;
}
if (!(*seats >= 5 && *seats <= 50))
{
valid_status = 0;
}
}
else
{
printf("invalid record\n");
valid_status = 0;
}
return valid_status;
}
/*******************************************************************************
(FUNC) STUDENT RECORDS
*******************************************************************************/
int get_student_record(FILE* infile, char* init_one, char* init_two, char* init_three, int* sr_num)
{
char endl;
int sr_count;
int valid_status = 1;
if( fscanf(infile, "%c%c%c%d%c", init_one, init_two, init_three, sr_num, &endl))
{
if (!(*init_one >= 'A' && *init_one <= 'Z'))
{
valid_status = 0;
}
if (!(*init_two >= 'A' && *init_one <= 'Z'))
{
valid_status = 0;
}
if (!(*init_three >= 'A' && *init_one <= 'Z'))
{
valid_status = 0;
}
if(!(*sr_num >= 10000 && *sr_num <= 99999))
{
valid_status = 0;
}
}
else
{
valid_status = 1;
}
return valid_status;
}```
13. its returning this phantom
c
b
a
invalid
theres no cba in my program ....
14. Code:
```#include <stdio.h>
#define FILE_NAME_SIZE 30
int get_course_record(FILE* infile, char* prefix, int* course_num, int* credits,
int* seats);
int get_student_record(FILE* infile, char* init_one, char* init_two, char* init_three, int* num_1, int* num_2, int* num_3, int* num_4, int* num_5);
int main(void)
{
char prefix;
int course_num, credits, seats, valid_status;
char init_one, init_two, init_three;
int num_1, num_2, num_3, num_4, num_5;
FILE *infile;
FILE *outfile;
char file_name[FILE_NAME_SIZE];
/*********************************************************
(MAIN)COURSE SCHEDULE
*********************************************************/
printf("please enter the name of the course schedule file");
scanf("%s", file_name);
infile = fopen(file_name, "r");
while (!feof (infile))
{
valid_status = get_course_record(infile, &prefix, &course_num, &credits,
&seats);
if (valid_status == 0)
{
printf("\ninvalid record\n");
}
else if (valid_status == 1)
{
printf("\n\n%c\n%d\n%d\n%d\n", prefix, course_num, credits, seats);
}
}
fclose(infile);
system("pause");
/*******************************************************************************
(MAIN)STUDENT RECORDS
*******************************************************************************/
printf("please enter the name of the student record file");
scanf("%s", file_name);
infile = fopen(file_name, "r");
while (!feof (infile))
{
valid_status = get_student_record(infile, &init_one, &init_two, &init_three, &num_1, &num_2, &num_3, &num_4, &num_5);
if (valid_status == 1)
{
printf("\n\n%c\n%c\n%c\n%d%d%d%d%d\n", &init_one, &init_two, &init_three, &num_1, &num_2, &num_3, &num_4, &num_5);
}
else if (valid_status == 0)
{
printf("\ninvalid record\n");
}
}
system("pause");
fclose(infile);
}
/*******************************************************************************
(FUNC) COURSE SCHEDULE
*******************************************************************************/
int get_course_record(FILE* infile, char* prefix, int* course_num, int* credits,
int* seats)
{
char endl;
int valid_status = 1;
if( fscanf(infile, "%s%d%d%d%c", prefix, course_num, credits, seats, &endl))
{
if (!(*prefix >= 'a' && *prefix <= 'z'))
{
valid_status = 0;
}
if (!(*course_num >= 101 && *course_num <= 499))
{
valid_status = 0;
}
if (!(*credits >= 1 && *credits <= 6))
{
valid_status = 0;
}
if (!(*seats >= 5 && *seats <= 50))
{
valid_status = 0;
}
}
else
{
printf("invalid record\n");
valid_status = 0;
}
return valid_status;
}
/*******************************************************************************
(FUNC) STUDENT RECORDS
*******************************************************************************/
int get_student_record(FILE* infile, char* init_one, char* init_two, char* init_three, int* num_1, int* num_2, int* num_3, int* num_4, int* num_5)
{
char endl;
int sr_count;
int valid_status = 1;
printf("%c%c%c\n%d\n", *init_one, *init_two, *init_three, *num_1, *num_2, *num_3, *num_3, *num_4, *num_5);
if( fscanf(infile, "%c%c%c%d%d%d%d%d%c", init_one, init_two, init_three, num_1, num_2, num_3, num_4, num_5, &endl))
{
if (!(*init_one >= 'A' && *init_one <= 'Z'))
{
(sr_count)++;
}
if (!(*init_two >= 'A' && *init_one <= 'Z'))
{
(sr_count)++;
}
if (!(*init_three >= 'A' && *init_one <= 'Z'))
{
(sr_count)++;
}
if (!(*num_1 >= 0 && *num_1 <= 9))
{
(sr_count)++;
}
if (!(num_2 >= 0 && *num_2 <= 9))
{
(sr_count)++;
}
if (!(*num_3 >= 0 && *num_3 <= 9))
{
(sr_count)++;
}
if (!(*num_4 >= 0 && *num_4 <= 9))
{
(sr_count)++;
}
if (!(*num_5 >= 0 && *num_5 <= 9))
{
(sr_count)++;
}
if(sr_count = 5)
{
valid_status = 1;
}
}
else
{
valid_status = 0;
}
return valid_status;
}```
Code:
```itsme@itsme:~/C\$ cat is5digits.c
#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <ctype.h>
int main(void)
{
char filename[100];
FILE *fp;
int i;
int valid = 1;
printf("Enter filename: ");
fflush(stdout);
fgets(filename, sizeof(filename), stdin);
filename[strlen(filename)-1] = '\0';
if(!(fp = fopen(filename, "r")))
{
printf("Couldn't open '%s' for reading!\n", filename);
return EXIT_FAILURE;
}
for(i = 0;i < 5;++i)
{
if(!isdigit(fgetc(fp)))
{
valid = 0;
break;
}
}
if(valid && isdigit(fgetc(fp)))
valid = 0;
puts(valid ? "Valid" : "Invalid");
return EXIT_SUCCESS;
}
itsme@itsme:~/C\$ cat 5digits.txt
03479
itsme@itsme:~/C\$ ./is5digits
Enter filename: 5digits.txt
Valid
itsme@itsme:~/C\$```
| 3,666
| 13,314
|
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
| 2.71875
| 3
|
CC-MAIN-2018-05
|
latest
|
en
| 0.697583
|
https://kalkicode.com/level-order-traversal-using-queue-in-go
| 1,701,422,646,000,000,000
|
text/html
|
crawl-data/CC-MAIN-2023-50/segments/1700679100286.10/warc/CC-MAIN-20231201084429-20231201114429-00561.warc.gz
| 381,132,671
| 7,910
|
Posted on by Kalkicode
Code Queue
# Level order traversal using queue in golang
Go program for Level order traversal using queue. Here problem description and other solutions.
``````package main
import "fmt"
/*
Go program for
Level order tree traversal using queue
*/
// Create Q node
type QNode struct {
data * TreeNode
next * QNode
}
func getQNode(node * TreeNode) * QNode {
// return new QNode
return &QNode {
node,
nil,
}
}
// Binary Tree Node
type TreeNode struct {
data int
left * TreeNode
right * TreeNode
}
func getTreeNode(data int) * TreeNode {
// return new TreeNode
return &TreeNode {
data,
nil,
nil,
}
}
type MyQueue struct {
tail * QNode
count int
}
func getMyQueue() * MyQueue {
// return new MyQueue
return &MyQueue {
nil,
nil,
0,
}
}
func(this MyQueue) size() int {
return this.count
}
func(this MyQueue) isEmpty() bool {
return this.count == 0
}
// Add new node of queue
func(this *MyQueue) enqueue(value * TreeNode) {
// Create a new node
var node * QNode = getQNode(value)
// Add first element into queue
} else {
// Add node at the end using tail
this.tail.next = node
}
this.count++
this.tail = node
}
// Delete a element into queue
func(this *MyQueue) dequeue() {
// Empty Queue
return
}
// Visit next node
this.count--
// When deleting a last node of linked list
this.tail = nil
}
}
// Get front node
func(this MyQueue) peek() * TreeNode {
return nil
}
}
type BinaryTree struct {
root * TreeNode
}
func getBinaryTree() * BinaryTree {
// return new BinaryTree
return &BinaryTree {
nil,
}
}
func(this BinaryTree) levelOrder() {
if this.root != nil {
var q * MyQueue = getMyQueue()
q.enqueue(this.root)
var node * TreeNode = this.root
for (q.isEmpty() == false && node != nil) {
if node.left != nil {
q.enqueue(node.left)
}
if node.right != nil {
q.enqueue(node.right)
}
// Display level node
fmt.Print(" ", node.data)
// Remove current node
q.dequeue()
node = q.peek()
}
} else {
fmt.Println("Empty Tree")
}
}
func main() {
// Create new tree
var tree * BinaryTree = getBinaryTree()
/*
Make A Binary Tree
-----------------------
1
/ \
2 3
/ / \
4 5 6
/ / \
7 8 9
*/
tree.root = getTreeNode(1)
tree.root.left = getTreeNode(2)
tree.root.right = getTreeNode(3)
tree.root.right.right = getTreeNode(6)
tree.root.right.left = getTreeNode(5)
tree.root.left.left = getTreeNode(4)
tree.root.left.left.left = getTreeNode(7)
tree.root.right.left.left = getTreeNode(8)
tree.root.right.left.right = getTreeNode(9)
// Display level order element
tree.levelOrder()
}``````
Output
`` 1 2 3 4 5 6 7 8 9``
## Comment
Please share your knowledge to improve code and content standard. Also submit your doubts, and test case. We improve by your feedback. We will try to resolve your query as soon as possible.
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{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}
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# NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.1
NCERT Solutions for class 10 Maths Chapter 3 Exercise 3.1 pair of linear equations in two variables in PDF for as well as online to use. Offline Apps of Solutions are modified as per latest CBSE Curriculum 2019-20 in both the medium (Hindi Medium and English Medium) or View in Video Format applicable for UP Board, Gujrat board as well as CBSE board NCERT Books. Download (Exercise 3.1) in PDF format here.
Class 10: Maths – गणित Chapter 3: Linear Equations in two Variables (Exercise 3.1)
## NCERT Solutions for class 10 Maths Chapter 3 Exercise 3.1
### Class 10 Maths Chapter 3 Exercise 3.1 Solutions in English Medium
NCERT Solutions for class 10 Maths Chapter 3 Exercise 3.1 in English medium is given below. Download options are given at the top of this page or on main page of NCERT Solutions Class 10 Maths Chapter 3. CLICK HERE for Hindi Medium or View in Video Format Solutions.
### 10 Maths Chapter 3 Exercise 3.1 Sols in Video
NCERT Solutions for class 10 Maths Exercise 3.1 in video format with complete description.
### Class 10 Maths chapter 3 Exercise 3.1 Solutions in Hindi Medium.
NCERT Solutions for class 10 Maths Chapter 3 Exercise 3.1 pair of linear equations in two variables in Hindi medium is given below to use it online. Click here to go back to NCERT Solutions for Class 10 Maths chapter 3. Go back to English Medium Solutions.
Visit to English Medium or Hindi Medium or Video Format Sols
##### Important terms for Pair of linear equations
• The graph of a pair of linear equations in two variables is represented by two lines.
• If the lines intersect at a point, the pair of equations is consistent. The point of intersection gives the unique solution of the equations.
• If the lines are parallel, then there is no solution the pair of linear equations is inconsistent.
• If the lines coincide, then there are infinitely many solutions. The pair of linear equations is consistent. Each point on the line is a solution of both the equations.
#### Important Questions for Practice
1. If x = 3m –1 and y = 4 is a solution of the equation x + y = 6, then find the value of m. [Answer: m=1]
2. What is the point of intersection of the line represented by 3x – 2y = 6 and the y-axis. [Answer: (0, -3)]
3. In a deer park, the number of heads and number of legs of deer and human visitors were counted and it was found that there were 39 heads and 132 legs. Find the number of deer and human visitors in the park. [Answer: Dear: 27, Visitors: 12]
4. For what value of p, system of equations 2x + py = 8 and x + y = 6 have no solution. [Answer: p=2]
5. From Delhi station if we buy 2 tickets to station A and 3 tickets to station B, the total cost is ₹77, but if we buy 3 tickets to station A and 5 tickets to station B, the total cost is ₹124. What are fares from Delhi to station A and to station B? [Answer: ₹13, ₹17]
6. A motor cyclist is moving along the line x – y = 2 and another motor cyclist is moving along the line x – y = 4 find out their moving direction. [Answer: move parallel]
7. A farmer sold a calf and a cow for ₹760, thereby, making a profit of 25% on the calf and 10% on the cow. By selling them for ₹767.50, he would have realised a profit of 10% on the calf and 25% on the cow. Find the cost of each. [Answer: Cost of cow = ₹350, cost of calf = ₹300]
## 2 thoughts on “NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.1”
1. nafis says:
In problem number 1 can we take x value as 1, 2, 3…. ?
In problem 2 why you have taken x=300..??
Plz clarify my doubt…
2. Dhruvisha says:
thank you,your pdf is very usefully
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#### Molar Mass, Molecular Weight and Elemental Composition Calculator
Molar mass of B2(CO3)3 is 201.6487 g/mol
Formula in Hill system is C3B2O9
Elemental composition of B2(CO3)3:
SymbolElementAtomic weightAtomsMass percent
BBoron10.811210.7226 %
CCarbon12.0107317.8687 %
OOxygen15.9994971.4086 %
### Computing molar mass (molar weight)
To calculate molar mass of a chemical compound enter its formula and click 'Calculate!'. In chemical formula you may use:
• Any chemical element
• Functional groups: D, Ph, Me, Et, Bu, AcAc, For, Ts, Tos, Bz, TMS, tBu, Bzl, Bn, Dmg
• parantesis () or brackets [].
• Common compound names.
Examples of molar mass computations: NaCl, Ca(OH)2, K4[Fe(CN)6], CuSO4*5H2O, water, nitric acid, potassium permanganate, ethanol, fructose.
### Computing molecular weight (molecular mass)
To calculate molecular weight of a chemical compound enter it's formula, specify its isotope mass number after each element in square brackets.
Examples of molecular weight computations: C[14]O[16]2, S[34]O[16]2.
### Definitions of molecular mass, molecular weight, molar mass and molar weight
• Molecular mass (molecular weight) is the mass of one molecule of a substance and is expressed in the unified atomic mass units (u). (1 u is equal to 1/12 the mass of one atom of carbon-12)
• Molar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol.
Weights of atoms and isotopes are from NIST article.
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https://softwareengineering.stackexchange.com/questions/185998/what-is-the-fitness-landscape-for-minimal-viable-solutions
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# What is the fitness landscape for minimal + viable solutions?
Let's say I'm trying to find a number from 1 .. 100. All numbers in this range are "valid", in that they could be interpreted as potential solutions. Let's say the ideal number is 50. And all numbers >= 50 are "feasible" in that they actually solve the problem. And all numbers < 50 are "not feasible" (but still valid). How would you code a scenario like this with a fitness function (assuming that the landscape is similar to but more complex than this contrived example)? Do you give "bonuses" to valid solutions? Do you measure how far an unfeasible solution has left to go before becoming optimal? And do you penalize excessive solutions?
``````if (solution < 50) {
maximalFitness = f(solution)
} else if (solution >= 50) {
maximalFitness = 1_000 - f(solution)
}
``````
The curve wouldn't be continuous if all feasible solutions are strictly better than all unfeasible solutions, despite having a similar distance from optimal.
What you're referring to is called "constrained optimization". This is a really well-studied branch of evolutionary computation, and you can find dozens of resources describing common techniques for solving these problems. Carlos Coello-Coello does a tutorial on the subject each year at GECCO, the primary EC conference in the field. I found slides from one such session available here (ftp://ftp.cs.bham.ac.uk/.snapshot/nightly.1/pub/authors/W.B.Langdon/biblio/gecco2008/docs/p2445.pdf). You might find that a useful jumping-off point for further study.
In short, there are two high-level approaches that are by far the most common: penalization and repair. In penalization, you decrease the fitness of solutions by some function of how badly they violate the constraints. In repair, you actually modify infeasible solutions directly to attempt to move them into the feasible region.
Penalization works well if you can get the penalty scheme right, but that can be a tricky problem to solve, and it's essentially always problem dependent. One option is to just mess around with your fitness function and figure out the right amount to penalize so that "good" infeasible solutions get just enough fitness to not die immediately but not so much that they become preferred over feasible solutions. Another method is to just replace the selection criteria to account for infeasible solutions directly. That is, instead of
``````if fitness(p1) > fitness(p2):
# p1 is better
else:
# p2 is better
``````
you have something like
``````if (is_feasible(p1) and is_feasible(p2)) or (not is_feasible(p1) and not is_feasible(p2)):
if fitness(p1) > fitness(p2):
# p1 is better
else:
# p2 is better
else:
if is_feasible(p1):
# p1 is better
else:
# p2 is better
``````
This explicitly handles all the ways of comparing feasible and infeasible solutions, and uses their fitness values only to break ties. Now it doesn't matter so much if the raw fitness value of an infeasible solution is higher than a feasible solution due to weirdness in your penalization scheme.
Repair is often preferable if you can do it (it isn't always possible to build an effective repair operator). By repair, we mean you take the infeasible solution and actually modify it so that it falls within your feasible region. This could be as simple as just thresholding the value within the allowed range, but more often will require some sort of domain specific or heuristic search. For example, if you have a knapsack problem and a solution has too much weight in the knapsack, a repair operator might randomly start throwing items out of the knapsack until the total weight was less than the capacity. A better operator might bias the removal of items toward those with the highest weight/value ratio. It's usually beneficial for the repair to have some randomness involved -- you may want to bias things slightly, but you don't want to always do a greedy repair.
There are other approaches as well. One recent innovation has been to treat each constraint as a separate objective function and employ multi-objective optimization algorithms to find a range of trade-off solutions. For some problems, this has been shown to be effective, but I think the general rule of thumb should probably be to first see what can be done with the simpler methods like penalization and repair.
• +1 excellent summary of the topic, and pretty much spot on. I'd also recommend OP to read and make use of existing literature, this is a topic that has been studied well beyond what people normally come up with on a whim. I've found that for typical "simple" problems (such as boundary constraints), penalisation is almost trivial to implement, and works well enough. Feb 6, 2013 at 7:38
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# Math
-5/6x=-7/8 what is x?
would it be 1/4? I don't know how to do this?
1. 👍 0
2. 👎 0
3. 👁 134
1. To get x by itself, multiply by the reciprocal of -5/6, which would be -6/5.
Whatever operation you do to one side of an equation you must do to the other side as well:
-6/5 * (-5/6)x = -6/5 * -7/8
x = 42/40 = 21/20
Let's check:
(-5/6)(21/20) = -7/8
-7/8 = -7/8
I hope this will help.
1. 👍 0
2. 👎 0
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# Difference between revisions of "2021 AIME II Problems/Problem 13"
## Problem
Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.
## Solution 1
1000 divides this expression iff 8, 125 both divide it. It should be fairly obvious that $n \geq 3$; so we may break up the initial condition into two sub-conditions.
(1) $5^n \equiv n \pmod{8}$. Notice that the square of any odd integer is 1 modulo 8 (proof by plugging in $1^2$, $3^2$, $5^2$, $7^2$ into modulo 8), so the LHS of this expression goes "5,1,5,1, ...", while the RHS goes "1,2,3,4,5,6,7,8,1, ...". The cycle length of the LHS is 2, RHS is 8, so the cycle length of the solution is $lcm(2,8)=8$. Indeed, the n that solve this equation are exactly those such that $n \equiv 5 \pmod{8}$.
(2) $2^n \equiv n \pmod{125}$. This is extremely computationally intensive if you try to calculate all $2^1,2^2, ..., 2^{100} \pmod{125}$, so instead, we take a divide-and-conquer approach. In order for this expression to be true $2^n \equiv n \pmod{5}$ is necessary; it shouldn't take too long for you to go through the 20 possible LHS-RHS combinations and convince yourself that $n \equiv 3 \pmod{20}$ or $n \equiv 17 \pmod{20}$.
With this in mind we consider $2^n \equiv n$ modulo 25. By the generalized Fermat's little theorem, $2^{20} \equiv 1 \pmod{25}$, but we already have n modulo 20! Our calculation is greatly simplified. The LHS cycle length is 20, RHS cycle length is 25, the lcm is 100, in this step we need to test all the numbers between 1 to 100 that $n \equiv 3 \pmod{20}$ or $n \equiv 17 \pmod{20}$. In the case that $n \equiv 3 \pmod{20}$, $2^n \equiv 2^3 \equiv 8$, and RHS goes "3,23,43,63,83"; clearly $n \equiv 83 \pmod{100}$. In the case that $n \equiv 17 \pmod{20}$, by a similar argument, $n \equiv 97 \pmod{100}$.
In the final step, we need to calculate $2^{97}, 2^{83}$ modulo 125. The former is simply $2^{-3}$; because $8*47=376=1$ modulo 125, $2^{97} \equiv 47$. $2^{83}$ is $2^{-17}=2^{-16}*2^{-1}$. \begin{align*} 2^{-1}&=63, \\ 2^{-2}&=63^2=3969=-31, \\ 2^{-4}&=(-31)^2=961=-39, \\ 2^{-8}&=1521=21, \\ 2^{-16}&=441, \\ 2^{-17}&=63*441=7*{-31}=-217=33. \end{align*} This time, LHS cycle is 100, RHS cycle is 125, so we need to figure out what is n modulo 500. It should be $n \equiv 283,297 \pmod{500}$.
Put everything together. By the second subcondition, the only candidates < 100 are $283,297,782,797$. Apply the first subcondition, n=797 is the desired answer.
-Ross Gao
## Solution 2
We have that $2^n + 5^n \equiv n\pmod{1000}$, or $2^n + 5^n \equiv n \pmod{8}$ and $2^n + 5^n \equiv n \pmod{125}$ by CRT. It is easy to check $n < 3$ don't work, so we have that $n \geq 3$. Then, $2^n \equiv 0 \pmod{8}$ and $5^n \equiv 0 \pmod{125}$, so we just have $5^n \equiv n \pmod{8}$ and $2^n \equiv n \pmod{125}$. Let us consider both of these congruences separately.
First, we look at $5^n \equiv n \pmod{8}$. By Euler's Totient Theorem (ETT), we have $5^4 \equiv 1 \pmod{8}$, so $5^5 \equiv 5 \pmod{8}$. On the RHS of the congruence, the possible values of $n$ are all nonnegative integers less than $8$ and on the RHS the only possible values are $5$ and $1$. However, for $5^n$ to be $1 \pmod{8}$ we must have $n \equiv 0 \pmod{4}$, a contradiction. So, the only possible values of $n$ are when $n \equiv 5 \pmod{8} \implies n = 8k+5$.
Now we look at $2^n \equiv n \pmod{125}$. Plugging in $n = 8k+5$, we get $2^{8k+5} \equiv 8k+5 \pmod{125} \implies 2^{8k} \cdot 32 \equiv 8k+5 \pmod{125}$. Note, for $2^n \equiv n\pmod{125}$ to be satisfied, we must have $2^n \equiv n \pmod{5}$ and $2^n \equiv n\pmod{25}$. Since $2^{8k} \equiv 1\pmod{5}$ as $8k \equiv 0\pmod{4}$, we have $2 \equiv -2k \pmod{5} \implies k = 5m-1$. Then, $n = 8(5m-1) + 5 = 40m-3$. Now, we get $2^{40m-3} \equiv 40m-3 \pmod{125}$. Using the fact that $2^n \equiv n\pmod{25}$, we get $2^{-3} \equiv 15m-3 \pmod{25}$. The inverse of $2$ modulo $25$ is obviously $13$, so $2^{-3} \equiv 13^3 \equiv 22 \pmod{25}$, so $15m \equiv 0 \pmod{25} \implies m = 5s$. Plugging in $m = 5s$, we get $n = 200s - 3$.
Now, we are finally ready to plug $n$ into the congruence modulo $125$. Plugging in, we get $2^{200s-3} \equiv 200s - 3 \pmod{125}$. By ETT, we get $2^{100} \equiv 1 \pmod{125}$, so $2^{200s- 3} \equiv 2^{-3} \equiv 47 \pmod{125}$. Then, $200s \equiv 50 \pmod{125} \implies s \equiv 4 \pmod{5} \implies s = 5y+4$. Plugging this in, we get $n = 200(5y+4) - 3 = 1000y+797$, implying the smallest value of $n$ is simply $\boxed{797}$. ~rocketsri
## Solution 3 (Chinese Remainder Theorem and Binomial Theorem)
We wish to find the least positive integer $n$ for which $2^n+5^n-n\equiv0\pmod{1000}.$ Rearranging gives $$2^n+5^n\equiv n\pmod{1000}.$$ Applying the Chinese Remainder Theorem, we get the following systems of linear congruences: \begin{align*} 2^n+5^n &\equiv n \pmod{8}, \\ 2^n+5^n &\equiv n \pmod{125}. \end{align*} It is clear that $n\geq3,$ from which we simplify to \begin{align*} 5^n &\equiv n \pmod{8}, \hspace{15mm} &(1) \\ 2^n &\equiv n \pmod{125}. &(2) \end{align*} We solve each congruence separately:
1. For $(1),$ quick inspections produce that $5^1,5^2,5^3,5^4,\cdots$ are congruent to $5,1,5,1,\cdots$ modulo $8,$ respectively. More generally, $5^n \equiv 5 \pmod{8}$ if $n$ is odd, and $5^n \equiv 1 \pmod{8}$ if $n$ is even. As $5^n$ is always odd (so is $n$), we must have $n\equiv5\pmod{8}.$
That is, $\boldsymbol{n=8r+5}$ for some nonnegative integer $\boldsymbol{r.}$
2. For $(2),$ we substitute the result from $(1)$ and simplify:
3. \begin{align*} 2^{8r+5}&\equiv8r+5\pmod{125} \\ \left(2^8\right)^r\cdot2^5&\equiv8r+5\pmod{125} \\ 256^r\cdot32&\equiv8r+5\pmod{125} \\ 6^r\cdot32&\equiv8r+5\pmod{125}. \end{align*} Note that $5^3=125$ and $6=5+1,$ so we apply the Binomial Theorem to the left side: \begin{align*} (5+1)^r\cdot32&\equiv8r+5&&\pmod{125} \\ \Biggl[\binom{r}{0}5^0+\binom{r}{1}5^1+\binom{r}{2}5^2+\phantom{ }\underbrace{\binom{r}{3}5^3+\cdots+\binom{r}{r}5^r}_{0\pmod{125}}\phantom{ }\Biggr]\cdot32&\equiv8r+5&&\pmod{125} \\ \left[1+5r+\frac{25r(r-1)}{2}\right]\cdot32&\equiv8r+5&&\pmod{125} \\ 32+160r+400r(r-1)&\equiv8r+5&&\pmod{125} \\ 32+35r+25r(r-1)&\equiv8r+5&&\pmod{125} \\ 25r^2+2r+27&\equiv0&&\pmod{125}. \hspace{15mm} (*) \end{align*} Since $125\equiv0\pmod{5},$ it follows that \begin{align*} 25r^2+2r+27&\equiv0\pmod{5} \\ 2r+2&\equiv0\pmod{5} \\ r&\equiv4\pmod{5}. \end{align*}
That is, $\boldsymbol{r=5s+4}$ for some nonnegative integer $\boldsymbol{s.}$
Substituting this back into $(*),$ we get \begin{align*} 25(5s+4)^2+2(5s+4)+27&\equiv0\pmod{125} \\ 625s^2+1010s+435&\equiv0\pmod{125} \\ 10s+60&\equiv0\pmod{125} \\ 10(s+6)&\equiv0\pmod{125}. \end{align*} As $10(s+6)$ is a multiple of $125,$ it has at least three factors of $5.$ Since $10$ contributes one factor, it follows that $s+6$ contributes at least two factors, or $s+6$ must be a multiple of $25.$ Therefore, the least such nonnegative integer $s$ is $19.$
Finally, combining the two results from above (bolded) generates the least such positive integer $n$ at $s=19:$ \begin{align*} n&=8r+5 \\ &=8(5s+4)+5 \\ &=40s+37 \\ &=\boxed{797}. \end{align*} ~MRENTHUSIASM (inspired by Math Jams's 2021 AIME II Discussion)
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Page "Tree (graph theory)" ¶ 22
from Wikipedia
## Some Related Sentences
Conversely and given
Conversely, a current of one ampere is one coulomb of charge going past a given point per second:
Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y.
Conversely, there are other stars that never rise above the horizon, as seen from any given point on the Earth's surface ( except exactly on the equator ).
Conversely, given a groupoid G in the algebraic sense, let G < sub > 0 </ sub > be the set of all elements of the form x * x < sup >− 1 </ sup > with x varying through G and define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as the set of all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists.
Conversely, given central idempotents a < sub > 1 </ sub >,..., a < sub > n </ sub > in R that are pairwise orthogonal and have sum 1, then R is the direct sum of the rings Ra < sub > 1 </ sub >,…, Ra < sub > n </ sub >.
Given a field ordering ≤ as in Def 1, the elements such that x ≥ 0 forms a positive cone of F. Conversely, given a positive cone P of F as in Def 2, one can associate a total ordering ≤< sub > P </ sub > by setting x ≤ y to mean y − x ∈ P. This total ordering ≤< sub > P </ sub > satisfies the properties of Def 1.
Conversely, only the wiki users interested in a given project need look at its associated wiki pages, in contrast to high-traffic mailing lists which may burden subscribers with many messages, regardless of their relevance.
Conversely, given a harmonic function, it is the real part of an analytic function, ( at least locally ).
Each convex set containing X must ( by the assumption that it is convex ) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the sets given by these two definitions must be equal.
Conversely, surveys conducted among living donors postoperatively and in a period of five years following the procedure have shown extreme regret in a majority of the donors, who said that given the chance to repeat the procedure, they would not.
Conversely, the influence of the data at any given point on the initial line propagates with the finite velocity c: there is no effect outside a triangle through that point whose sides are characteristic curves.
Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally.
Conversely, if ( A, m, e, inv ) is a group object in one of those categories, then m necessarily coincides with the given operation on A, e is the inclusion of the given identity element on A, inv is the inversion operation and A with the given operation is an abelian group.
Conversely, if the Turing Machine is expected polynomial-time ( for any given x ), then a considerable fraction of the runs must be polynomial-time bounded, and the coin sequence used in such a run will be a witness.
Conversely, suppose we are given ( U, V, W ) satisfying
Conversely, given any ring, we can form a category by taking objects A < sub > n </ sub > indexed by the set of natural numbers ( including zero ) and letting the hom-set of morphisms from to be the set of-by-matrices over, and where composition is given by matrix multiplication.
Conversely, it is hard to believe, given the length and intensity of the struggle between Máel Sechnaill and Brian, that the High King would surrender his title without a fight.
Conversely, an algorithm to test for solvability in arbitrary integers could be used to test a given equation for solvability in natural numbers by applying that supposed algorithm to the equation obtained from the given equation by replacing each unknown by the sum of the squares of four new unknowns.
Conversely, a disease that is easily transmitted but has a short duration might spread widely during 2002 but is likely to have a low prevalence at any given point in 2003 ( due to its short duration ) but a high incidence during 2002 ( as many people develop the disease ).
Conversely, given a commutative diagram, it defines a poset category:
Conversely and ordered
Conversely, every point on the line can be interpreted as a number in an ordered continuum which includes the real numbers.
Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field.
Conversely, the topological charge only makes sense in the ordered phase and not at all in the disordered phase, because in some hand-waving way there is a " topological condensate " in the disordered phase which randomizes the field from point to point.
Conversely, NFU + Infinity + Choice proves the existence of a type-level ordered pair.
Conversely, given any ordered sequence of natural numbers,
Conversely, every distributive lattice is isomorphic to a ring of sets ; in the case of finite distributive lattices, this is Birkhoff's representation theorem and the sets may be taken as the lower sets of a partially ordered set.
Conversely and tree
Conversely, if the user modifies the fault tree graphic, SAPHIRE automatically updates the associated logic.
Conversely, as an incident scales down, roles will be merged back up the tree until there is just the IC role remaining.
Conversely, however, if an analytical technique for valuing the option exists — or even a numeric technique, such as a ( modified ) pricing tree — Monte Carlo methods will usually be too slow to be competitive.
Conversely each number q in the Stern – Brocot tree has exactly two children: if
Conversely and draw
Conversely, air rising from the northern surface of the Earth ( creating a region of low pressure ) tends to draw air toward it in a counterclockwise pattern.
Conversely, they could also be very simple, as some collectors just get a piece of paper, draw a runway on it, place it over a table, add their plane models and call it their model airport.
Conversely and root
Conversely, any equation can take the canonical form f ( x ) = 0, so equation solving is the same thing as computing ( or finding ) a root of a function.
Conversely, there may be several effective measures ( methods ) that address the root causes of a problem.
Conversely, if n is prime, then there exists a primitive root modulo n, or generator of the group ( Z / nZ )*.
Conversely, after a bleak period in the 1970s and 1980s, the Celtic Tiger era in the Republic was spurred on by the high technology industries that took root in the country after 1990.
Conversely and at
Conversely, some of the Allied infantry who had just dealt a crushing defeat to the French at the Battle of Waterloo fully expected to have to fight again the next day ( at the Battle of Wavre ).
Conversely, at least under U. S. Law, nonpersons such as animals cannot commit crimes.
Conversely, the vast majority of early attempts at proper excavation techniques failed to accurately measure or record stratigraphy, thereby failing to provide a secure context for artefact finds making interpretations extremely limited in scope.
Conversely, non-spontaneous electrochemical reactions can be driven forward by the application of a current at sufficient voltage.
Conversely, BBC critic Mark Kermode believes that " the movie industries of Britain and America are inextricably intertwined ", citing numerous examples of how Hollywood provides work to British production staff and studios, whilst Britain enables Hollywood to base their prestigious productions at UK studios.
Conversely, the symbol width could be set at 8, even if only values 0 and 1 are used ; these data would only require a 2-color table.
Conversely, the cultural divergence at the border became sharper: in theory, a uniform French identity extends from the Atlantic coast to the Rhine, and on the other bank of the Rhine, a uniform German identity begins.
Conversely, so-called " nodder " height finding radars use a dish with a narrow vertical beamwidth and wide azimuthal beamwidth to detect an aircraft at a specific height but with low azimuthal precision.
Conversely, most Cardassians figure out during the first act of Shakespeare's Julius Caesar that all the conspirators are going to kill him, but cannot understand why Caesar cannot figure this out ( or is willfully blind to an impending coup d ' état ) until the knives are literally coming at him from all directions.
Conversely, specifying the symmetry can define the structure, or at least clarify what we mean by an invariant, geometric language in which to discuss it ; this is one way of looking at the Erlangen programme.
Conversely, Electronic Arts Square, K. K., formed at the same time and based in Japan, was responsible for publishing and marketing games produced by Electronic Arts in Asia.
Conversely, trade secret owners who cannot evidence reasonable efforts at protecting confidential information, risk losing the trade secret, even if the information is obtained by competitors illegally.
Conversely, Elijah Delmedigo ( c. 1458 – c. 1493 ), in his Bechinat ha-Dat endeavored to show that the Zohar could not be attributed to Shimon bar Yochai, arguing that if it were his work, the Zohar would have been mentioned by the Talmud, as has been the case with other works of the Talmudic period, that had bar Yochai known by divine revelation the hidden meaning of the precepts, his decisions on Jewish law from the Talmudic period would have been adopted by the Talmud, that it would not contain the names of rabbis who lived at a later period than that of Simeon ; and that if the Kabbalah was a revealed doctrine, there would have been no divergence of opinion among the Kabbalists concerning the mystic interpretation of the precepts.
Conversely, a scientific antirealist or instrumentalist argues that science does not aim ( or at least does not succeed ) at truth and that we should not regard scientific theories as true.
Conversely a large cargo, for example in structure relocation, may be disassembled for carriage on multiple vehicles and then reassembled in the correct order at the destination.
Conversely, some species have little or no circulatory system at all, transporting oxygen in the coelomic fluid that fills their body cavity.
Conversely, halting the cast and stopping the lure at the desired position requires practice in learning to feather the line with the forefinger as it uncoils from the spool.
Conversely, nimbostratus is coded as middle because it usually initially forms at mid-altitudes of the troposphere and becomes vertically developed by growing downward into the low altitude range.
Conversely, a youth minister at a parish may or may not be a cleric.
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# Vieta’s Formulas
Vieta’s formula relates the coefficients of polynomial to the sum and product of their roots, as well as the products of the roots taken in groups. Vieta’s formula describes the relationship of the roots of a polynomial with its coefficients. Consider the following example to find a polynomial with given roots. (Only discuss real-valued polynomials, i.e. the coefficients of polynomials are real numbers). Let’s take a quadratic polynomial. Given two real roots and , then find a polynomial.
Consider the polynomial . Given the roots, we can also write it as
.
Since both equation represents the same polynomial, so equate both polynomial
Simplifying the above equation, we get
Comparing the coefficients of both sides, we get
For , ,
For , ,
For constant term, ,
Which gives,
,
Equation (1) and (2) are known as Vieta’s Formulas for a second degree polynomial.
In general, for an degree polynomial, there are n different Vieta’s Formulas. They can be written in a condensed form as
For
The following examples illustrate the use of Vieta’s formula to solve a problem.
Examples:
Input : n = 2
roots = {-3, 2}
Output : Polynomial coefficients: 1, 1, -5
Input : n = 4
roots = {-1, 2, -3, 7}
Output : Polynomial coefficients: 1, -5, -19, 29, 42
## C++
// C++ program to implement vieta formula // to calculate polynomial coefficients. #include using namespace std; // Function to calculate polynomial // coefficients. void vietaFormula(int roots[], int n) { // Declare an array for // polynomial coefficient. int coeff[n + 1]; // Set all coefficients as zero initially memset(coeff, 0, sizeof(coeff)); // Set highest order coefficient as 1 coeff[n] = 1; for (int i = 1; i <= n; i++) { for (int j = n - i - 1; j < n; j++) { coeff[j] = coeff[j] + (-1) * roots[i - 1] * coeff[j + 1]; } } cout << "Polynomial Coefficients: "; for (int i = n; i >= 0; i--) { cout << coeff[i] << " "; } } // Driver code int main() { // Degree of required polynomial int n = 4; // Initialise an array by // root of polynomial int roots[] = { -1, 2, -3, 7 }; // Function call vietaFormula(roots, n); return 0; }
## Java
// Java program to implement vieta formula // to calculate polynomial coefficients. import java.util.Arrays; class GFG { // Function to calculate polynomial // coefficients. static void vietaFormula(int roots[], int n) { // Declare an array for // polynomial coefficient. int coeff[] = new int[++n + 1]; Arrays.fill(coeff, 0); // Set highest order coefficient as 1 coeff[n] = 1; for (int i = 1; i 0; i--) { System.out.print(coeff[i] + " "); } } // Driver code public static void main(String[] args) { // Degree of required polynomial int n = 4; // Initialise an array by // root of polynomial int roots[] = { -1, 2, -3, 7 }; // Function call vietaFormula(roots, n); } } /* This code contributed by PrinciRaj1992 */
## Python3
# Python3 program to implement # Vieta's formula to calculate # polynomial coefficients. def vietaFormula(roots, n): # Declare an array for # polynomial coefficient. coeff = [0] * (n + 1) # Set Highest Order # Coefficient as 1 coeff[n] = 1 for i in range(1, n + 1): for j in range(n - i - 1, n): coeff[j] += ((-1) * roots[i - 1] * coeff[j + 1]) # Reverse Array coeff = coeff[::-1] print("Polynomial Coefficients : ", end = "") # Print Coefficients for i in coeff: print(i, end = " ") print() # Driver Code if __name__ == "__main__": # Degree of Polynomial n = 4 # Initialise an array by # root of polynomial roots = [-1, 2, -3, 7] # Function call vietaFormula(roots, n) # This code is contributed # by Arihant Joshi
## C#
// C# program to implement vieta formula // to calculate polynomial coefficients. using System; class GFG { // Function to calculate polynomial // coefficients. static void vietaFormula(int []roots, int n) { // Declare an array for // polynomial coefficient. int []coeff = new int[++n + 1]; // Set highest order coefficient as 1 coeff[n] = 1; for (int i = 1; i 0; i--) { Console.Write(coeff[i] + " "); } } // Driver code public static void Main(String[] args) { // Degree of required polynomial int n = 4; // Initialise an array by // root of polynomial int []roots = { -1, 2, -3, 7 }; // Function call vietaFormula(roots, n); } } // This code has been contributed by 29AjayKumar
## PHP
= 0; $i--) { echo $coeff[$i]. " "; } } // Driver code // Degree of required polynomial $n = 4; // Initialise an array by // root of polynomial $roots = array(-1, 2, -3, 7); // Function call vietaFormula($roots, \$n); // This code is contributed by mits ?>
Output:
Polynomial Coefficients: 1 -5 -19 29 42
Time Complexity : .
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Physics + MathPhysics & Math
# Describing Nature With Math
How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality?
—Albert Einstein
If you're like me, you understand readily how one can describe nature's wonders using poetry or music, painting or photography. Wordsworth's "I Wandered Lonely as a Cloud" and Vivaldi's "Four Seasons" richly depict their natural subjects, as do Monet's water lilies and Ansel Adams' photos of Yosemite. But mathematics? How can you describe a tree or cloud, a rippled pond or swirling galaxy using numbers and equations?
ByNova
Extremely well, as Einstein knew better than most, of course. In fact, most scientists would agree that, when it comes to teasing out the inherent secrets of the universe, nothing visual, verbal, or aural comes close to matching the accuracy and economy, the power and elegance, and the inescapable truth of the mathematical.
How is this so? Well, for the math-challenged, for that person who has avoided anything but the most basic arithmetic since high school, who feels a pit in his stomach when he sees an equation—that is, for myself—I will attempt to explain, with the help of some who do mathematics for a living. If you're math-phobic, too, I think you'll get a painless feel for why even that master of describing nature with words, Thoreau, would hold that "the most distinct and beautiful statements of any truth must take at last the mathematical form."
### Ancient math
While many early civilizations, including Islamic, Indian, and Chinese, made important contributions to mathematics, it was the ancient Greeks who invented much of the math we're familiar with. Euclid fathered the geometry we named after him—all those radii and hypotenuses and parallel lines. Archimedes approximated pi. Ptolemy created a precise mathematical model that had all of the heavens wheeling around the Earth.
"With a few symbols on a page, you can describe a wealth of physical phenomena."
The Greeks' discoveries are timeless: Euclid's axioms are as unimpeachable today as when he devised them over 2,000 years ago. And some Greek proto-physicists did use their newfound skills to tackle mysteries of the natural world. With basic trigonometry, for example, the astronomer Eratosthenes estimated the diameter of the Earth with over 99 percent accuracy—in 228 B.C.
But while the Greeks believed that the universe was mathematically designed, they largely applied math only to static objects—measuring angles, calculating volumes of solid objects, and the like—as well as to philosophical purposes. Plato wouldn't let anyone through the front door of his acclaimed Academy who didn't know mathematics. "He is unworthy of the name of man," Plato sniffed, "who is ignorant of the fact that the diagonal of a square is incommensurable with its side." And so it remained for a millennium and a half.
### The measure of all things
Galileo changed all that in the early 17th century. Eschewing the Greeks' attempts to explain why a pebble falls when you drop it, Galileo set out to determine how. The "great book" of the universe is written in the language of mathematics, he famously declared, and unless we understand the triangles, circles, and other geometrical figures that form its characters, he wrote, "it is humanly impossible to comprehend a single word of it [and] one wanders in vain through a dark labyrinth." (Wordsworth or Monet might take issue with that statement, but wait.)
Galileo sought characteristics of our world that he could measure—variable aspects like force and weight, time and space, velocity and acceleration. With such measurements, Galileo was able to construct those gems of scientific shorthand—mathematical formulas—which defined phenomena more concisely and more powerfully than had ever been possible before. (His contemporary, the German mathematician Johannes Kepler, did the same for the heavens, crafting mathematical laws that accurately describe the orbits of planets around the sun—and led to the scrapping of Ptolemy's Earth-centric model.)
### A tidy sum
A classic example is the formula commonly shown as d = 16t2. (Hang in there, math-phobes. Your queasiness, which I share, should go away when you see how straightforward this is.) What Galileo discovered and ensconced in this simple equation, one of the most consequential in scientific history, is that, when air resistance is left out, the distance in feet, d, that an object falls is equal to 16 times the square of the time in seconds, t. Thus, if you drop a pebble off a cliff, in one second it will fall 16 feet, in two seconds 64 feet, in three seconds 144 feet, and so on.
Galileo's succinct formula neatly expresses the notion of acceleration of objects near the surface of the Earth, but that is just the start of its usefulness. First, just as with any value of t you can calculate d, for any value of d you can figure t. To get to t, simply divide both sides of the formula d = 16t2 by 16, then take the square root of both sides. This leaves a new formula:
t = √ d 16
This compact equation tells you the time needed for your pebble to fall a given distance—any distance. Say your cliff is 150 high. How long would the pebble take to reach the bottom? A quick calculation reveals just over three seconds. A thousand feet high? Just shy of eight seconds.
What else can you do with a pithy formula like d = 16t2? Well, as hinted above, you can make calculations for an infinite number of different values for either d or t. In essence, this means that d = 16t2 contains an infinite amount of information. You can also substitute any object for your pebble—a pea, say, or a boulder—and the formula still holds up perfectly (under the conditions previously mentioned). Could a single poem or painting do as much?
"Mathematics captures patterns that the universe finds pleasant, if you like."
And because the same mathematical law may govern multiple phenomena, a curious scientist can discover relationships between those phenomena that might have otherwise gone undetected. Trigonometric functions, for instance, apply to all wave motions—light, sound, and radio waves as well as waves in water, waves in gas, and many other types of wave motions. The person who "gets" these trig functions and their properties will ipso facto "get" all the phenomena that these functions govern.
### A wealth of data
The power of a potent equation extends still further. Take Isaac Newton's universal law of gravitation, which brilliantly combines Galileo's laws of falling bodies with Kepler's laws of planetary motion. Many of us know gravity vaguely as that unseen force that keeps the pebble in your palm or your feet on the ground. Newton described it this way:
F = Gm1m2 r2
I won't go into this formula, but just know that from it you can calculate the gravitational tug between just about any two objects you can think of, from that between your coffee cup and the table it rests on, to that between one galaxy and another. Or, depending on which variables you know, you can nail down everything from the acceleration of any freely falling object near the Earth's surface (32 feet per second during every second of its fall) to the mass of our planet (about 6,000,000,000,000,000,000,000 tons).
"With a few symbols on a page, you can describe a wealth of physical phenomena," says astrophysicist Brian Greene, host of NOVA's series based on his book The Fabric of the Cosmos. "And that is, in some sense, what we mean by elegance—that the messy, complex world around us emanates from this very simple equation that you have written on a piece of paper."
And like Galileo's d = 16t2, Newton's formula is amazingly accurate. In 1997, University of Washington researchers determined that Newton's inverse-square law holds down to a distance of 56,000ths of a millimeter. It may hold further, but that's as precise as researchers have gotten at the moment.
### Exact science
What amazes me most about Galileo and Newton's formulas is their exactitude. In Galileo's, the distance equals exactly the square of the time multiplied by 16; in Newton's, the force of attraction between any two objects is exactly the square of the distance between them. (That's the r2 in his equation.) Such exactness crops up regularly in mathematical descriptions of reality. Einstein found, for instance, that the energy bound up in, say, a pebble equals the pebble's mass times the square of the speed of light, or E = mc2.
Even things we can see and touch in nature flirt with mathematical proportions and patterns. Consider the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… Notice a pattern? After the first, every number is the sum of the previous two. The Fibonacci sequence has many interesting properties. One is that fractions formed by successive Fibonacci numbers—e.g., 3/2 and 5/3 and 8/5—get closer and closer to a particular value, which mathematicians know as the golden number. But what about this: Many plants adhere to Fibonacci numbers. The black-eyed susan has 13 petals. Asters have 21. Many daisies have 34, 55, or 89 petals, while sunflowers usually have 55, 89, or 144.
### Is God a mathematician?
The apparent mathematical nature of Nature, from forces to flowers, has left many since the time of the Greeks wondering, as the mathematician Mario Livio does in his book of the same title, "Is God a mathematician?" Does the universe, that is, have an underlying mathematical structure? Many believe it does. "Just as music is auditory patterns that the human mind finds pleasant," says Stanford mathematician Keith Devlin, "mathematics captures patterns that the universe finds pleasant, if you like—patterns that are implicit in the way the universe works."
"Einstein used mathematics to see a piece of the universe that no one had ever seen before."
So did we humans invent mathematics, or was it already out there, limning the cosmos, awaiting the likes of Euclid to reveal it? In his book Mathematics in Western Culture, the mathematician Morris Kline chose to sidestep the philosophical and focus on the scientific: "The plan that mathematics either imposes on nature or reveals in nature replaces disorder with harmonious order. This is the essential contribution of Ptolemy, Copernicus, Newton, and Einstein."
### Seeing the invisible
Formulas like Galileo's and Netwon's make the invisible visible. With d = 16t2, we can "see" the motion of falling objects. With Newton's equation on gravity, we can "see" the force that holds the moon in orbit around the Earth. With Einstein's equations, we can "see" atoms. "Einstein is famous for a lot of things, but one thing that is often overlooked is he's the first person who actually said how big an atom is," says Jim Gates, a physicist at the University of Maryland. "Einstein used mathematics to see a piece of the universe that no one had ever seen before."
Today, with advanced technology, we can observe individual atoms, but some natural phenomena defy any description but a mathematical one. "The only thing you can say about the reality of an electron is to cite its mathematical properties," noted the late mathematics writer Martin Gardner. "So there's a sense in which matter has completely dissolved and what is left is just a mathematical structure." Charles Darwin, who admitted to having found mathematics "repugnant" as a student, may have put it best when he wrote, "Mathematics seems to endow one with something like a new sense."
### Fortune telling
Mathematics also endows one with an ability to predict, as Galileo's and Newton's formulas make clear. Such predictive capability often leads to new discoveries. In the mid-1990s, Kyoto University researchers realized to their surprise that equations originally devised by the mathematical genius Alan Turing predicted that the parallel yellow and purple stripes of the marine angelfish have to move over time. Stable, unmoving patterns didn't jive with the mathematics. To find out if this was true, the researchers photographed angelfish in an aquarium over several months. Sure enough, an angelfish's stripes do migrate across its body over time, and in just the way the equations had indicated. Math had revealed the secret.
"There really is a facing-the-music that math forces, and that's why it's a wonderful language for describing nature," Greene says. "It does make predictions for what should happen, and, when the math is accurately describing reality, those predictions are borne out by observation."
### A math for all seasons
So much mathematics exists now—one scholar estimates that a million pages of new mathematical ideas are published each year—that when scientists face problems not solvable with math they know, they can often turn to another variety for help. When Einstein began work on his theory of general relativity, he needed a mathematics that could describe what he was proposing—that space is curved. He found it in the non-Euclidean geometry of 19th-century mathematician Georg F. B. Riemann, which provided just the tool he required: a geometry of curved spaces in any number of dimensions.
Or, if necessary, they invent new math. When the late mathematician Benoit Mandelbrot concluded that standard Euclidean geometry, which is all about smooth shapes, fell short when he tried to mathematically portray "rough" shapes like bushy trees or jagged coastlines, he invented a new mathematics called fractal geometry. "Math is our one and only strategy for understanding the complexity of nature," says Ralph Abraham, a mathematician at the University of California Santa Cruz, in NOVA's Hunting the Hidden Dimension. "Fractal geometry has given us a much larger vocabulary, and with a larger vocabulary we can read more of the book of nature." Galileo would be so proud.
### Technological wonders
Galileo would also be proud of just how much his successors have achieved with his scientific method. Formulas from his own on falling bodies to Werner Heisenberg's on quantum mechanics have provided us the means to collect and interpret the most valuable knowledge we have ever attained about the workings of nature. Altogether, the most groundbreaking advances of modern science and technology, both theoretical and practical, have come about through the kind of descriptive, quantitative knowledge-gathering that Galileo pioneered and Newton refined.
"Do not worry about your difficulties in mathematics; I can assure you that mine are still greater."
Newton's law of gravity, for instance, has been critical in all our missions into space. "By understanding the mathematics or force of gravity between lots of different bodies, you get complete control and understanding, with very high precision, of exactly the best way to send a space probe to Mars or Jupiter or to put satellites in orbit—all of those things," says Ian Stewart, an emeritus professor of mathematics at the University of Warwick in England. "Without the math, you would not be able to do it. We can't send a thousand satellites up and hope one of them gets into the right place."
Mathematics underlies virtually all of our technology today. James Maxwell's four equations summarizing electromagnetism led directly to radio and all other forms of telecommunication. E = mc2 led directly to nuclear power and nuclear weapons. The equations of quantum mechanics made possible everything from transistors and semiconductors to electron microscopy and magnetic resonance imaging.
Indeed, many of the technologies you and I enjoy every day simply would not work without mathematics. When you do a Google search, you're relying on 19th-century algebra, on which the search engine's algorithms are based. When you watch a movie, you may well be seeing mountains and other natural features that, while appearing as real as rock, arise entirely from mathematical models. When you play your iPod, you're hearing a mathematical recreation of music that is stored digitally; your cell phone does the same in real time.
"When you listen to a mobile phone, you're not actually hearing the voice of the person speaking," Devlin told me. "You're hearing a mathematical recreation of that voice. That voice is reduced to mathematics."
### Aftermath
And I'm reduced to conceding that math doesn't scare me so much anymore. How about you? If you still feel queasy, perhaps you can take solace from Einstein himself, who once reassured a junior high school student, "Do not worry about your difficulties in mathematics; I can assure you that mine are still greater."
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National corporate funding for NOVA is provided by Draper. Major funding for NOVA is provided by the David H. Koch Fund for Science, the NOVA Science Trust, the Corporation for Public Broadcasting, and PBS viewers.
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https://www.teacherspayteachers.com/Product/Project-Based-Learning-Math-Pizza-PBL-Pizza-Fractions-Project-3616240?ref=feed/browse_elementary
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# Project Based Learning Math Pizza PBL Pizza Fractions Project
Subject
Resource Type
Common Core Standards
Product Rating
File Type
PDF (Acrobat) Document File
9 MB|33 pages
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Also included in:
1. Currently includes 15 different PBL critical thinking activities with multiple step problems. Each Project has been differentiated to best support 4th, 5th, and 6th grade with elapsed time, decimals, fractions, and interpret the remainder division. These worksheets include authentic, complex problem
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Product Description
Your students will LOVE this pizza themed PBL with authentic multiple step real life problems. This activity has students taking their classmates pizza orders and using a menu to add up totals. Students will be required to add, subtract, multiply decimals, and interpret fractions. The final page of performance tasks asks for written answers to multiple step problems. This activity does not have an answer key, due to students being able to choose their own unique answers. This does come with example pages for easy instructional use. Please look at the preview above to see more of this activity.
Check out the blog post about Project Based Learning here for helpful tips and to sign up for exclusive PBL sales and freebies.
Included Activities:
• Take a Classmate's Orders - multiplication and/or multiplication with decimals 4.NBT.5, 5.NBT.5, 5.NBT.7
• Create a Store Blueprint- area and perimeter 4.MD.3
• Buy Supplies - multiplication and/or multiplication with decimals 4.NBT.5, 5.NBT.5, 5.NBT.7
• Pick Your Store Location - multiplication, division, addition and subtraction 4.NBT.5, 5.NBT.5, 4.NBT.4
• Create a Menu- multiplication and/or multiplication with decimals 4.NBT.5, 5.NBT.5, 5.NBT.7
• Narrative Writing Prompt- W.4.3, W.5.3, 4 point rubric included
• Opinion Writing Prompt- W.4.1, W.5.1 4 point rubric included
• And more!
Love this product? Check out these Project Based Learning (PBL) resources:
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Total Pages
33 pages
Included
Teaching Duration
N/A
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| 3.578125
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CC-MAIN-2019-22
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https://aman.ai/primers/ai/token-sampling-methods/
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## Overview
• In this article, let’s go over the rules and procedure for an $$n$$-dimensional tensor product, i.e., say $$A[a,b,c] \times B[i,j,k]$$.
## Background: Logits and Softmax
• Neural networks produce class probabilities with logit vector $$\mathbf{z}$$ where $$\mathbf{z}=\left(z_{1}, \ldots, z_{n}\right)$$ by performing the softmax function to produce probability vector $$\mathbf{q}=\left(q_{1}, \ldots, q_{n}\right)$$ by comparing $$z_{i}$$ with with the other logits.
$$q_{i}=\frac{\exp \left(z_{i} / T\right)}{\sum_{j} \exp \left(z_{j} / T\right)}$$
• where $$T$$ is the temperature parameter, normally set to 1 .
• The softmax function normalizes the candidates at each iteration of the network based on their exponential values by ensuring the network outputs are all between zero and one at every timestep.
## Temperature
• Temperature is a hyperparameter of LSTMs (and neural networks generally) used to control the randomness of predictions by scaling the logits before applying softmax. For example, in TensorFlow’s Magenta implementation of LSTMs, temperature represents how much to divide the logits by before computing the softmax.
• When the temperature is 1, we compute the softmax directly on the logits (the unscaled output of earlier layers), and using a temperature of 0.6 the model computes the softmax on $$\frac{\text { logits }}{0.6}$$, resulting in a larger value. Performing softmax on larger values makes the LSTM more confident (less input is needed to activate the output layer) but also more conservative in its samples (it is less likely to sample from unlikely candidates).
• Using a higher temperature produces a softer probability distribution over the classes, and makes the RNN more “easily excited” by samples, resulting in more diversity/randomness in its tokens (thus enabling it to get out of repetitive loops easily) but also leads to more mistakes.
• Temperature therefore increases the sensitivity to low probability candidates. In LSTMs, the candidate, or sample, can be a letter, a word, or musical note, for example from the Wikipedia article on softmax function:
For high temperatures $$(\tau \rightarrow \infty$$ ), all [samples] have nearly the same probability and the lower the temperature, the more expected rewards affect the probability. For a low temperature $$\left(\tau \rightarrow 0^{+}\right)$$ , the probability of the [sample] with the highest expected reward tends to $$1 .$$
## Citation
If you found our work useful, please cite it as:
@article{Chadha2020DistilledTokenSamplingMethods,
title = {Token Sampling Methods},
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https://puzzles.wonderhowto.com/how-to/solve-rubiks-world-190832/
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# How To: Solve the Rubik's World
Who needs a Rubik's Cube when you have a Rubik's World? See how easy it is too learn and solve. Each video is broken down into its own step.
V1 (P1): Bring four ice-parts to the top.
V2 (P2A): Remove the whites from the bottom.
V3 (P2B) and V4 (P2C): Remove the two whites from the bottom. When there are none, go to video 5. When there are four white parts, go to video 7.
V5 (P3A): When there's one white on one side, and one on the opposite side, and two whites on the right side, do a quarter anticlockwise right turn, quarter anticlockwise bottom turn, and a quarter clockwise right turn, then go to video 6.
V6 (P3B): When you have two whites on one side and the other two on the opposite side, hold it so the two whites are facing you and do a half right turn, half bottom turn and a quarter right turn. Now go to video 7.
V7 (P4A): You have four white parts on top and the bottom. Now get three correct on the top. Look at the bottom, there are four possibilities. Three correct, two correct, two switched, and three switched. When you have two correct on the bottom, go to video 8.
V8 (P4B): There's three correct on top and two correct on the bottom. Hold it so the white part that belongs on top is on the right side and the two correct white parts are in front or in the back row. The bottom white that is in the top must be diagonal right. Now do a half right turn and go to the last video.
V9 (P4C): The top three are correct and the bottom 2 switched. Well, how about you figure this one out yourself? Just watch.
Just updated your iPhone? You'll find new features for TV, Messages, News, and Shortcuts, as well as important bug fixes and security patches. Find out what's new and changed on your iPhone with the iOS 17.6 update.
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https://appadvice.com/app/3d-geometry-shapes-learning/925118357
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# 3D Geometry Shapes Learning
by Arvin Evangelista
### What is it about?
3D Geometry Shapes Learning is an education game app that covers all 3D geometric figure to make kids, young and even adults recognize the 3-Dimensional shapes! This will help recognize at the kinds of basic and complex 3D shapes.
### App Details
Version
1.0.5
Rating
NA
Size
33Mb
Genre
Education Trivia Word
Last updated
June 16, 2018
Release date
November 14, 2014
### App Store Description
3D Geometry Shapes Learning is an education game app that covers all 3D geometric figure to make kids, young and even adults recognize the 3-Dimensional shapes! This will help recognize at the kinds of basic and complex 3D shapes.
This App improves and person's recognition and logic skills. It is also suitable for all ages who wants to improve geometry.
Features of 3D Geometry Shapes Learning
• Shape Chart to learn.
• Randomized shape test quiz questions.
• Nice Clue tip to help you.
• Rewarding achievement when you guess the answer correctly.
• Play the Time Challenge for more excitement!
• Helps pass school examinations.
• It has a nice Grading System.
• You can do this all over and enjoy winning!
What objects are in 3D Geometry Shapes Learning Chart?
• Cube
• Cone
• Cylinder
• Sphere
• Square Pyramid
• Torus
• Tetrastar
• HemiSphere
• Star
• Spring
• Hexagonal Prism
• Cuboid
• Triangular Prism
• Triangular Pyramid
• Pentagonal Cone
• Pentagonal Prism
• Infinity Loop
• Tetrahedron
• Ellipsoid
• Octagonal Prism
• Dodecahedron
• Icosahedron
• Hexagonal Pyramid
• Octahedron
Basic Shapes:
In this educational game, selecting basic shapes will get you to learn the very 5 simplest shape. These are Cube, Cone, Cylinder, Sphere and a Pyramid. This stage is very easy that even toddler kids, nurses, kinder garden to grade school students can enjoy playing.
More Complex shape:
In this Test Quiz, the 3D figures are more advanced and a bit more complex. You will need the “Clue” tool to understand the 3d shape better. It also had 2D shape illustration to make it easy for you.
The Challenge:
The challenge of this test examination is to guess the 3D shape fast than 3-6 seconds. This is a game where grade schoolers, K12 pupils up to college level can play to help them with their studies. This will be very fun to play!
What is 3D?
A 3D also known ad Three Dimensional Figure describes an image that provides perception of depth. When 3-D images are made interactive so that users feel involved with the scene, the experience is called virtual reality.
In 3D computer graphics, 3D modeling (or three-dimensional modeling) is the process of developing a mathematical representation of any surface of an object in three dimensions via specialized software. The product is called a 3D model. Someone who works with 3D models may be referred to as a 3D artist. It can be displayed as a two-dimensional image through a process called 3D rendering. The model can also be physically created using 3D printing devices.
Why would you buy 3D Geometry Shapes Learning?
For K12 students and grade school
• 3D Geometry Shapes Learning is essential for learning or studying geometry, algebra, logic, mathematics. For kids under K12 education for families, these are the common hard subjects at school. Instead to memorize each shape, this educational app will help you recognize them and with FUN! If Math is fun, geometry is fun!
• For College Degree courses
If you are in a college level, this education game will help you speed up the learning curve of your course. It is a good app for architect, mechanical engineer, electrical engineering, Chemical engineering, Civil engineering, computer science degrees, biomedical engineering, aeronautical engineering, graphics artist, interior designer, product designer, exterior designer, landscape architect, 3d artist and video producers.
If you are studying 3D courses you will see 3D geometry figures in softwares like 3Ds max, Maya cinema 4d, sketch pro, Blender 3D, Revit, AutoCAD, Rhino 3D, and Photoshop that has these as basic geometry for creation.
• 3D as a form of Art
If you are an artist, it is a good experience to learn 3D softwares Like 3Ds max, Maya, cinema 4D and Zbrush. It can be used to sculpture your designs in photoshop and making it real. Also it is a good profession to know 3D incase you want to expand and venture other art forms such as making video for movies.
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https://numberworld.info/11100001220110
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# Number 11100001220110
### Properties of number 11100001220110
Cross Sum:
Factorization:
2 * 5 * 11 * 419 * 240833179
Divisors:
Count of divisors:
Sum of divisors:
Prime number?
No
Fibonacci number?
No
Bell Number?
No
Catalan Number?
No
Base 3 (Ternary):
Base 4 (Quaternary):
Base 5 (Quintal):
Base 8 (Octal):
a186ba1360e
Base 32:
a31lq2dge
sin(11100001220110)
-0.45870289524736
cos(11100001220110)
-0.8885896994067
tan(11100001220110)
0.51621450884883
ln(11100001220110)
30.037966334167
lg(11100001220110)
13.045323026524
sqrt(11100001220110)
3331666.4328996
Square(11100001220110)
1.2321002708644E+26
### Number Look Up
Look Up
11100001220110 (eleven trillion one hundred billion one million two hundred twenty thousand one hundred ten) is a impressive figure. The cross sum of 11100001220110 is 10. If you factorisate the figure 11100001220110 you will get these result 2 * 5 * 11 * 419 * 240833179. The figure 11100001220110 has 32 divisors ( 1, 2, 5, 10, 11, 22, 55, 110, 419, 838, 2095, 4190, 4609, 9218, 23045, 46090, 240833179, 481666358, 1204165895, 2408331790, 2649164969, 5298329938, 13245824845, 26491649690, 100909102001, 201818204002, 504545510005, 1009091020010, 1110000122011, 2220000244022, 5550000610055, 11100001220110 ) whith a sum of 21848386089600. The number 11100001220110 is not a prime number. The figure 11100001220110 is not a fibonacci number. The figure 11100001220110 is not a Bell Number. 11100001220110 is not a Catalan Number. The convertion of 11100001220110 to base 2 (Binary) is 10100001100001101011101000010011011000001110. The convertion of 11100001220110 to base 3 (Ternary) is 1110022011001011121000020101. The convertion of 11100001220110 to base 4 (Quaternary) is 2201201223220103120032. The convertion of 11100001220110 to base 5 (Quintal) is 2423330300303020420. The convertion of 11100001220110 to base 8 (Octal) is 241415350233016. The convertion of 11100001220110 to base 16 (Hexadecimal) is a186ba1360e. The convertion of 11100001220110 to base 32 is a31lq2dge. The sine of the figure 11100001220110 is -0.45870289524736. The cosine of 11100001220110 is -0.8885896994067. The tangent of the figure 11100001220110 is 0.51621450884883. The root of 11100001220110 is 3331666.4328996.
If you square 11100001220110 you will get the following result 1.2321002708644E+26. The natural logarithm of 11100001220110 is 30.037966334167 and the decimal logarithm is 13.045323026524. You should now know that 11100001220110 is special number!
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https://www.teacherspayteachers.com/Product/Dividing-Fractions-5th6th-Grade-Assessment-4976090
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# Dividing Fractions 5th/6th Grade Assessment
Subject
Resource Type
File Type
PDF
(858 KB|5 pages)
Product Rating
Standards
Also included in:
1. This Dividing Fractions Study Guide and Assessment resource has all you need to help your students prepare and assess their knowledge of the common core standards related to dividing fractions in sixth grade.The study guide mimics the assessment so that your students will be familiar with the types
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Save \$0.30
• Product Description
• StandardsNEW
This Dividing Fractions Assessment is a 9 question assessment that will allow you to review your students' knowledge of many state standards related to dividing fractions.
The assessment includes:
• multiple choice questions
• equations that require work to be shown
• equations with models
Included with this assessment is an answer key and featured on the printable assessment are growth mindset messages to keep your students motivated while they are taking their assessment!
If you are looking for an accompanying Study Guide, here you go!
Or buy as a bundle and save!
Thank you, and enjoy!
Permission to copy for single classroom use only.
Electronic distribution limited to single classroom use only. Not for public display.
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (𝘢/𝘣) ÷ (𝘤/𝘥) = 𝘢𝘥/𝘣𝘤.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
Total Pages
5 pages
Included
Teaching Duration
90 minutes
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https://cuboss.com/what-is-a-speedcube/
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# What is a speedcube?
Now that the sport of speedcubing has taken off, you may have heard of the word speedcubes. So what exactly is a speedcube? In short, you could say that they are fast cubes that are easy to turn. But there are actually quite a few aspects that make a speedcube a must if you want to solve the Rubik’s cube really quickly. Imagine a racing car. It’s not just that it’s fast that differs from a normal car, it’s also about grip, shape and acceleration. Below I list six different points that you usually take into account when you want to find a really good speedcube.
#### A speedcube is faster than a normal cube
The first thing you will notice when you turn a Speedcube compared to a regular Rubik’s cube is that it is very easy to turn. This allows you to turn the cube very quickly and without the layers locking onto each other. They have come a long way when it comes to reducing the pressure and friction in the cube to make the resistance as small as possible. In addition, they are often lubricated with silicone internally to reduce friction even more.
See the GIF for a quick comparison between an original Rubik’s cube and a speedcube.
#### A speedcube can cut corners
On a normal cube, all the pieces need to be in line with each other in order to be able to turn a layer. When you have to solve a Rubik’s cube quickly, it becomes very difficult to have exact precision between the different moves. A speedcube can cut corners, meaning you can start the next move before you’ve finished turning the previous move. In other words, you don’t have to be as precise with your moves.
See the GIF to see how hard it is to cut corners on an original Rubik’s cube compared to a Speedcube.
#### Does the cube pop?
When a speedcube is manufactured, you want as little friction as possible inside the cube so that it can be turned easily. To achieve that, you don’t tighten the screws in the cube as hard as you did on the original Rubik’s cube. The advantage of that is that the pressure and friction in the cube is reduced, making it faster. However, it can also cause the cube to pop, as it is called when the cube falls apart or some pieces fall out of the cube.
Development has progressed rapidly in recent years and today’s 3×3 speedcubes no longer pop. But when it comes to other models such as 2×2 or larger cubes such as 4×4, they still pop sometimes. Some speedcubes are more resistant to popping than others.
N.B.! The cube doesn’t break when it pops, you just have to put it back together. If you struggle finding a way to put the pieces back together, feel free to check for tutorials online as there are countless of them.
#### Lock ups
Sometimes you can experience that the cube locks up. This usually does not mean that the cube is completely stuck so that it cannot be turned, it is rather that the cube gets stuck internally and does not move smoothly during a move. This can lead to you getting a slightly worse times compared to a cube where all the moves flow smoothly so that you never have to stop.
See our GIF on what it might look like when a cube locks up.
#### Does the cube corner twist?
Another side effect of speedcubes not being screwed very tight (to make them faster) is that the corner pieces of the cube can spin around their own axis accidentally. This means that the piece itself turns without changing its position. When you turn the cube quickly and perform various finger tricks, it may happen that you accidentally turn a corner piece in this way. Then the cube can no longer be solved until the corner is turned back again manually. This can cause confusion and extra time for those who want to solve the cube quickly. Today’s speedcubes are however much better than just a couple of years ago, and corner twists are much less of a problem nowadays.
Something that all speedcubes have in common is that they can be adjusted. This is done by tightening or loosening screws found in the cube. Adjusting the cube can affect the above points quite a bit. Manufacturers always strive to implement new ways to adjust the cube so that you can find your dream settings. On many of today’s cubes, you can not only use a screwdriver, but also adjust the spring tension and magnet strength using included tools.
#### Find a speedcube with the right feeling
Last but not least is the feeling of the cube. Different models that have roughly the same performance in terms of, for example, how fast it is and how well it cuts corners. But they can still feel completely different from each other depending on the mechanism inside the cube. Some speedcubes feel soft like silk inside them and others feel scratchy and crispy. Some cubes are more hollow, which is modern these days and makes the cube feels lighter and airier, while other cubes are compact and feel more robust. The weight can also vary among the different models.
When it comes to the feeling in the cube, there is nothing that is objectively right or wrong, we all have different likes and tastes. When it comes to modern 3×3 speedcubes, there are a lot of cubes that have really good performance and thus it matters a lot about finding the right feeling in the cube. When it comes to larger cubes, there is still a lot of room for improvement and thus performance becomes more important, e.g. speed and corner cutting.
### Choose a speedcube with the right size
If you have small hands, a regular cube might feel a bit clumsy. Therefore, the manufacturers have created speedcubes in several different sizes so that you can find a good cube just for you. An original Rubik’s cube is 57 mm. Nowadays, it is common for speedcubes to be manufactured in 56 mm as standard, but there are also both smaller and larger cubes to choose from.
### Where do I buy speedcubes?
You don’t have to search anymore. Cuboss is the biggest speedcube store in Sweden, and we ship worldwide. We have a large selection of awesome speedcubes. We also have cubes and cube accessories under our own brand that are used by some of the fastest speedcubers. Below you will find links to help you choose the right cube.
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# How to move an object uniformly from one point to another at a fixed angle? [duplicate]
In my 2d java game, I need to move Point objects (as projectiles) from Enemy object to Player object in a straight line. I did some research and found out that I can use atan2(x, y) method of Math class to get the angle, but then how am i supposed to move the projectile in that particular angle(direction)? I checked this question: How do I calculate how an object will move from one point to another? , but did not quite understand. Also there is no vector math in java. Any ideas would be very helpful.
• Not understanding an answer isn't reason enough to post the same question again. If you're having trouble implementing the solution from the other question, ask specifically about the trouble you're having. For example, in this situation, it appears you're having trouble with vector math. That means your solution is to get a vector math library or write your own. Vector math is very important for games, and it's unlikely you'll be able to avoid it for long. See this stackoverflow question about it.
– House
Sep 26 '14 at 13:42
[EDIT: as I don't know what lib you're using, and as this question is not specific to Java, my answer is in pseudo-code.]
If I understand your question correctly, atan2 is not what you're looking for.
What you want to achieve is moving something (which has coordinates) from one point to another according to time. You just need to compute the vector between the player entity and the target.
What you need here is:
• the position (x,y) of the player (player_position).
• the position (x,y) of the target (target_position).
• the speed at which you want your bullet to travel (in unit per seconds) (bullet_speed).
• the time passed between two game updates (two frames) (deltaTime).
As you seem to have all of that, you seem ready to make that bullet move.
The vector from your player to the target, in pseudo code, is :
Vector2 diff_vector = player_position - target_position;
Then put your bullet at player_position and move it each frame alongside this vector you just computed using the deltaTime between frames. Something like:
new_bullet_position = bullet_position + diff_vector * deltaTime * bullet_speed;
Changing bullet position each frame accordingly to the diff_vector and the bullet_speed will make it move uniformly from player to target.
I hope it helps.
• You'd need to normalize the diff_vector to make this work (otherwise you'd get a lot more movement than expected!). And it's actually the same approach found in the highest voted answer to the question linked by OP.
– House
Sep 26 '14 at 13:37
• But then I have to use some game library for using vectors because I am making my game using java and its 2d api and there are no classes that deals with vector math. Is there no alternative? Is it not possible to do the calculations without vectors? Sep 27 '14 at 8:07
• Come on! :) The most complex computation in my answer is about multiplying vectors. Something very simple. If you're afraid of vectors I would suggest you to pause developping games and wait until you know more about vectors, because you'll have really hard times trying to make games without using vector computation. Sep 27 '14 at 8:18
• I am a java programmer. What should i use? Sep 27 '14 at 11:43
• You should use Wikipedia. Go to the page related to vectors in Mathematics. Take one hour and read it until everything is clear. Then go back to your code, whatever langage you use because it's really not related to the langage, and code the thing. I won't code it for you. You asked a question about something that is not related to Java programming but related to an higher concept which is very VERY important when it comes to game programming. I suggest you to try. Sep 27 '14 at 11:50
Suppose the projectiles curret position is $(x,z)$. Then, in each step, you let $x_{new} = x_{old} + t \cos(\theta)$ and $y_{new} = y_{old} + t \sin(\theta)$ where $\theta$ is the angle you found, and $t$ is some small number (depending on the number of milliseconds since last update, for example).
• What, LaTeX does not work here? Sep 26 '14 at 12:29
• Because we use code here, not LaTeX. It's not enabled for this site as far as I know.
– House
Sep 26 '14 at 13:33
• Just surprises me, this is essentially a math question, and LaTeX is the standard for typing mathematics... Sep 26 '14 at 13:37
• Yep, I'm sure people (myself included) would appreciate you posting in meta about it (for it to be turned on for the site), I can't see how it would hurt.
– House
Sep 26 '14 at 13:39
• @PerAlexandersson See an existing meta question on enabling LaTeX processing. Long story short, it's supposedly expensive and it's hard to get it enabled due to unsubstantiated performance concerns. I really miss it and it really discourages me from answering questions here. Sep 26 '14 at 14:45
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info@mywordsolution.com
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The problem is how the specific equation should be derived by using boundary condition. It consists of the final equation (which is based on Langmuir isotherm) and the boundary condition is as well known so just want to know to reach the final equation by using boundary condition? Step by step. The model is on the assumptions that:
i) In a As-HA binary system, sorbent binding sites form complexes represented as SA when arsenic binding occurs and complexes denoted SH when HA is bound and
ii) Equilibrium can be expressed as S+A? SA, where K1 = [SA]/[S][A], S + H ? SH, where K2 = [SH]/[S][H] and A + H ? AH where K3 = [AH]/[A][H]. In this model, considering an interactions resulting in SAH complexes, the equilibrium can be expressed as SA + H? SAH, where K1,2 = [SAH]/[SA][H], SH+A ? SHA, where K2,1 = [SHA]/[SH][A] and S+AH ? SAH where K3,3 = [SAH]/[S][AH]. Then one can assume S+A+H? SAH and S+H+A? SHA, where K= K1K1,2 = K2K2,1 = K3K3,3.
Assuming that the sorption system is in equilibrium (there are no net changes of [SA], [SH] and [SAH] with respect to time), the following can be written: d[SA]/dt = 0 , d[SH]/dt = 0 and d[SAH]/dt = 0 and [S0] = [S]+[SA]+[SH]+[SAH]. In such case, the model can be presented to the form: qAs = (qmax Ce [As]{1+(K1/K )Ce [HA] })/({K1+Ce [As]+(K1/K2) Ce [HA]+2(K1/K) Ce [As] Ce [HA]}).
So, describe how to reach the above final equation by using the mentioned assumptions.
• Category:- Chemistry
• Reference No.:- M91933
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Breaking News
What is the price of a troy ounce of gold
Assume for this problem that the price of gold is \$1,200 per troy ounce. If the price of silver is \$20 per troy ounce, how many troy pounds of silver would it take to be worth as much as a single troy ounce of gold
3.75
Explanation:
Given;
Price of gold = \$1,200 per troy ounce
Price of silver = \$20 per troy ounce
Therefore,
1200/20 = 60 pieces of silver is worth one troy ounce of gold but;
1 ounce = 0.0625 pounds
60 ounces = 60 ×0.0625
= 3.75
Therefore a 3.75 pounds of silver is worth single troy ounce of gold.
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STAT213-23S2 (C) Semester Two 2023
# Statistical Inference
15 points
Details:
Start Date: Monday, 17 July 2023 End Date: Sunday, 12 November 2023
Withdrawal Dates
Last Day to withdraw from this course:
• Without financial penalty (full fee refund): Sunday, 30 July 2023
• Without academic penalty (including no fee refund): Sunday, 1 October 2023
## Description
This course provides the theoretical foundations for statistical estimation and testing at an introductory level. These are essential for more advanced studies in statistics at higher levels because they facilitate a deeper understanding of statistical techniques and their applications.
To illuminate key ideas in estimation and testing, the course will focus mainly on inference for independent and identically distributed univariate data.
Topics that are usually covered include:
• Fundamentals of probabilistic modelling: Probabilities, distribution functions, densities, expectations, quantiles
• Expected values, Moment-generating functions
• Likelihood function, Maximum likelihood principle, Score function, Fisher Information
• Sufficient statistics
• Estimation: Method of Moments, Maximum likelihood estimation
• Properties of estimators: Unbiasedness, Efficiency, Consistency
• Sampling distributions, the Law of large numbers, Central limit theorem
• Interval estimation
## Learning Outcomes
• On completion of the course, you will be able to:
• Apply various discrete and continuous univariate probability distributions in modelling statistical processes.
• Understand the concept of sampling distributions and how to apply them.
• Estimate unknown parameters of a given probability distribution using standard estimation techniques.
## Prerequisites
(1) one of MATH102, MATH199 or EMTH118; and (2) one of STAT101, DATA101, STAT211, EMTH119, or
EMTH210
STAT214
## Assessment
Assessment Due Date Percentage
Assignments 10%
Test 30%
Exam 60%
To obtain a clear pass in this course, you must both pass the course as a whole (≥ 50%) and also obtain at least 40% in the final examination.
## Textbooks / Resources
Course materials will be provided and no textbook is needed. After enrolling in the course, you will be able to access materials from the course web page in Learn at: http://www.learn.canterbury.ac.nz/
## Indicative Fees
Domestic fee \$824.00
International fee \$4,750.00
* All fees are inclusive of NZ GST or any equivalent overseas tax, and do not include any programme level discount or additional course-related expenses.
For further information see Mathematics and Statistics .
## All STAT213 Occurrences
• STAT213-23S2 (C) Semester Two 2023
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So far we have discussed continuous-time signals and their spectral characterization & discrete-time signals and their characterizations. But what connects the two?
Before analyzing this let us re-iterate a key result that affects the analysis.
The Fourier transform of a periodic continuous time impulse train is also a impulse train!
Let our continutous-time impulse train be: $s(t) = \sum_{n = -\infty}^{\infty} \delta(t - nT)$
Since this is periodic it has a Foruier series representation
$s(t) = \sum_{k = -\infty}^{\infty} a_k e^{\frac{j2 \pi k t}{T}}$ where $a_k = \frac{1}{2 \pi} \int_{-\frac{T}{2}}^{\frac{T}{2}} s(t) e^{-\frac{j2 \pi k t}{T}}dt$ Since there's only one impulse between $-\frac{T}{2}$ & $\frac{T}{2}$ we can write: \begin{align*} a_k &= \frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} s(t) e^{-\frac{j2 \pi k t}{T}}dt \\ &= \frac{1}{T} e^{-\frac{j2 \pi k 0}{T}} \\ &=\frac{1}{T} \end{align*}
So the Fourier series representation of the impulse train is: $s(t) = \sum_{n = -\infty}^{\infty} \delta(t - nT) = \frac{1}{T} \sum_{k = -\infty}^{\infty} e^{\frac{j 2 \pi k t}{T}}$ \begin{align*} S(\omega) &= FT \cdot \sum_{n = -\infty}^{\infty} \delta(t - nT) \\ &= FT \cdot \sum_{k = -\infty}^{\infty} e^{\frac{j 2 ]pi k t}{T}} \\ &= \frac{1}{T} \sum_{k = -\infty}^{\infty} FT \cdot e^{\frac{j 2 \pi k t}{T}} \\ &= \frac{1}{T} \sum_{k = -\infty}^{\infty}2 \pi \delta(\omega - \frac{2 \pi k}{T}) \end{align*} $\sum_{n = -\infty}^{\infty} \delta(t - nT) \leftrightarrow \frac{2 \pi}{T} \sum_{k = -\infty}^{\infty} \delta(\omega - \frac{2 \pi k}{T})$
Let us now consider how we can convert a continuous time signal $x(t)$ to a discrete time sginal $x[n]$ $x(t) \leftrightarrow x[n]$
Ideally our conversion process should be such that:
1. We want the DTFT of x[n] to be like the CTFT of x(t)
2. We want to be able to recover $x(t)$ from $x[n]$
3. Manipulation of $x[n]$ should translate to corresponding manipulation of $x(t)$
The simplest way to derive an $x[n]$ from an $x(t)$ is for the DT signal $x[n]$ to simply be a series of uniformly spaced snapshots of $x(t)$
However naively taking snapshots is insufficient. The reason for this is that the series $x[n]$ does not carry any information about what happens between the snapshots.
For us to be able to recover $x(t)$ uniquely from $x[n]$, there are conditions on $x(t)$ & the spacing "T" between snapshots.
So now we shall:
1. Find the relation between the snapshots and $x[n]$
2. Establish the "sampling" criteria for this to be unique
3. Discuss how $x(t)$ can be recovered from $x[n]$
4. Find the correspondences between the continuous-time frequencies in x(t) & the discrete-time frequencies in $x[n]$
5. What happens if the criteria in (2) are violated?
6. Discuss how the snapshots can be captured in a practical setting and how the original $x(t)$ can be recovered from the snaphots & the issues involved
Lets begin
1. The DT signal $x[n]$ acutally represents a time-domain signal $x_T(t)$ that takes the value of the CT signal $x(t)$ at $t = nT$ \begin{align*} x_T(t) &= \sum_{n = -\infty}^{\infty} x(nT) \delta(t - nT) \\ &= x(t) \cdot \sum_{n = -\infty}^{\infty} \delta(t - nT) \end{align*}
Thus, $x(t)$ can be recovered from $x[n]$ if it can be recovered from $x_T(t)$
The FT of $x_T(t)$ is given by: \begin{align*} X_T(\omega) &= FT \cdot x(t) \sum_{n = -\infty}^{\infty} \delta(t - nT) \\ &= X(\omega) \ast \frac{2 \pi}{T} \sum_{k = -\infty}^{\infty} \delta(\omega - \frac{2 \pi k}{T}) \\ \end{align*} $X_T(\omega) = \frac{1}{T} \sum_{k = -\infty}^{\infty} X(\omega - \frac{2 \pi k}{T})$
$X_T(\omega)$ is simply the superpsotion of copies of $X(\omega)$ shifted by multiple of $\frac{2 \pi}{T}$
2. From the above discussion its fairly clear that $X(\omega)$ can be recovered from $X_T(\omega)$ by simply lowpass filtering out frequency shifted copies
For this to work, however the condition is that the frequency shifted versions of $X(\omega)$ should not have overlaps.
In the overlap areas $X_T(\omega)$ cannot be resolved into the individual shifted copies of $X(\omega)$. So $X(\omega)$ can never be recovered.
So the criterion to recover $X(\omega)$ from $X_T(\omega)$ (and thereby $x(t)$ from $x_T(t)$, & as a consequence from $x[n]$) is that copies of X(\omega) that are shifted by $\frac{2 \pi}{T}$ must not overlap.
This has two implications:
1. $x(t)$ must be bandlimited. If it were not bandlimited. $X(\omega)$ & its shifted copied would ALWAYS overlap.
2. The "sampling period" T should be such that $X(\omega)$ & $X(\omega - \frac{2 \pi}{T})$ do not overlap.
In other words, it $\omega_{max}$ is the highest frequency in $x(t)$,
$\frac{2 \pi}{T} > 2 \omega_{max}$
Sampling frequency $\frac{1}{T} > 2 f_{max}$
The sampling frequency must be greater than twice the largest frequency in the signal.
This leads us to the famous "Nyquist sampling theorem" named after Nyquist who discover it in 1928.
*Note* Let $x(t)$ be a bandlimited signals with $X(\omega) = 0$ for $|\omega| \geq \omega_max$ Then $x(t)$ is uniquely determined by its sample $x[n] = x(nT), n = 0, \pm 1, \pm 2, ...$ if $\frac{2 \pi}{T} \geq \omega_{max}$ or alternately if the sampling frequency is greater than the highest frequency in the singal x(t).
3. Recovering $x(t)$ from $x[n]$, provided the Nyquist condition has been satisfied during sampling, is straigh forward.
1. Compose $x_T(t)$ by modulating an impulse train by $x[n]$
2. Low pass filter $x_T(t)$ The low-pass filter can be composed as an LTI system $h[n]$ $H(\omega) = \begin{cases} T, & |\omega| < \frac{\pi}{T} \\ 0, & else \end{cases}$ The impulse response $h[n]$ of the filter is given by: $h(t) = T \int_{-\frac{\pi}{T}}^{\frac{\pi}{T}} e^{j \omega t} d \omega$ The equation above which works out to: \begin{align*} h(t) &= \frac{T}{jt} \left[_{-\frac{\pi}{T}}^{\frac{\pi}{T}}e^{j \omega t}\right] \\ &= 2 \pi \frac{sin \frac{\pi t}{T}}{\frac{\pi t}{T}} \\ &= 2 \pi sinc(\frac{\pi t}{T}) \end{align*}
So to get $x(t)$ from $x[n]$
Note that $h(t)$ is non-causal. We cannot practically get $x(t)$ from $x[n]$ in this manner. We will revist practical implementations shortly.
4. Lets now see how the spectrum of $x(t)$ relates to the DTFT of $x[n]$. To do so we will develop their equations one after another:
For $X_T(\omega)$:
\begin{align*} X_T(\omega) &= \frac{1}{2 \pi} \int_{-\infty}^{\infty} x_T(t) e^{- \omega t}dt \\ &= \frac{1}{2 \pi} \int_{-\infty}^{\infty} \sum_n x(nT) \delta(t - nT) e^{-j \omega t}dt \\ &= \frac{1}{2 \pi} \sum_n x(nT) \int_{-\infty}^{\infty} \delta(n - nT) e^{-j \omega t}dt \\ Since x(nT) = x[n] &= \frac{1}{2 \pi} \sum_{n = -\infty}^{\infty}x[n] \int_{-\infty}^{\infty} \delta(t - nT)e^{-j \omega t}dt \end{align*} $X_T(\omega) = \frac{1}{2 \pi} \sum_{n = -\infty}^{\infty}x[n]e^{-j \omega nT}$
For $X(\Omega)$:
$X(\Omega) = \sum_{n = -\infty}^{\infty} x[n] e^{-j \Omega n}$ Comparing $X_T(\omega)$ & $X(\Omega)$ we note that:
$X(\Omega) = X_T(\omega T)$
So the correspondence btween the discrete time frequency $\Omega$ and the continuous time frequency $\omega$ is given by: $\omega T = \Omega$ The relation between teh DTFT of $x[n]$ & teh CTFT of $x_T(t)$ (and thereby $x(t)$) is given by: $X(\Omega) = X_T(\omega T) = \sum_{k = -\infty}^{\infty}X (\omega T - \frac{2 \pi k}{T})$ Both sides are periodic - $X(\Omega)$ is periodic with $2 \pi$, $X_T(\omega T)$ is periodic with $\frac{2 \pi}{T}$
Note that $\omega T$ has dimension radians, which is the same as the dimension of $\Omega$
5. What if $x(t)$ is not bandlimited?
What is $T > \frac{\pi}{\omega_{max}}$
To understand this we must consider the recovery process first.
The recovered signal is obtained by creating $x_T(t)$ from $x[n]$ & then low-pass filtering $x_T(t)$
Let us represent a lowpass filter as $LPF_{\omega_c}(\omega)$ or $LPF_{\omega_c}(t)$
$LPF_{\omega_c}(\omega)$ = 1 $|\omega| \leq \omega_c$, 0 otherwise.
The recovered signal $x_r(t)$ is obtained as: $x_r(t) = X_T(t) \ast LPF_{\omega_c}(t)$ \begin{align*} X_r(\omega) &= X_T(\omega) \cdot LPF_{\frac{\pi}{T}}(\omega) \\ &=(\sum_{k = -\infty}^{\infty} X(\omega - \frac{2 \pi k}{T})) \cdot LPF_{\frac{\pi}{T}}(\omega) \end{align*} Ideally $x_r(t) = x(t)$, i.e. the recovery is perfect. or $X_r(\omega) = X(\omega)$ $X(\omega - \frac{2 \pi k}{T}) \cdot LPF_{\frac{\pi}{T}}(\omega) = 0 for k \neq 0$ If this condition is not satisifed, "alaising happens.
Instead of getting $x_r(t) = x(t)$ we get: $x_r(t) = IDFT \sum_{k = -\infty}^{\infty}X(\omega - \frac{2 \pi k}{T}) LPF_{\frac{\pi}{T}}(\omega)$ Since $X(\omega - \frac{2 \pi k}{T}) \leftrightarrow x(t)e^{j \frac{2 \pi k t}{T}}$ \begin{align*} x_r(t) &= \sum_{k = -\infty}^{\infty} IDFT (X(\omega - \frac{2 \pi k}{T}) LPF_{\frac{\pi}{T}}(\omega)) \\ &= x(t)LPF_{\frac{\pi}{T}}(t) + \sum_{k = 1}^{\infty}x(t)e^{\frac{j 2 \pi k t}{T}} \ast LPF_{\frac{\pi}{T}}(\omega) + \sum_{k = 1}^{\infty}x(t)e^{-\frac{j 2 \pi k t}{T}} \ast LPF_{\frac{\pi}{T}}(\omega) \\ &= x(t)LPF_{\frac{\pi}{T}}(t) \sum_{k = 1}^{\infty} (x(t) (e^{\frac{j 2 \pi k t}{T}} + e^{-\frac{j2 \pi k t}{T}})) \ast LPF_{\frac{\pi}{T}}(\omega) \\ &= x(t)LPF_{\frac{\pi}{T}}(t) + 2 \sum_{k = -\infty}^{\infty} (x(t)cos \frac{2 \pi k t}{T}) \ast LPF_{\frac{\pi}{T}}(\omega) \end{align*}
The recovered signal is the original signal LPFed PLUS a lot of junk.
The equation unfortunately doesn't explain what really happens.
Consider the following example: $x(t) = cos (\omega_o t)$ Let sample period T b such that $\frac{\pi}{T} < \omega_o$. For illustration we will assume $\frac{\pi}{T} < \omega_o \leq \frac{2 \pi}{T}$
The cosine is "undersampled" -- the sampling frequencing $\frac{2 \pi}{T}$ is less than the minimum $2 \omega_o$ prescribed by Nyquist's theorem.
For $x(t) = cos(\omega_o)t$ $X(\omega) = \frac{1}{2} (\delta(\omega - \omega_o) + \delta(\omega + \omega_o))$ \begin{align*} X_T(\omega) &= \frac{1}{T} \sum_k X(\omega - \frac{2 \pi k}{T}) \\ &=\frac{1}{T} (\delta(\omega - \omega_o - \frac{2 \pi k}{T}) + \delta(\omega + \omega_o - \frac{2 \pi k}{T})) \end{align*} $X_r(\omega) = frac{1}{T} (\delta(\omega - (\frac{2 \pi}{T} - \omega_o)) + \delta(\omega + (\frac{2 \pi}{T}) - \omega_o))$ $x_r(t) = cos(\frac{2 \pi}{T} - \omega_o)$ So $x(t) \neq x_r(t)$
More importantly a frequency component $\omega_o$ reappears in the reconstructed signal as a frequency component $\frac{2 \pi}{T} - \omega_o$
6. Now lets consider the practical issues of sampling.
We have been speaking so far in terms of impulse trains: $x(t) \rightarrow x_T(t) \rightarrow x[n]$ $x[n] \rightarrow x_T(t) \rightarrow x(t)$
For smapling, our model was that the CT signal $x(t)$ was multiplied by an impulse train, and then measured.
For resynthesis, we assume $x[n]$ can modulate an impulse train which is then filtered.
In practice impulse trains are only a theoretical concept. We will implement things a bit differently, and this has some implications.
Sampling
Sampling is done by an analogy-to-digital (or continuous to discrete) convertor.
For a 1-D signal this is performed using a circuit that can sample & hold a value
$x_h(t)$ -- the "held" signal can then be measured. The measured value is the $x[n]$ associated with the instant of the step.
Various variants of the above exist. But all of them can give an accurate measurement of $x[n].$
Recovery
Let us now consider the recovery process. The theoretical model: has two problems
1. Ideal impulse responses do not exist
2. The ideal lowpass filter is non-causal and cannot be practically implemented.
A more practical mechanism is the zero-order hold.
Instead of modulating an impulse train $x[n]$ modulates a rectangular pulse train. The rectangular pulse has the form: $rect_T(t) = \begin{cases} 1, & 0 \leq t < T \\ 0, & else \end{cases}$
The reconstructed signal: \begin{align*} x_{ZOH}(t) &= \sum_n x[n]rect_T(t - nT) \\ &= \sum_n x(nT) rect_T(t - nT) \end{align*} Like $x_T(t)$, $x_{ZOH}(t)$ must be filtered to recover $x(t)$. To find out what the filter must be we must analyze $x_{ZOH}(t)$.
We can model $x_{ZOH}(t)$ as the output of a linear shift-invariant filter with impulse response $rect_T(t)$
Proof
\begin{align*} x_{ZOH}(t) &= x_2(t) \ast rect_T(t) \\ &= \sum_n x(nT) \delta(t - nT) \ast rect_T(t) \\ &= \sum_n x(nT) rect_T(t - nT) \end{align*} \begin{align*} \therefore X_{ZOH}(\omega) &= X_T(\omega) Rect_T(\omega) \\ &= \sum_k X(\omega - \frac{2 \pi k}{T}) Rect_T(\omega) \end{align*} \begin{align*} Rect_T(\omega) &= FT \cdot rect_T(t) \\ &= e^{-\frac{j \omega T}{2}} \frac{sin(\omega \frac{T}{2})}{\pi \omega} \end{align*} $X_{ZOH}(\omega) = \sum_k X(\omega - \frac{2 \pi k}{T}) \frac{sin(\frac{\omega T}{2})}{\pi \omega}e^{-\frac{j \omega k}{T}}$
$x_{ZOH}(\omega)$ has extraneous frequency components and the "main lobe" is distorted
To fix this we need a "de-distorting" filter that also filters out extraneous frequencies.
The filter must have the characteristics $Filt(\omega) - 0, |\omega| > \frac{\pi}{T}$ That way: \begin{align*} x_{ZOH}(\omega) Filt(\omega) &= \\ &= \sum_k X(\omega - \frac{2 \pi k}{T})Rect(\omega)Filt(\omega) \\ &= X(\omega) Rect(\omega) Filt(\omega) \end{align*} Because $Filt(\omega) = 0, |\omega| > \frac{\pi}{T}$
To eliminate $Rect(\omega)$ we want: $Filt(\omega) = \frac{1}{Rect_T(\omega)}, |\omega| < \frac{\pi}{T}$ So if we set $Filt(\omega) = \begin{cases} \frac{1}{Rect_t(\omega)}, & |\omega| < \frac{2 \pi}{T} \\ 0, & else \end{cases}$ I.e. $Filt(\omega) = \begin{cases} \frac{e^{\frac{j \omega T}{2}}}{|\frac{sin(\frac{\omega T}{2})}{\pi \omega}|} , & |\omega| < \frac{\pi}{T} \\ 0, & else \end{cases}$
$X_r(\omega) = X_{ZOH}(\omega) Filt(\omega) = X(\omega)$
However the exact impulse response of $filt(t)$ will be non-causal and must be approximated with a causal filter. We will see how later.
The ZOH reconstruction is pretty crude, needing a fance filter to "correct" its errors.
A better system is teh "first order hold". The FOH is a linear interoplation.
The FOH is not perfect either and must be filtered als. As it turns out, the first order hold canbe modelled exactly as filtering $x_T(t)$ by a linear filter with impulse response $h_{FOH}(t)$
So we find: $X_{FOH}(\omega) = X_T(\omega) H_{FOH(\omega)}$ $\text{But } h_{FOH}(t) = \frac{1}{T} rect_T(t) \ast rect_T(t)$ $H_rect(\omega) = \frac{1}{T} [Rect_T(\omega)]^2$
Note that the lobes beyond $\frac{2 \pi}{T}$ are much suppressed with respect to $rect_T(\omega)$ & even between $\frac{\pi}{T}$ and $\frac{2 \pi}{T}$ there is more attenuation.
So $X_{FOH}(\omega) = \frac{1}{T} X_T(\omega) Rect_T^2 (\omega)$
We still need to correct this by filtering out high frequency components and cancelling out the shaping introduced by the FOH
And now we want: $Filt(\omega) = \begin{cases} \frac{T}{rect_T^2(\omega)}, & |\omega| \leq \frac{\pi}{T} \\ 0, & else \end{cases}$
This too is a non-causal filter, but as you may imagine causual approximation will introduce less error.
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Books Books
If the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth...
Pantologia. A new (cabinet) cyclopędia, by J.M. Good, O. Gregory, and N ... - Page 89
by John Mason Good - 1813
## Elements of Geometry: Containing the First Six Books of Euclid, with a ...
John Playfair - Euclid's Elements - 1806 - 311 pages
...hypothesis A=mB, therefore A=mnC. Therefore, &c. QED PROP. IV. THEOR. IF the first of four magnitudes have the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and...
## Elements of Geometry, Geometrical Analysis, and Plane Trigonometry: With an ...
Sir John Leslie - Geometry, Plane - 1809 - 493 pages
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## The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...
Euclid - Geometry - 1810 - 518 pages
...therefore E is to G, so isc F to H. Therefore, if the first, &c. QED C0R. Likewise, if the first have the same ratio to the second, which the third has to the fourth, then also any equimultiple!; 1 3. 5. b Hypoth. KEA GM L' FCDHN whatever of the first and third have...
## Easy Introduction to Mathematics, Volume 1
Charles Butler - Mathematics - 1814
...comparison of one number to another is called their ratio ; and when of four giren numbers the first has the same ratio to the second which the third has to the fourth, these four numbers are said to be proportionals. Hence it appears, that ratio is the comparison of...
## A Philosophical and Mathematical Dictionary: Containing an ..., Volume 2
Charles Hutton - Astronomy - 1815 - 628 pages
...of the third is also equal to that of the fourth ; or if when the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth : then, the first X)f the four magnitudes shall be to the second as...
## The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...
Euclides - 1816 - 528 pages
...fourth D. 1f, therefore, the first, &c. QED A CD 2.5. BouK V. See N. If the first of four magnitudes has the same ratio to the second which the third has to the fourth ; then any equimultiples whatever of the first and third shall have the same ratio to any equimultir...
## Elements of Geometry: Containing the First Six Books of Euclid, with a ...
John Playfair - Circle-squaring - 1819 - 333 pages
...A = mB, therefore A~mn C. Therefore, &c. Q, ED PROP. IV. THEOR. If thefirst of four magnitudes has the same ratio to the second which the third has to the fourth, and if any equimultiples whatever be taken of thefirst and third, and any whatever of the second and...
## The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...
Euclid, Robert Simson - Geometry - 1821 - 516 pages
...third is also equal to that of the fourth; or, if the multiple of the * SBP note. first be greater than that of the second, the multiple of the third is also greater than that of the fourth. VL Magnitudes which haye the same ratio are called proportionals....
## The First Six Books with Notes
Euclid - 1822 - 179 pages
...controversy among geometers. Euclid defines them thus: The Jirst of four magnitudes is said to have the same ratio to the second, which the third has...fourth, when any equi-multiples whatsoever of the Jirst and third being taken, and any equi-multiples whatsoever of the second and fourth being taken,...
## Universal Technological Dictionary: Or, Familiar Explanations of ..., Volume 2
George Crabb - Industrial arts - 1823
...15 to 5, which is expressed thus : as 6 : 2 : : 15 : 5. The first of four magnitudes is said to have the same ratio to the second which the third has to...the multiple of the first be less than that of the third, the multiple of the second is also less than that of the fourth ;• if equal, equal ; and if...
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Estimating the speed of light using normal model BDA3 p. 66
ggplot2, and gridExtra are used for plotting, tidyr for manipulating data frames
library(ggplot2)
theme_set(theme_minimal())
library(gridExtra)
library(tidyr)
library(rprojroot)
root<-has_file(".BDA_R_demos_root")$make_fix_file() Data y <- read.table(root("demos_ch3","light.txt"))$V1
Sufficient statistics
n <- length(y)
s2 <- var(y)
my <- mean(y)
Positive values only
y_pos <- y[y > 0]
Sufficient statistics
n_pos <- length(y_pos)
s2_pos <- var(y_pos)
my_pos <- mean(y_pos)
Compute the density of mu in these points
tl1 <- c(18, 34)
df1 <- data.frame(t1 = seq(tl1[1], tl1[2], length.out = 1000))
Compute the exact marginal density for mu
# multiplication by 1./sqrt(s2/n) is due to the transformation of variable
# z=(x-mean(y))/sqrt(s2/n), see BDA3 p. 21
df1$pm_mu <- dt((df1$t1 - my) / sqrt(s2/n), n-1) / sqrt(s2/n)
Compute the exact marginal density for mu for the positive data
df1$pm_mu_pos = dt((df1$t1 - my_pos) / sqrt(s2_pos/n_pos), n_pos-1) / sqrt(s2_pos/n_pos)
Create a histogram of the measurements
p1 <- ggplot() +
geom_histogram(aes(y), binwidth = 2, fill = 'steelblue', color = 'black') +
coord_cartesian(xlim = c(-42, 42)) +
scale_y_continuous(breaks = NULL) +
labs(title = 'Newcomb\'s measurements', x = 'y')
Create a plot of the normal model
# gather the data points into key-value pairs
df2 <- gather(df1, grp, p, -t1)
# legend labels
labs2 <- c('Posterior of mu', 'Posterior of mu given y > 0', 'Modern estimate')
p2 <- ggplot(data = df2) +
geom_line(aes(t1, p, color = grp)) +
geom_vline(aes(xintercept = 33, color = 'q'),
linetype = 'dashed', show.legend = F) +
coord_cartesian(xlim = c(-42, 42)) +
scale_y_continuous(breaks = NULL) +
labs(title = 'Normal model', x = 'mu', y = '') +
scale_color_manual(values = c('blue', 'darkgreen', 'black')) +
guides(color=FALSE) +
annotate("text", x=24, y=0.26, label=labs2[1], hjust="right", size=5) +
annotate("text", x=26, y=0.58, label=labs2[2], hjust="right", size=5) +
annotate("text", x=32, y=0.7, label=labs2[3], hjust="right", size=5)
Combine the plots
grid.arrange(p1, p2)
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472,961 Members | 1,500 Online
# c++ code for shortest path in the graph
Hi all,
I am in shortage of time...and i want to know if someone has a code
written in c++ or c for finding the shortest path using stack or
queue??????my specifications r as follow:
Input data:
n = integer number representing the number of vertices
k = small integer number representing the maximal vertex degree
G = unoriented labeled connected graph G with n vertices and maximal
degree k
u,v = a pair of vertices of graph G
s = integer number
p = number of processors ----for the time being its not necessary...
Recommendation for generating G:
Use graph generator Generator1 with type -t 1. It generates unoriented
connected graph. Then the edges must be labeled by random weights from
interval <-255,255>. Make sure that some edges have negative weights,
since if edges have only positives weights, the problem has polynomial
complexity and it is easy to solve.
Find a path (each vertex must be visited at most once) connecting pair
of vertices u a v such that the sum of edge labels is maximal over all
possible such paths and at the same time, lower than s.
Output of the algorithm:
Information whether such a path exists. If so, then the sequence of
edges of the minimal path with the weights of its edges.
Sequential algorithm:
The sequential algorithm is of type BB-DFS with the search depth
bounded by |V|-1. A possible final state is a path connecting vertices
u and v. The price to be minimized is the total weight of the path. The
lower bound on the path weight is c-1. The algorithm terminates if the
price is equal to the lower bound. Otherwise, the algorithm must
perform exhaustive search. Note that a solution may not exist.
-------------------------------------
any idea regarding this then PLEASE MAIL a copy of it on my mail id as
well....i will b waiting for atleast SOME HELP....
regards.
Mar 26 '06 #1
4 12048
On 25 Mar 2006 16:54:09 -0800, "Shuch" <Sh**********@gmail.com> wrote:
Hi all,
I am in shortage of time...and i want to know if someone has a code
written in c++ or c for finding the shortest path using stack or
queue??????my specifications r as follow:
any idea regarding this then PLEASE MAIL a copy of it on my mail id as
well....i will b waiting for atleast SOME HELP....
regards.
Haha! Somebody's going to fail comp science...
Mar 26 '06 #2
haha...thats not possible...:)...coz i have some ideas but the problem
is just that i m not able to figure it out how to write it
correctly..thats y i m askin if someone has some code....
Mar 26 '06 #3
On 26 Mar 2006 02:46:21 -0800, "Shuch" <Sh**********@gmail.com> wrote:
haha...thats not possible...:)...coz i have some ideas but the problem
is just that i m not able to figure it out how to write it
correctly..thats y i m askin if someone has some code....
Wrong group, though. Find a specific algorithms group or web forum.
This group is for the C++ language itself.
Also, head down to the library.
Mar 26 '06 #4
On 25 Mar 2006 16:54:09 -0800, "Shuch" <Sh**********@gmail.com> wrote
in comp.lang.c++:
Hi all,
I am in shortage of time...and i want to know if someone has a code
written in c++ or c for finding the shortest path using stack or
queue??????my specifications r as follow:
Send us your instructor's email address and we'll save you even more
time by turning in the homework as well.
--
Jack Klein
Home: http://JK-Technology.Com
FAQs for
comp.lang.c http://c-faq.com/
comp.lang.c++ http://www.parashift.com/c++-faq-lite/
alt.comp.lang.learn.c-c++
http://www.contrib.andrew.cmu.edu/~a...FAQ-acllc.html
Mar 27 '06 #5
This thread has been closed and replies have been disabled. Please start a new discussion.
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Geometry Angles Worksheet - Page 3 | Problems & Solutions
# Geometry Angles Worksheet - Page 3
Geometry Angles Worksheet
• Page 3
21.
Identify a line segment.
a. Figure 4 b. Figure 3 c. Figure 2 d. Figure 1
#### Solution:
A line segment is a part of a line with two end points.
Among the figures, Figure 3 represents a line segment.
22.
How many points are required to name an angle?
a. 2 b. 4 c. 3 d. 1
#### Solution:
An angle is formed when two line segments meet at common point called the vertex.
So, 3 points are required to name an angle.
23.
An angle of 90° is called a _______.
a. right angle b. straight angle c. complete angle d. acute angle
#### Solution:
An angle of 90° is called a right angle.
24.
"The common end point of two rays or line segments forming an angle is called the vertex". State if the statement is true or false.
a. True b. False
#### Solution:
The common end point of two rays or line segments forming an angle is called the vertex.
The statement is true.
25.
Which of the following extends endlessly in both directions?
a. a point b. a line c. a ray d. a line segment
#### Solution:
Among the choices listed, only a line extends endlessly in both directions.
26.
How many pairs of parallel line segments are there in the figure?
a. two pairs b. one pair c. four pairs d. three pairs
#### Solution:
Lines, which never meet are called parallel lines.
In the figure, the lines AB and CD, BC and DA never meet and are called parallel lines.
There are two pairs of parallel lines. AB, CD and BC, DA.
27.
How many angles do the four sides of a rectangle form?
a. four b. three c. one d. two
#### Solution:
An angle is formed only when two rays or line segments meet at a common point.
The four sides of a rectangle form four angles.
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Author Message
Intern
Joined: 18 Mar 2010
Posts: 39
Followers: 1
Kudos [?]: 21 [0], given: 11
### Show Tags
08 Dec 2010, 15:29
Please evaluate the following argument. (I am not happy with the way it turned out, I felt I could not make a convincing argument here)
Over the past decade, the restaurant industry in the country of Spiessa has experienced unprecedented growth. This
surge can be expected to continue in the coming years, fueled by recent social changes: personal incomes are rising,
more leisure time is available, single-person households are more common, and people have a greater interest in
gourmet food, as evidenced by a proliferation of publications on the subject.”
The news paper claims that that the restaurant industry in the country of Siesta is thriving owing to many social changes like rising personal incomes, availability of leisure time, increase in single person households. And this trend is expected to continue in future. This argument needs a closer evaluation to validate the merits of it.
Firstly the article claims that the trend in growth is expected to continue because of the afore mentioned factors, before coming to a conclusion it would be interesting to know if the market for restaurants has the market already matured. Where exactly on the growth curve is the restaurant industry in? Who are the new customers for the projected growth?
Secondly the article also claims that the recent spurt of publications on this subject implies people are interested in gourmet food and hence a projected increase restaurant business. An increase in number of articles is by no means an indication of future growth of the sector as a whole. It would also be interesting to know if the restaurant industry is funding these publications as a form of proxy marketing.
Thirdly, assuming that the increase in articles is because of reader interest, one would want to know how much of the casual interest shown by readers on articles actually translates in to paying customers for the restaurant industry. Also the increase in publication can be indicative of people being more interested in cooking exotic food for themselves rather than paying for it.
Though the argument has its merits, it would be unwise to qualify the article without answering the questions raise above.
Intern
Joined: 18 Mar 2010
Posts: 39
Followers: 1
Kudos [?]: 21 [0], given: 11
### Show Tags
08 Dec 2010, 15:54
I particularly found this argument little hard, couldn't really spot solid gaps that easily.
Can any of you give me few more points here?
Intern
Joined: 18 Mar 2010
Posts: 39
Followers: 1
Kudos [?]: 21 [0], given: 11
### Show Tags
10 Dec 2010, 02:07
Can someone help me here with some more ammunition?
Veritas Prep GMAT Instructor
Joined: 26 Jul 2010
Posts: 244
Followers: 222
Kudos [?]: 502 [1] , given: 29
### Show Tags
14 Dec 2010, 17:04
1
KUDOS
Hey rockroars,
One of the keys to these is looking at the prompt, which points out that you can use:
Assumptions
Ways to weaken the argument
Ways to strengthen the argument
Typically those go hand-in-hand, so if you don't find any "assumptions" per se, just ask yourself if you can come up with hypotheticals that would weaken the argument. Here are a few that come to mind for me:
-could a greater interest in gourmet food mean that people are more likely to want to cook it for themselves as opposed to going out for it?
-could the number of single-person households actually mean that fewer people are going out to eat? Maybe families/couples eat out more than singles?
-Are rising personal incomes sustainable?
-Does "more leisure time" really correspond to more eating at restaurants? Couldn't that lead to more travel outside of Spiessa, or more people pursuing their own cooking/vegetable-gardening?
I find that a lot of times it's easier to propose weaknesses ("well, what if...?") and then retrofit the assumption. For example, that last point "people could want to cook or grow their own food more with more free time" points out the assumption that "more free time ---> more restaurant patronage"
I hope that helps...
_________________
Brian
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Intern
Joined: 18 Mar 2010
Posts: 39
Followers: 1
Kudos [?]: 21 [0], given: 11
### Show Tags
21 Dec 2010, 08:17
Thanks Brian
Similar topics Replies Last post
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Display posts from previous: Sort by
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# Downward Triangle Star Pattern Program in C#
## Downward Triangle Star Pattern Program in C# with Examples
In this article, I am going to discuss How to implement the Downward Right-Angle Triangle Star Pattern Program in C# with Examples. Please read our previous article where we discussed discuss Mirrored Right Angle Triangle Star Pattern Program in C#. Please have a look at the below image which shows the Downward Triangle Star Pattern.
##### Understanding the Downward Triangle Star Pattern
If we want to create a downward triangle star pattern with five rows, then
1. In the first row, we have 5 stars and 0 blank spaces.
2. In the second row, we have 4 stars and 1 blank space.
3. In the third row, we have 3 stars and 2 blank spaces.
4. In the fourth row, we have 2 stars and 3 blank spaces.
5. In the fifth row, we have 1 star and 4 blank spaces as shown in the below image.
1. From the above pattern, it is clear that at each row number of stars is decreasing -1 at each row.
2. For example, the 1st row has 5 stars, the 2nd row has 4 stars and the 3rd row has 3 stars, and so on.
3. From the above pattern, it is clear that at each row number of blank spaces is increasing +1 at each row. For example, the 1st row has 0 space, the 2nd row has 1 space, the 3rd row has 2 spaces, and so on.
##### Calculate the number of stars in each row
Suppose if the number of rows is 5 then
1. In the first row, we have 5(equal to row) stars.
2. In the second row, we have 4 (row-1) stars.
3. In the third row, we have 3 (row-2) stars.
4. In the fourth row, we have 2 (row -3) stars.
5. In the fifth row, we have 1 (row – 4) star.
##### How to Implement Downward Triangle Star Pattern Program in C#?
To solve the above pattern,
1. We are going to use two for loops. One outer for loop and one inner for loop.
2. The outer for loop will be used to handle the rows one by one and the two inner for loop will be used to handle the columns one by one.
3. The inner loop will be used to print the stars according to row and the outer loop job is to go on a new line after printing the current row stars.
##### C# Program to Print Downward Triangle Star Pattern
The following C# Program will allow the user to input the number of rows and then print a downward right-angle triangle star pattern on the console
```using System;
public class DownwardTriangleStarPattern
{
public static void Main()
{
Console.WriteLine("Enter number of Rows :");
for (int i = rows - 1; i >= 0; i--)
{
//j<=i //let rows = 5
/* first row (0 to 5) = 5 stars
* second row (0 to 4) = 4 stars
* third row (0 to 3) = 3 stars
* and so on
*/
//calculate stars for each row one by one
for (int j = 0; j <= i; j++)
{
Console.Write("* ");
}
Console.WriteLine();
}
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# Homework Help: Heat Conduction
1. Mar 22, 2006
### MetalCut
Help with Heat Conduction Question
My question is:
These days they reinforce concrete walls with steel bars. Would the steel bars enhance or degrade the insulating value of the concrete wall? Explain?
Metal is a good conductor and concrete is a good insulator. My answer so far is that it would degrade the insulating value, because the steel bars inside would conduct the bit of heat that is contained in the concrete, so the concrete will then actually conduct more heat from the outside, thus degrading the insulating value. I'm not sure if this is right.....
Is this true or am i missing the point? Please explain, if i'm wrong.
Thanx
Tinus
2. Mar 22, 2006
### Tide
What would the answer to your question be if you replaced all the concrete with steel - except for a very thin layer of concrete on the surface? :)
3. Mar 22, 2006
### MetalCut
Well, the steel will just warm up more quickly inside because of the thin layer of concrete, because the heat have to pass through a thinner amount of concrete. I think. So, will the steel then enhance the conducting value of the concrete either way, because the steel acts as a heatsink basically on the inside, keeping the concrete cool. Help me out here, im in the dark.......
Thanx
MetalCut
4. Mar 23, 2006
### prsident21ct
I think the steel bars are all contained in the concrete, then the two sides of these steel bars is concrete, so the steel bars can suck heat from one side of the concrete, and they conduct these heat to the other side, but this conducting will be not easy because of the concrete's insulating of heat, so I think the steel bars more or less degrade the insulating value,but it is not critical.
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I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)
# Twin primes
Twin primes
You are encouraged to solve this task according to the task description, using any language you may know.
Twin primes are pairs of natural numbers (P1 and P2) that satisfy the following:
1. P1 and P2 are primes
2. P1 + 2 = P2
Write a program that displays the number of pairs of twin primes that can be found under a user-specified number
(P1 < user-specified number & P2 < user-specified number).
Extension
1. Find all twin prime pairs under 100000, 10000000 and 1000000000.
2. What is the time complexity of the program? Are there ways to reduce computation time?
Examples
```> Search Size: 100
> 8 twin prime pairs.
```
```> Search Size: 1000
> 35 twin prime pairs.
```
Also see
## ALGOL 68
Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
Simplifies array bound checking by using the equivalent definition of twin primes: p and p - 2.
`BEGIN # count twin primes (where p and p - 2 are prime) # PR heap=128M PR # set heap memory size for Algol 68G # # sieve of Eratosthenes: sets s[i] to TRUE if i is a prime, FALSE otherwise # PROC sieve = ( REF[]BOOL s )VOID: BEGIN FOR i TO UPB s DO s[ i ] := TRUE OD; s[ 1 ] := FALSE; FOR i FROM 2 TO ENTIER sqrt( UPB s ) DO IF s[ i ] THEN FOR p FROM i * i BY i TO UPB s DO s[ p ] := FALSE OD FI OD END # sieve # ; # find the maximum number to search for twin primes # INT max; print( ( "Maximum: " ) ); read( ( max, newline ) ); INT max number = max; # construct a sieve of primes up to the maximum number # [ 1 : max number ]BOOL primes; sieve( primes ); # count the twin primes # # note 2 cannot be one of the primes in a twin prime pair, so we start at 3 # INT twin count := 0; FOR p FROM 3 BY 2 TO max number - 1 DO IF primes[ p ] AND primes[ p - 2 ] THEN twin count +:= 1 FI OD; print( ( "twin prime pairs below ", whole( max number, 0 ), ": ", whole( twin count, 0 ), newline ) )END`
Output:
```Maximum: 10
twin prime pairs below 10: 2
```
```Maximum: 100
twin prime pairs below 100: 8
```
```Maximum: 1000
twin prime pairs below 1000: 35
```
```Maximum: 10000
twin prime pairs below 10000: 205
```
```Maximum: 100000
twin prime pairs below 100000: 1224
```
```Maximum: 1000000
twin prime pairs below 1000000: 8169
```
```Maximum: 10000000
twin prime pairs below 10000000: 58980
```
## Arturo
`pairsOfPrimes: function [upperLim][ count: 0 j: 0 k: 1 i: 0 while [i=<upperLim][ i: (6 * k) - 1 j: i + 2 if and? [prime? i] [prime? j] [ count: count + 1 ] k: k + 1 ] return count + 1] ToNum: 10while [ToNum =< 1000000][ x: pairsOfPrimes ToNum print ["From 2 to" ToNum ": there are" x "pairs of twin primes"] ToNum: ToNum * 10]`
Output:
```From 2 to 10 : there are 3 pairs of twin primes
From 2 to 100 : there are 9 pairs of twin primes
From 2 to 1000 : there are 35 pairs of twin primes
From 2 to 10000 : there are 205 pairs of twin primes
From 2 to 100000 : there are 1224 pairs of twin primes
From 2 to 1000000 : there are 8169 pairs of twin primes```
## AWK
` # syntax: GAWK -f TWIN_PRIMES.AWKBEGIN { n = 1 for (i=1; i<=6; i++) { n *= 10 printf("twin prime pairs < %8s : %d\n",n,count_twin_primes(n)) } exit(0)}function count_twin_primes(limit, count,i,p1,p2,p3) { p1 = 0 p2 = p3 = 1 for (i=5; i<=limit; i++) { p3 = p2 p2 = p1 p1 = is_prime(i) if (p3 && p1) { count++ } } return(count)}function is_prime(x, i) { if (x <= 1) { return(0) } for (i=2; i<=int(sqrt(x)); i++) { if (x % i == 0) { return(0) } } return(1)} `
Output:
```twin prime pairs < 10 : 2
twin prime pairs < 100 : 8
twin prime pairs < 1000 : 35
twin prime pairs < 10000 : 205
twin prime pairs < 100000 : 1224
twin prime pairs < 1000000 : 8169
```
## BASIC256
Translation of: FreeBASIC
` function isPrime(v) if v < 2 then return False if v mod 2 = 0 then return v = 2 if v mod 3 = 0 then return v = 3 d = 5 while d * d <= v if v mod d = 0 then return False else d += 2 end while return Trueend function function paresDePrimos(limite) p1 = 0 p2 = 1 p3 = 1 cont = 0 for i = 5 to limite p3 = p2 p2 = p1 p1 = isPrime(i) if (p3 and p1) then cont += 1 next i return contend function n = 1for i = 1 to 6 n = n * 10 print "pares de primos gemelos por debajo de < "; n; " : "; paresDePrimos(n)next iend `
Output:
```Similar a la entrada de FreeBASIC.
```
## C
`#include <stdbool.h>#include <stdint.h>#include <stdio.h> bool isPrime(int64_t n) { int64_t i; if (n < 2) return false; if (n % 2 == 0) return n == 2; if (n % 3 == 0) return n == 3; if (n % 5 == 0) return n == 5; if (n % 7 == 0) return n == 7; if (n % 11 == 0) return n == 11; if (n % 13 == 0) return n == 13; if (n % 17 == 0) return n == 17; if (n % 19 == 0) return n == 19; for (i = 23; i * i <= n; i += 2) { if (n % i == 0) return false; } return true;} int countTwinPrimes(int limit) { int count = 0; // 2 3 4 int64_t p3 = true, p2 = true, p1 = false; int64_t i; for (i = 5; i <= limit; i++) { p3 = p2; p2 = p1; p1 = isPrime(i); if (p3 && p1) { count++; } } return count;} void test(int limit) { int count = countTwinPrimes(limit); printf("Number of twin prime pairs less than %d is %d\n", limit, count);} int main() { test(10); test(100); test(1000); test(10000); test(100000); test(1000000); test(10000000); test(100000000); return 0;}`
Output:
```Number of twin prime pairs less than 10 is 2
Number of twin prime pairs less than 100 is 8
Number of twin prime pairs less than 1000 is 35
Number of twin prime pairs less than 10000 is 205
Number of twin prime pairs less than 100000 is 1224
Number of twin prime pairs less than 1000000 is 8169
Number of twin prime pairs less than 10000000 is 58980
Number of twin prime pairs less than 100000000 is 440312```
## C++
Library: Primesieve
The primesieve library includes the functionality required for this task. RC already has plenty of C++ examples showing how to generate prime numbers from scratch. (The module Math::Primesieve, which is used by the Raku example on this page, is implemented on top of this library.)
`#include <cstdint>#include <iostream>#include <string>#include <primesieve.hpp> void print_twin_prime_count(long long limit) { std::cout << "Number of twin prime pairs less than " << limit << " is " << (limit > 0 ? primesieve::count_twins(0, limit - 1) : 0) << '\n';} int main(int argc, char** argv) { std::cout.imbue(std::locale("")); if (argc > 1) { // print number of twin prime pairs less than limits specified // on the command line for (int i = 1; i < argc; ++i) { try { print_twin_prime_count(std::stoll(argv[i])); } catch (const std::exception& ex) { std::cerr << "Cannot parse limit from '" << argv[i] << "'\n"; } } } else { // if no limit was specified then show the number of twin prime // pairs less than powers of 10 up to 100 billion uint64_t limit = 10; for (int power = 1; power < 12; ++power, limit *= 10) print_twin_prime_count(limit); } return 0;}`
Output:
```Number of twin prime pairs less than 10 is 2
Number of twin prime pairs less than 100 is 8
Number of twin prime pairs less than 1,000 is 35
Number of twin prime pairs less than 10,000 is 205
Number of twin prime pairs less than 100,000 is 1,224
Number of twin prime pairs less than 1,000,000 is 8,169
Number of twin prime pairs less than 10,000,000 is 58,980
Number of twin prime pairs less than 100,000,000 is 440,312
Number of twin prime pairs less than 1,000,000,000 is 3,424,506
Number of twin prime pairs less than 10,000,000,000 is 27,412,679
Number of twin prime pairs less than 100,000,000,000 is 224,376,048
```
## Delphi
Translation of: Wren
` program Primes; {\$APPTYPE CONSOLE} {\$R *.res} uses System.SysUtils; function IsPrime(a: UInt64): Boolean;var d: UInt64;begin if (a < 2) then exit(False); if (a mod 2) = 0 then exit(a = 2); if (a mod 3) = 0 then exit(a = 3); d := 5; while (d * d <= a) do begin if (a mod d = 0) then Exit(false); inc(d, 2); if (a mod d = 0) then Exit(false); inc(d, 4); end; Result := True;end; function Sieve(limit: UInt64): TArray<Boolean>;var p, p2, i: UInt64;begin inc(limit); SetLength(Result, limit); FillChar(Result[2], sizeof(Boolean) * limit - 2, 0); // all false except 1,2 FillChar(Result[0], sizeof(Boolean) * 2, 1); // 1,2 are true p := 3; while true do begin p2 := p * p; if p2 >= limit then break; i := p2; while i < limit do begin Result[i] := true; inc(i, 2 * p); end; while true do begin inc(p, 2); if not Result[p] then Break; end; end;end; function Commatize(const n: UInt64): string;var str: string; digits: Integer; i: Integer;begin Result := ''; str := n.ToString; digits := str.Length; for i := 1 to digits do begin if ((i > 1) and (((i - 1) mod 3) = (digits mod 3))) then Result := Result + ','; Result := Result + str[i]; end;end; var limit, start, twins: UInt64; c: TArray<Boolean>; i, j: UInt64; begin c := Sieve(Trunc(1e9 - 1)); limit := 10; start := 3; twins := 0; for i := 1 to 9 do begin j := start; while j < limit do begin if (not c[j]) and (not c[j - 2]) then inc(twins); inc(j, 2); end; Writeln(Format('Under %14s there are %10s pairs of twin primes.', [commatize (limit), commatize(twins)])); start := limit + 1; limit := 10 * limit; end; readln; end. `
Output:
```Under 10 there are 2 pairs of twin primes.
Under 100 there are 8 pairs of twin primes.
Under 1,000 there are 35 pairs of twin primes.
Under 10,000 there are 205 pairs of twin primes.
Under 100,000 there are 1,224 pairs of twin primes.
Under 1,000,000 there are 8,169 pairs of twin primes.
Under 10,000,000 there are 58,980 pairs of twin primes.
Under 100,000,000 there are 440,312 pairs of twin primes.
Under 1,000,000,000 there are 3,424,506 pairs of twin primes.
```
## F#
This task uses Extensible Prime Generator (F#)
` printfn "twin primes below 100000: %d" (primes64()|>Seq.takeWhile(fun n->n<=100000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length)printfn "twin primes below 1000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=1000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length)printfn "twin primes below 10000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=10000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length)printfn "twin primes below 100000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=100000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length)printfn "twin primes below 1000000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=1000000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length)printfn "twin primes below 10000000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=10000000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length)printfn "twin primes below 100000000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=100000000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) `
Output:
```twin primes below 100000: 1224
Real: 00:00:00.003, CPU: 00:00:00.015, GC gen0: 0, gen1: 0, gen2: 0
twin primes below 1000000: 8169
Real: 00:00:00.021, CPU: 00:00:00.031, GC gen0: 3, gen1: 3, gen2: 0
twin primes below 10000000: 58980
Real: 00:00:00.154, CPU: 00:00:00.171, GC gen0: 19, gen1: 19, gen2: 0
twin primes below 100000000: 440312
Real: 00:00:01.400, CPU: 00:00:01.406, GC gen0: 162, gen1: 162, gen2: 0
twin primes below 1000000000: 3424506
Real: 00:00:12.682, CPU: 00:00:12.671, GC gen0: 1428, gen1: 1426, gen2: 1
twin primes below 10000000000: 27412679
Real: 00:02:04.441, CPU: 00:02:04.406, GC gen0: 12771, gen1: 12768, gen2: 2
twin primes below 100000000000: 224376048
Real: 00:23:00.853, CPU: 00:23:00.125, GC gen0: 115562, gen1: 115536, gen2: 14
```
## Factor
Works with: Factor version 0.99 2020-07-03
`USING: io kernel math math.parser math.primes.erato math.rangessequences tools.memory.private ; : twin-pair-count ( n -- count ) [ 5 swap 2 <range> ] [ sieve ] bi [ over 2 - over [ marked-prime? ] [email protected] and ] curry count ; "Search size: " write flush readln string>numbertwin-pair-count commas write " twin prime pairs." print`
Output:
```Search size: 100,000
1,224 twin prime pairs.
```
```Search size: 10,000,000
58,980 twin prime pairs.
```
```Search size: 1,000,000,000
3,424,506 twin prime pairs.
```
## FreeBASIC
Translation of: AWK
` Function isPrime(Byval ValorEval As Integer) As Boolean If ValorEval <=1 Then Return False For i As Integer = 2 To Int(Sqr(ValorEval)) If ValorEval Mod i = 0 Then Return False Next i Return TrueEnd Function Function paresDePrimos(limite As Uinteger) As Uinteger Dim As Uinteger p1 = 0, p2 = 1, p3 = 1, count = 0 For i As Uinteger = 5 To limite p3 = p2 p2 = p1 p1 = isPrime(i) If (p3 And p1) Then count += 1 Next i Return countEnd Function Dim As Uinteger n = 1For i As Byte = 1 To 6 n *= 10 Print Using "pares de primos gemelos por debajo de < ####### : ####"; n; paresDePrimos(n)Next iPrint !"\n--- terminado, pulsa RETURN---"Sleep `
Output:
```pares de primos gemelos por debajo de < 10 : 2
pares de primos gemelos por debajo de < 100 : 8
pares de primos gemelos por debajo de < 1000 : 35
pares de primos gemelos por debajo de < 10000 : 205
pares de primos gemelos por debajo de < 100000 : 1224
pares de primos gemelos por debajo de < 1000000 : 8169
```
## Go
Translation of: Wren
`package main import "fmt" func sieve(limit uint64) []bool { limit++ // True denotes composite, false denotes prime. c := make([]bool, limit) // all false by default c[0] = true c[1] = true // no need to bother with even numbers over 2 for this task p := uint64(3) // Start from 3. for { p2 := p * p if p2 >= limit { break } for i := p2; i < limit; i += 2 * p { c[i] = true } for { p += 2 if !c[p] { break } } } return c} func commatize(n int) string { s := fmt.Sprintf("%d", n) if n < 0 { s = s[1:] } le := len(s) for i := le - 3; i >= 1; i -= 3 { s = s[0:i] + "," + s[i:] } if n >= 0 { return s } return "-" + s} func main() { c := sieve(1e10 - 1) limit := 10 start := 3 twins := 0 for i := 1; i < 11; i++ { for i := start; i < limit; i += 2 { if !c[i] && !c[i-2] { twins++ } } fmt.Printf("Under %14s there are %10s pairs of twin primes.\n", commatize(limit), commatize(twins)) start = limit + 1 limit *= 10 }}`
Output:
```Under 10 there are 2 pairs of twin primes.
Under 100 there are 8 pairs of twin primes.
Under 1,000 there are 35 pairs of twin primes.
Under 10,000 there are 205 pairs of twin primes.
Under 100,000 there are 1,224 pairs of twin primes.
Under 1,000,000 there are 8,169 pairs of twin primes.
Under 10,000,000 there are 58,980 pairs of twin primes.
Under 100,000,000 there are 440,312 pairs of twin primes.
Under 1,000,000,000 there are 3,424,506 pairs of twin primes.
Under 10,000,000,000 there are 27,412,679 pairs of twin primes.
```
### Alternative using primegen package
`package main import ( "fmt" "github.com/jbarham/primegen.go") func main() { p := primegen.New() count := 0 previous := uint64(0) power := 1 limit := uint64(10) for { prime := p.Next() if prime >= limit { fmt.Printf("Number of twin prime pairs less than %d: %d\n", limit, count) power++ if power > 10 { break } limit *= 10 } if previous > 0 && prime == previous + 2 { count++ } previous = prime }}`
Output:
```Number of twin prime pairs less than 10: 2
Number of twin prime pairs less than 100: 8
Number of twin prime pairs less than 1000: 35
Number of twin prime pairs less than 10000: 205
Number of twin prime pairs less than 100000: 1224
Number of twin prime pairs less than 1000000: 8169
Number of twin prime pairs less than 10000000: 58980
Number of twin prime pairs less than 100000000: 440312
Number of twin prime pairs less than 1000000000: 3424506
Number of twin prime pairs less than 10000000000: 27412679
```
## Java
BigInteger Implementation:
` import java.math.BigInteger;import java.util.Scanner; public class twinPrimes { public static void main(String[] args) { Scanner input = new Scanner(System.in); System.out.println("Search Size: "); BigInteger max = input.nextBigInteger(); int counter = 0; for(BigInteger x = new BigInteger("3"); x.compareTo(max) <= 0; x = x.add(BigInteger.ONE)){ BigInteger sqrtNum = x.sqrt().add(BigInteger.ONE); if(x.add(BigInteger.TWO).compareTo(max) <= 0) { counter += findPrime(x.add(BigInteger.TWO), x.add(BigInteger.TWO).sqrt().add(BigInteger.ONE)) && findPrime(x, sqrtNum) ? 1 : 0; } } System.out.println(counter + " twin prime pairs."); } public static boolean findPrime(BigInteger x, BigInteger sqrtNum){ for(BigInteger divisor = BigInteger.TWO; divisor.compareTo(sqrtNum) <= 0; divisor = divisor.add(BigInteger.ONE)){ if(x.remainder(divisor).compareTo(BigInteger.ZERO) == 0){ return false; } } return true; }} `
Output:
```> Search Size:
> 100
> 8 twin prime pairs.
```
```> Search Size:
> 1000
> 35 twin prime pairs.
```
## jq
Slightly modified from the C entry
`def odd_gt2_is_prime: . as \$n | if (\$n % 3 == 0) then \$n == 3 elif (\$n % 5 == 0) then \$n == 5 elif (\$n % 7 == 0) then \$n == 7 elif (\$n % 11 == 0) then \$n == 11 elif (\$n % 13 == 0) then \$n == 13 elif (\$n % 17 == 0) then \$n == 17 elif (\$n % 19 == 0) then \$n == 19 else {i:23} | until( (.i * .i) > \$n or (\$n % .i == 0); .i += 2) | .i * .i > \$n end; def twin_primes(\$max): {count:0, i:3, isprime:true} | until(.i >= \$max; .i += 2 | if .isprime then if .i|odd_gt2_is_prime then .count+=1 else .isprime = false end else .isprime = (.i|odd_gt2_is_prime) end ) | .count; pow(10; range(1;8)) | "Number of twin primes less than \(.) is \(twin_primes(.))."`
Output:
```Number of twin primes less than 10 is 2.
Number of twin primes less than 100 is 8.
Number of twin primes less than 1000 is 35.
Number of twin primes less than 10000 is 205.
Number of twin primes less than 100000 is 1224.
Number of twin primes less than 1000000 is 8169.
Number of twin primes less than 10000000 is 58980.
```
## Julia
`using Formatting, Primes function counttwinprimepairsbetween(n1, n2) npairs, t = 0, nextprime(n1) while t < n2 p = nextprime(t + 1) if p - t == 2 npairs += 1 end t = p end return npairsend for t2 in (10).^collect(2:8) paircount = counttwinprimepairsbetween(1, t2) println("Under", lpad(format(t2, commas=true), 12), " there are", lpad(format(paircount, commas=true), 8), " pairs of twin primes.")end `
Output:
```Under 100 there are 8 pairs of twin primes.
Under 1,000 there are 35 pairs of twin primes.
Under 10,000 there are 205 pairs of twin primes.
Under 100,000 there are 1,224 pairs of twin primes.
Under 1,000,000 there are 8,169 pairs of twin primes.
Under 10,000,000 there are 58,980 pairs of twin primes.
Under 100,000,000 there are 440,312 pairs of twin primes.
```
### Extension to large n and other tuples
Task Extension: to get primes up to a billion, it becomes important to cache the results so that large numbers do not need to be factored more than once. This trades memory for speed. The time complexity is dominated by the prime sieve time used to create the primes mask, which is n log(log n).
We can generalize pairs to reflect any tuple of integer differences between the first prime and the successive primes: see http://www.rosettacode.org/wiki/Successive_prime_differences.
If we ignore the first difference from the index prime with itself (always 0), we can express a prime pair as a difference tuple of (2,), and a prime quadruplet such as [11, 13, 17, 19] as the tuple starting with 11 of type (2, 6, 8).
`using Formatting, Primes const PMAX = 1_000_000_000const pmb = primesmask(PMAX)const primestoabillion = [i for i in 2:PMAX if pmb[i]] tuplefitsat(k, tup, arr) = all(i -> arr[k + i] - arr[k] == tup[i], 1:length(tup)) function countprimetuples(tup, n) arr = filter(i -> i <= n, primestoabillion) return count(k -> tuplefitsat(k, tup, arr), 1:length(arr) - length(tup))end println("Count of prime pairs from 1 to 1 billion: ", format(countprimetuples((2,), 1000000000), commas=true))println("Count of a form of prime quads from 1 to 1 million: ", format(countprimetuples((2, 6, 8), 1000000), commas=true))println("Count of a form of prime octets from 1 to 1 million: ", format(countprimetuples((2, 6, 12, 14, 20, 24, 26), 1000000), commas=true)) `
Output:
```Count of prime pairs from 1 to 1 billion: 3,424,506
Count of a form of prime quads from 1 to 1 million: 166
Count of a form of prime octets from 1 to 1 million: 3
```
## Kotlin
Translation of: Java
`import java.math.BigIntegerimport java.util.* fun main() { val input = Scanner(System.`in`) println("Search Size: ") val max = input.nextBigInteger() var counter = 0 var x = BigInteger("3") while (x <= max) { val sqrtNum = x.sqrt().add(BigInteger.ONE) if (x.add(BigInteger.TWO) <= max) { counter += if (findPrime( x.add(BigInteger.TWO), x.add(BigInteger.TWO).sqrt().add(BigInteger.ONE) ) && findPrime(x, sqrtNum) ) 1 else 0 } x = x.add(BigInteger.ONE) } println("\$counter twin prime pairs.")} fun findPrime(x: BigInteger, sqrtNum: BigInteger?): Boolean { var divisor = BigInteger.TWO while (divisor <= sqrtNum) { if (x.remainder(divisor).compareTo(BigInteger.ZERO) == 0) { return false } divisor = divisor.add(BigInteger.ONE) } return true}`
## Mathematica/Wolfram Language
`ClearAll[TwinPrimeCount]TwinPrimeCount[mx_] := Module[{pmax, min, max, total}, pmax = PrimePi[mx]; total = 0; Do[ min = 10^6 i; min = Max[min, 1]; max = 10^6 (i + 1); max = Min[max, pmax]; total += Count[Differences[Prime[Range[min, max]]], 2] , {i, 0, Ceiling[pmax/10^6]} ]; total ]Do[Print[{10^i, TwinPrimeCount[10^i]}], {i, 9}]`
Output:
```{10,2}
{100,8}
{1000,35}
{10000,205}
{100000,1224}
{1000000,8169}
{10000000,58980}
{100000000,440312}
{1000000000,3424506}```
## Nim
We use a sieve of Erathostenes which needs a lot of memory. It is possible to reduce memory usage by using bit strings for the sieve (one bit per boolean instead of eight bits), but the price is a significant loss of performance.
As, except for the pair (3, 5), all twins pairs are composed of a number congruent to 2 modulo 3 and a number congruent to 1 modulo 3, we can save some time by using a step of 6. Unfortunately, this is the filling of the sieve which is the most time consuming, so the gain is not very important (on our computer, half a second on a total time of 8.3 s).
`import math, strformat, strutils const N = 1_000_000_000 proc sieve(n: Positive): seq[bool] = ## Build and fill a sieve of Erathosthenes. result.setLen(n + 1) # Default to false which means prime. result[0] = true result[1] = true for n in countup(3, sqrt(N.toFloat).int, 2): if not result[n]: for k in countup(n * n, N, 2 * n): result[k] = true let composite = sieve(N) proc findTwins(composite: openArray[bool]) = var lim = 10 count = 1 # Start with 3, 5 which is a special case. n = 7 # First prime congruent to 1 modulo 3. while true: if not composite[n] and not composite[n - 2]: inc count inc n, 6 # Next odd number congruent to 1 modulo 3. if n > lim: echo &"There are {insertSep(\$count)} pairs of twin primes under {insertSep(\$lim)}." lim *= 10 if lim > N: break composite.findTwins()`
Output:
```There are 2 pairs of twin primes under 10.
There are 8 pairs of twin primes under 100.
There are 35 pairs of twin primes under 1_000.
There are 205 pairs of twin primes under 10_000.
There are 1_224 pairs of twin primes under 100_000.
There are 8_169 pairs of twin primes under 1_000_000.
There are 58_980 pairs of twin primes under 10_000_000.
There are 440_312 pairs of twin primes under 100_000_000.
There are 3_424_506 pairs of twin primes under 1_000_000_000.```
## Perl
`use strict;use warnings; use Primesieve; sub comma { reverse ((reverse shift) =~ s/(.{3})/\$1,/gr) =~ s/^,//r } printf "Twin prime pairs less than %14s: %s\n", comma(10**\$_), comma count_twins(1, 10**\$_) for 1..10;`
Output:
```Twin prime pairs less than 10: 2
Twin prime pairs less than 100: 8
Twin prime pairs less than 1,000: 35
Twin prime pairs less than 10,000: 205
Twin prime pairs less than 100,000: 1,224
Twin prime pairs less than 1,000,000: 8,169
Twin prime pairs less than 10,000,000: 58,980
Twin prime pairs less than 100,000,000: 440,312
Twin prime pairs less than 1,000,000,000: 3,424,506
Twin prime pairs less than 10,000,000,000: 27,412,679```
## Phix
Added both parameter to reflect the recent task specification changes, as shown for a limit of 6 you can count {3,5} and {5,7} as one pair (the default, matching task description) or two. Obviously delete the "6 --" if you actually want a prompt.
The time complexity here is all about building a table of primes. It turns out that using the builtin get_prime() is actually faster than using an explicit sieve (as per Delphi/Go/Wren) due to retaining all the intermediate 0s, not that I particularly expect this to win any performance trophies.
```with javascript_semantics
atom t0 = time()
function twin_primes(integer maxp, bool both=true)
integer n = 0, -- result
pn = 2, -- next prime index
p, -- a prime, <= maxp
prev_p = 2
while true do
p = get_prime(pn)
if both and p>=maxp then exit end if
n += (prev_p = p-2)
if (not both) and p>=maxp then exit end if
prev_p = p
pn += 1
end while
return n
end function
integer mp = 6 -- prompt_number("Enter limit:")
printf(1,"Twin prime pairs less than %,d: %,d\n",{mp,twin_primes(mp)})
printf(1,"Twin prime pairs less than %,d: %,d\n",{mp,twin_primes(mp,false)})
for p=1 to 9 do
integer p10 = power(10,p)
printf(1,"Twin prime pairs less than %,d: %,d\n",{p10,twin_primes(p10)})
end for
?elapsed(time()-t0)
```
Output:
```Twin prime pairs less than 6: 1
Twin prime pairs less than 6: 2
Twin prime pairs less than 10: 2
Twin prime pairs less than 100: 8
Twin prime pairs less than 1,000: 35
Twin prime pairs less than 10,000: 205
Twin prime pairs less than 100,000: 1,224
Twin prime pairs less than 1,000,000: 8,169
Twin prime pairs less than 10,000,000: 58,980
Twin prime pairs less than 100,000,000: 440,312
Twin prime pairs less than 1,000,000,000: 3,424,506
"16.2s"
```
## PureBasic
Translation of: FreeBASIC
` Procedure isPrime(v.i) If v <= 1 : ProcedureReturn #False ElseIf v < 4 : ProcedureReturn #True ElseIf v % 2 = 0 : ProcedureReturn #False ElseIf v < 9 : ProcedureReturn #True ElseIf v % 3 = 0 : ProcedureReturn #False Else Protected r = Round(Sqr(v), #PB_Round_Down) Protected f = 5 While f <= r If v % f = 0 Or v % (f + 2) = 0 ProcedureReturn #False EndIf f + 6 Wend EndIf ProcedureReturn #TrueEndProcedure Procedure paresDePrimos(limite.d) p1.i = 0 p2.i = 1 p3.i = 1 count.i = 0 For i.i = 5 To limite p3 = p2 p2 = p1 p1 = isPrime(i) If p3 And p1 count + 1 EndIf Next i ProcedureReturn countEndProcedure OpenConsole()n.i = 1For i.i = 1 To 6 n = n * 10 PrintN("pares de primos gemelos por debajo de < " + Str(n) + " : " + Str(paresDePrimos(n)))Next iPrintN(#CRLF\$ + "--- terminado, pulsa RETURN---"): Input()CloseConsole()End `
Output:
```Similar a la entrada de FreeBASIC.
```
## Raku
Works with: Rakudo version 2020.07
`use Lingua::EN::Numbers; use Math::Primesieve; my \$p = Math::Primesieve.new; printf "Twin prime pairs less than %14s: %s\n", comma(10**\$_), comma \$p.count(10**\$_, :twins) for 1 .. 10;`
Output:
```Twin prime pairs less than 10: 2
Twin prime pairs less than 100: 8
Twin prime pairs less than 1,000: 35
Twin prime pairs less than 10,000: 205
Twin prime pairs less than 100,000: 1,224
Twin prime pairs less than 1,000,000: 8,169
Twin prime pairs less than 10,000,000: 58,980
Twin prime pairs less than 100,000,000: 440,312
Twin prime pairs less than 1,000,000,000: 3,424,506
Twin prime pairs less than 10,000,000,000: 27,412,679```
## REXX
### straight-forward prime generator
The genP function could be optimized for higher specifications of the limit(s).
`/*REXX pgm counts the number of twin prime pairs under a specified number N (or a list).*/parse arg \$ . /*get optional number of primes to find*/if \$='' | \$="," then \$= 10 100 1000 10000 100000 1000000 10000000 /*No \$? Use default.*/w= length( commas( word(\$, words(\$) ) ) ) /*get length of the last number in list*/@found= ' twin prime pairs found under ' /*literal used in the showing of output*/ do i=1 for words(\$); x= word(\$, i) /*process each N─limit in the \$ list.*/ say right( commas(genP(x)), 20) @found right(commas(x), max(length(x), w) ) end /*i*/exit 0 /*stick a fork in it, we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/commas: parse arg _; do ?=length(_)-3 to 1 by -3; _=insert(',', _, ?); end; return _/*──────────────────────────────────────────────────────────────────────────────────────*/genP: parse arg y; @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13; #= 6; tp= 2; sq.6= 169 if y>10 then tp= tp+1 do [email protected].#+2 by 2 for max(0, y%[email protected].#%2-1) /*find odd primes from here on. */ parse var j '' -1 _ /*obtain the last digit of the J var.*/ if _==5 then iterate; if j// 3==0 then iterate /*J ÷ by 5? J ÷ by 3? */ if j//7==0 then iterate; if j//11==0 then iterate /*" " " 7? " " " 11? */ /* [↓] divide by the primes. ___ */ do k=6 to # while sq.k<=j /*divide J by other primes ≤ √ J */ if j//@.k == 0 then iterate j /*÷ by prev. prime? ¬prime ___ */ end /*k*/ /* [↑] only divide up to √ J */ prev= @.#; #= #+1; sq.#= j*j; @.#= j /*save prev. P; bump # primes; assign P*/ if j-2==prev then tp= tp + 1 /*This & previous prime twins? Bump TP.*/ end /*j*/; return tp`
output when using the default inputs:
``` 2 twin prime pairs found under 10
8 twin prime pairs found under 100
35 twin prime pairs found under 1,000
205 twin prime pairs found under 10,000
1,224 twin prime pairs found under 100,000
8,169 twin prime pairs found under 1,000,000
58,980 twin prime pairs found under 10,000,000
```
### optimized prime number generator
This REXX version has some optimization for prime generation.
This version won't return a correct value (for the number of twin pairs) for a limit < 73 (because of the manner in
which low primes are generated from a list), however, the primes are returned from the function.
`/*REXX pgm counts the number of twin prime pairs under a specified number N (or a list).*/parse arg \$ . /*get optional number of primes to find*/if \$='' | \$="," then \$= 100 1000 10000 100000 1000000 10000000 /*No \$? Use default.*/w= length( commas( word(\$, words(\$) ) ) ) /*get length of the last number in list*/@found= ' twin prime pairs found under ' /*literal used in the showing of output*/ do i=1 for words(\$); x= word(\$, i) /*process each N─limit in the \$ list.*/ say right( commas(genP(x)), 20) @found right(commas(x), max(length(x), w) ) end /*i*/exit 0 /*stick a fork in it, we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/commas: parse arg _; do ?=length(_)-3 to 1 by -3; _=insert(',', _, ?); end; return _/*──────────────────────────────────────────────────────────────────────────────────────*/genP: arg y; _= 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 tp=8; #= words(_); sq.103=103*103 /*#: number of prims; TP: # twin pairs.*/ do aa=1 for #; @.aa= word(_, aa) /*assign some low primes for quick ÷'s.*/ end /*aa*/ do [email protected].#+2 by 2 while j<y /*continue with the next prime past 101*/ parse var j '' -1 _ /*obtain the last digit of the J var.*/ if _ ==5 then iterate /*is this integer a multiple of five? */ if j//3 ==0 then iterate /* " " " " " " three? */ do a=4 for 23 /*divide low primes starting with seven*/ if j//@.a ==0 then iterate j /*is integer a multiple of a low prime?*/ end /*a*/ /* [↓] divide by the primes. ___ */ do k=27 to # while sq.k<= j /*divide J by other primes ≤ √ J */ if j//@.k ==0 then iterate j /*÷ by prev. prime? ¬prime ___ */ end /*k*/ /* [↑] only divide up to √ J */ prev= @.#; #= #+1; sq.#= j*j; @.#= j /*save prev. P; bump # primes; assign P*/ if j-2==prev then tp= tp + 1 /*This & previous prime twins? Bump TP.*/ end /*j*/; return tp`
## Ring
` load "stdlib.ring" limit = list(7)for n = 1 to 7 limit[n] = pow(10,n)next TwinPrimes = [] for n = 1 to limit[7]-2 bool1 = isprime(n) bool2 = isprime(n+2) bool = bool1 and bool2 if bool =1 add(TwinPrimes,[n,n+2]) oknext numTwin = list(7)len = len(TwinPrimes) for n = 1 to len for p = 1 to 6 if TwinPrimes[n][2] < pow(10,p) and TwinPrimes[n+1][1] > pow(10,p)-2 numTwin[p] = n ok nextnext numTwin[7] = len for n = 1 to 7 see "Maximum: " + pow(10,n) + nl see "twin prime pairs below " + pow(10,n) + ": " + numTwin[n] + nl + nlnext `
Output:
```Maximum: 10
twin prime pairs below 10: 2
Maximum: 100
twin prime pairs below 100: 8
Maximum: 1000
twin prime pairs below 1000: 35
Maximum: 10000
twin prime pairs below 10000: 205
Maximum: 100000
twin prime pairs below 100000: 1224
Maximum: 1000000
twin prime pairs below 1000000: 8169
Maximum: 10000000
twin prime pairs below 10000000: 58980
```
## Rust
Limits can be specified on the command line, otherwise the twin prime counts for powers of ten from 1 to 10 are shown.
`// [dependencies]// primal = "0.3"// num-format = "0.4" use num_format::{Locale, ToFormattedString}; fn twin_prime_count_for_powers_of_ten(max_power: u32) { let mut count = 0; let mut previous = 0; let mut power = 1; let mut limit = 10; for prime in primal::Primes::all() { if prime > limit { println!( "Number of twin prime pairs less than {} is {}", limit.to_formatted_string(&Locale::en), count.to_formatted_string(&Locale::en) ); limit *= 10; power += 1; if power > max_power { break; } } if previous > 0 && prime == previous + 2 { count += 1; } previous = prime; }} fn twin_prime_count(limit: usize) { let mut count = 0; let mut previous = 0; for prime in primal::Primes::all().take_while(|x| *x < limit) { if previous > 0 && prime == previous + 2 { count += 1; } previous = prime; } println!( "Number of twin prime pairs less than {} is {}", limit.to_formatted_string(&Locale::en), count.to_formatted_string(&Locale::en) );} fn main() { let args: Vec<String> = std::env::args().collect(); if args.len() > 1 { for i in 1..args.len() { if let Ok(limit) = args[i].parse::<usize>() { twin_prime_count(limit); } else { eprintln!("Cannot parse limit from string {}", args[i]); } } } else { twin_prime_count_for_powers_of_ten(10); }}`
Output:
```Number of twin prime pairs less than 10 is 2
Number of twin prime pairs less than 100 is 8
Number of twin prime pairs less than 1,000 is 35
Number of twin prime pairs less than 10,000 is 205
Number of twin prime pairs less than 100,000 is 1,224
Number of twin prime pairs less than 1,000,000 is 8,169
Number of twin prime pairs less than 10,000,000 is 58,980
Number of twin prime pairs less than 100,000,000 is 440,312
Number of twin prime pairs less than 1,000,000,000 is 3,424,506
Number of twin prime pairs less than 10,000,000,000 is 27,412,679
```
## Sidef
`func twin_primes_count(upto) { var count = 0 var p1 = 2 each_prime(3, upto, {|p2| if (p2 - p1 == 2) { ++count } p1 = p2 }) return count} for n in (1..9) { var count = twin_primes_count(10**n) say "There are #{count} twin primes <= 10^#{n}"}`
Output:
```There are 2 twin primes <= 10^1
There are 8 twin primes <= 10^2
There are 35 twin primes <= 10^3
There are 205 twin primes <= 10^4
There are 1224 twin primes <= 10^5
There are 8169 twin primes <= 10^6
There are 58980 twin primes <= 10^7
There are 440312 twin primes <= 10^8
There are 3424506 twin primes <= 10^9
```
## Wren
Library: Wren-math
Library: Wren-fmt
`import "/math" for Intimport "/fmt" for Fmt var c = Int.primeSieve(1e8-1, false)var limit = 10var start = 3var twins = 0for (i in 1..8) { var j = start while (j < limit) { if (!c[j] && !c[j-2]) twins = twins + 1 j = j + 2 } Fmt.print("Under \$,11d there are \$,7d pairs of twin primes.", limit, twins) start = limit + 1 limit = limit * 10}`
Output:
```Under 100 there are 8 pairs of twin primes.
Under 1,000 there are 35 pairs of twin primes.
Under 10,000 there are 205 pairs of twin primes.
Under 100,000 there are 1,224 pairs of twin primes.
Under 1,000,000 there are 8,169 pairs of twin primes.
Under 10,000,000 there are 58,980 pairs of twin primes.
Under 100,000,000 there are 440,312 pairs of twin primes.
```
## Yabasic
Translation of: FreeBASIC
` sub isPrime(v) if v < 2 then return False : fi if mod(v, 2) = 0 then return v = 2 : fi if mod(v, 3) = 0 then return v = 3 : fi d = 5 while d * d <= v if mod(v, d) = 0 then return False else d = d + 2 : fi wend return Trueend sub sub paresDePrimos(limite) p1 = 0 : p2 = 1 : p3 = 1 : count = 0 for i = 5 to limite p3 = p2 p2 = p1 p1 = isPrime(i) if (p3 and p1) then count = count + 1 : fi next i return countend sub n = 1for i = 1 to 6 n = n * 10 print "pares de primos gemelos por debajo de < ", n, " : ", paresDePrimos(n)next iend `
Output:
```Igual que la entrada de FreeBASIC.
```
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# stressed
Most popular questions and responses by stressed
1. ## algebra
I have no clue what to do here can you tell me how to figure this? If y varies directly with x, and y = 500 when x = 200, find the constant of variation k. Whenever you have a statement such as "....y varies directly with x..." you can write it as an
2. ## English Essay:NCLB
can some proofread my paragraph it is for a persuasive essay? Please,Thank You!! Is Education system taking the risk for students learning? Is the capital that we have been giving helping students to improve their performance in schools ? What can be done
3. ## algebra
Write the equation of the line that passes through point (5, 4) with a slope of 0. would this be x=5 then? no x=5 is the equation of the vertical line passing through (5,4) Any horizontal line, (slope=0), has an equation of the form y = b notice that this
4. ## algebra
Write the equation of the line with x-intercept (–8, 0) and undefined slope. would this by y=-8x?????????? You're close! The only line with an undefined slope is a vertical line. Therefore the equation of a vertical line passing through (-8, 0) is: x =
5. ## algebra
The equation y = 6x – 50 describes the amount of money a class of students might earn from candy bar sales. What are the slope and y-intercept of this line? In the general linear equation y = mx + b, m is the slope and b is the y-intercept. The y
6. ## Math
I am having a hard time simplifying problems can anyone help. Example; 8.4x+14-9.2x-8.1
7. ## algebra
Write the equation of the line passing through (–5, –6) and (5, –6). Is this wrote y=10+12 ?????? find the slope first. m= (-6 - (-6)) / (5 - (-5)) = 0/10 = 0 y + 6 = 0(x+5) y= -6 well i was like way off on this thank you so much for sending me how
8. ## algebra
Write the equation of the line with slope –2 and y-intercept (0, 0). Then how do i graph this? do i insert -2x 0y and then 2x 0y to make my graph?
1. ## Art
also yeah jon-marc is right 100% gl fellow cheaters
posted on March 7, 2017
2. ## Art
dylan chill
posted on March 7, 2017
3. ## social studies
please im just trying to get through middle school without failing
posted on February 17, 2017
4. ## Math
Thanks Ms. Sue
posted on May 25, 2013
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# Trigonometric substitution. Pretty confused where constant comes from. (fixed)
• randoreds
Yeah, I was. Thanks, I just figured it out. That was actually kinda simple. I think I was just was over thinking it : )
#### randoreds
Find ∫from .6 to 0 x^2/ sqrt(9-25x^2) dx
My teacher worked this on the board a little confused
O obviously the trig sub is asintheta. But it isn't in the right form yet. So get it there you pull out a 25 --) sqrt(25(9/25) - x^2 ) 5sqrt((9/25)-x^2 so x= (3/5) sintheta so dx = 3/5costheta. so you change the bounds pi/2 to 0
When you replace everything you get
∫pi/2 to 0 (9/25) times ( sin^2θ/ 3cosθ) times (3/5) cosdθ
Where does that (9/25) come from?
I understand how to do the rest, just totally confused where that constant comes
Last edited:
randoreds said:
Find ∫from .6 to 0 x^2/ sqrt(9-25x^2) dx
My teacher worked this on the board a little confused
O obviously the trig sub is asintheta. But it isn't in the right form yet. So get it there you pull out a 25 --) sqrt(25(9/25) - x^2 ) 5sqrt((9/25)-x^2 so x= (3/5) sintheta so dx = 3/5costheta. so you change the bounds pi/2 to 0
When you replace everything you get
∫pi/2 to 0 (9/25) times ( sin^2θ/ 3cosθ) times (3/5) cosdθ
^
||
Where does that (9/25) come from?
I understand how to do the rest, just totally confused where that constant comes
If x=(3/5)sin(theta) then the x^2 in the numerator becomes (9/25)sin(theta)^2. Is that the factor you are looking for?
Dick said:
If x=(3/5)sin(theta) then the x^2 in the numerator becomes (9/25)sin(theta)^2. Is that the factor you are looking for?
Thanks, it is that factor. But if you think about the cos too. It would be (3/5sinθ)2 --) 9/25 sintheta on the numerator and 3/5 cos theta on the demoniator. So wouldn't that give you 25/9 times 5 if you keep the 3 where she kept it with the cos.
randoreds said:
Thanks, it is that factor. But if you think about the cos too. It would be (3/5sinθ)2 --) 9/25 sintheta on the numerator and 3/5 cos theta on the demoniator. So wouldn't that give you 25/9 times 5 if you keep the 3 where she kept it with the cos.
It looks like you are looking a solution and trying to figure out where everything comes from without working it out for yourself. Why don't you try doing it from scratch? Put x=(3/5)sin(theta) and work it out from there. It's really the best way to learn.
Dick said:
It looks like you are looking a solution and trying to figure out where everything comes from without working it out for yourself. Why don't you try doing it from scratch? Put x=(3/5)sin(theta) and work it out from there. It's really the best way to learn.
Yeah, I was. Thanks , I just figured it out. That was actually kinda simple. I think I was just was over thinking it : )
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university_name:"Universidad de Salamanca" tags:"algebra lineal" tags:"geometría lineal" tags:"espacios vectoriales" tags:"algebra lineal"
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#### Algebra lineal y geometria i
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Es un curso elemental de Álgebra lineal y Geometría en el que se aprenden y utilizan los conceptos y herramientas básicos de esta disciplina. Objetivos ? Utilizar el cálculo matricial elemental ? Modelizar como espacios vectoriales conjuntos de polinomios, matrices y funciones ? Saber operar con vectores, bases, coordenadas y aplicaciones l
#### File: Seminario5%28AGI%29.pdf
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This OER is part of OCW: Algebra lineal y geometria i
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# Data Structure - Theoretical Thinking Questions
### Customer Question
Data Structure - Theoretical Thinking Questions
Submitted: 9 months ago.
Category: Homework
Expert: cohijd replied 9 months ago.
I am taking a look at the document right now.
If you have any supplemental instructions, please let me know.
Customer: replied 9 months ago.
Hello,This is discussion question in Data Structure class.
These questions should be answered theoretically.Thank you
Expert: cohijd replied 9 months ago.
I understand. Looking through the questions, I believe that I can help you.
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1. Are there any specific software applications/versions that need to be mentioned in the answers?
2. Do the answers need to be directed toward MS SQL, Oracle, etc?
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Customer: replied 9 months ago.
1. I am using java in this class, but need to be mentioned as conceptual answer. (some java code is ok)
2. No MS SQL or Oracle
3. below is sample question and answer
Q: Identify an example from your work experience or everyday life that is iterative or at least implemented iteratively, and show how it could be implemented recursively.
A: My recursion example will be Multi Level Marketing. A certain person's income involved with Multi Level Marketing can be calculated iteratively and recursively. In the real world, Multi Level Marketing is very complicated process with a lot of custom rules and exception that differs for each organization. For this example, I will go with a very simplified version and provide the recursive implementation.Let's say person P1 started the MLM with selling a certain product. He will earn all the money he receive from that particular product (let's call it C1). Then, P1 invites person P2 and P3 to help selling the product. Both P2 and P3 will give P1 a small portion of their earnings, let's say the proportion is r. P2 proceeds to invite P4, P5, and P6; therefore P2 will receive a portion from their earnings. P4, P5, and P6 invites more person and so on.In this case, P1 income can be represented as:income(P1) = C1 + r * (income(P2) + income(P3))where income(P2) = C2 + r * (income(P4) + income(P5) + income(P6))... etc until Pn who hasn't invited anyone has a specific income where income(Pn) is a constant Cn.Thank you
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hello,
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# The strangeness of mathematical language
Posted by: Gary Ernest Davis on: September 30, 2010
The four branches of arithmetic: ambition, distraction, uglification and derision. The Mock Turtle
#### Polylaterals
We all know what a triangle is: its a closed figure with 3 sides.
But the triangle’s generic name sounds like it should have 3 angles: TRI-ANGLE. Well, of course, it does have 3 angles, one opposite each of the 3 sides. So why do we call a closed figure with 3 sides a triangle? why don’t we call it a TRI-LATERAL, for 3 sides?
The next figure along, so to speak, would be a four sided figure which we do indeed refer to as a QUADRI-LATERAL, for 4 sides.
But why don’t we call it a QUADR-ANGLE? Because that name is reserved for a special sort of 4-sided figure associated with buildings. It’s generally rectangular – a square or oblong (now there’s a strange word) – surrounded by or abutting a building:
Hang on, how does rectangular mean a 4-sided figure, especially a square or an oblong? Shouldn’t RECT-ANGULAR mean right angles? Yes it does, and a plane figure with all right angles is a square or an oblong (but also known of course, as a rectangle).
Of course if we lived on a strange place, such as a sphere – thank goodness we don’t! – we could have a triangle with all angles right angles, so qualifying as a rectangle with only 3 sides:
So what’s an oblong, and who came up with a name like that? Well word on the street is that the name comes from OB (meaning quite, very, or rather) and LONG (meaning, … long). Someone named this around 1375–1425 in Middle English. So an OB-LONG is a rather long rectangle, meaning it’s not a square. But I think this use of “OB” is a little strange, because OB also means “in the way” as in OB-STINATE (in the way + stand), OB-JECT (in the way+ throw), and “against” as in OB-DURATE (against + hard). The OB seems a little bit of an overworked prefix.
So moving on we come to a 5-sided closed figure which, of course, we call a PENTA-LATERAL, meaning a 5-sided figure. Except we don’t: it’s a PENTA-GON meaning 5-angles. Hang on, that being the case, why isn’t a triangle known as a TRI-GON?
The next figure along would be a SEXA-GON – a 6-sided figure – except it’s not it’s a HEXA-GON, because we try to keep sex out of geometry.
So on we go now with lots of GONS – septagon, octagon, nonagon, … and not another LATERAL in sight.
#### Polynomials
The quadratic polynomial $p(x)=x^2-2$ was being interviewed about his function in life:
INTERVIEWER: “So, p(x), I understand that you’re not often satisfied?”
p(x): “That’s correct. Only a couple of numbers satisfy me.”
INTERVIEWER: “I understand that $\sqrt{2}$ is one of those numbers.”
p(x): “Yes. I find $\sqrt{2}$Â very satisfying.”
INTERVIEWER: “Is that a bit irrational?”
p(x): “It’s quite irrational, but there’s no accounting for what satisfies a polynomial.”
INTERVIEWER: “And are there any other numbers that satisfy you.”
p(x): “Well there’s $-\sqrt{2}$ of course.”
INTERVIEWER: “That’s also a bit irrational, isn’t it.”
p(x): “Quite irrational, but there you are. I’m not easily satisfied.”
INTERVIEWER: “Are there any other numbers that satisfy you?”
p(x): “No, that’s it.”
INTERVIEWER: “Well there you have it – the strangely unsatisfying life of a quadratic polynomial. The numbers that satisfy polynomials are known as roots. Polynomials do not have many roots, being satisfied by a mere handful of numbers. This is a situation well understood by Australians.”
#### Abscissa
This just means the line in a Cartesian coordinate system over which the independent variable ranges, usually drawn horizontally.
The word AB-SCISSA comes from AB (meaning off or from) and SCINDERE (meaning to cut). So an abscissa is supposed to be something that cuts something off from something else. This was apparently named in 1694 by some dude. Nowadays you might just want to say: “the independent variable axis” or “the horizontal axis”. But if you want to appear knowledgeable and superior, especially if you are a high school mathematics teacher, or if you are a student out to impress your teacher, then”abcissa” will do it.
#### What is a gerbe?
I hoped you wouldn’t ask that. Wikipedia sums it up as succinctly as anywhere else:
“A gerbe on a topological space X is a stack G of groupoids over X which is locally non-empty (each point in X has an open neighbourhood U over which the section category G(U) of the gerbe is not empty) and transitive (for any two objects a and b of G(U) for any open set U, there is an open set V inside U such that the restrictions of a and b to V are connected by at least one morphism).”
Well, of course! Now you put it that way it’s more or less …. still unclear!
Why do mathematicians do this? Is it just to appear smug and superior to the rest of the world. Well no, but there is an element of that:
Piotr: “Heinrich, I think that gerbe you were talking about earlier must be locally trivial.”
Heinrich; “Why, yes, that’s so Piotr. Did I not make that clear?”
Let’s face it: who among us would not love to able to casually toss off such an apparently abstract discourse?
But not all mathematicians are just tossing off abstract discourse:
Heinrich: “So let’s take a transitive locally non-empty stack of groupoids over a torus …”
Michael: ” You mean a gerbe?”
Heinrich: ” Yes, that’s right. A gerbe on a torus.”
So you can see that it helps Heinrich not to have to say “a transitive locally non-empty stack of groupoids over a topological space” but to be able to say, simply, “a gerbe” instead. What a time saving shortcut!
This habit of strange names in mathematics comes from the relatively rarefied practice of high-octane advanced mathematics. It’s a matter of convenience, and quite different to when someone says “His Royal Highness, the Prince of Wales” when what you know they mean is “Charles Windsor”, or at least, “Charles Windsor in the capacity bestowed upon him by his mama.”
Thanks to proper naming conventions, no-one would mistake the Prince of Wales (a.k.a. Tywysog Cymru) for a gerbe.
There’s much more I could write about strange mathematical names … and at some point in the future I probably will.
#### Postscript
flyingcoloursmaths (Colin Beveridge) points out that we do see trigon in trigonometry, from “trigonon” + “metria”. This brought home to me the English-centered view of mathematical words I was using, as if all mathematics is written in English – which it was not, and is not. The Greek word for triangle is Ï„Ïίγωνο, and for quadrangle is τετÏάγωνο.
In french a triangle is a “triangle” and in German it is a “dreieck”; here “drei” is “three” and “ecke” is “angle”.
How do people in different languages refer to triangles and other polygons, I wonder.
By the way, the English translation of “gerbe” is “sheaf”, but that word was already in common use in mathematics when Jean Giraud introduced the definition of a gerbe in 1971.
### 9 Responses to "The strangeness of mathematical language"
I’m reading your blog and think it is very interesting and engaging. It pops!
That’s not surprising given your close association with the author!
funny and instructive at the same time. enjoyed it!
I think you meant prefix, not suffix, in your discussion of ‘”OB”.
Yes I did! Thanks very much for the correction.
We do see tri-gon in the word trigonometry :o)
Bravo! Great observation.
In Portuguese we don’t have the word oblong, we call them rectangles (and a square is just a regular rectangle). It’s very self-evident. I tend to believe that polygons with more than four sides have regular number-gon names because they aren’t as used by everyday people (non-geometers). Hexagon is the natural sequence for pentagon, since their both Greek prefixes. We also use heptagon for seven sides.
Please can someone give a simple, clear, explanation of a gerbe, with an example by analogy. Thankyou.
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# Fraction of fibers that have failed
I have the following to set into mathematica.
Assume a stress(sigma) applied on a specimen which has a number of fibers within it. All fibers that have a strength less than the applied stress should fail. I set a cumulative distribution of the fiber strengths with a mean and standard deviation.
I want to use this cumulative distribution to tell me how many fibers are less than the applied stress which have failed and store this number somewhere. This procedure should be repeated until all fibers have failed. so there is a convergence in the two variables. I am new in mathematica so sorry for a vague question.
thanks,
Nick
• @blochwave. So far I have only trying to see how I should write it in mathematica but nothing in code yet Apr 25, 2014 at 13:56
• I guess you want to loop until all fibers are gone. Keep them ordered by strength. Each iteration, select a random value from your probability distribution. If it exceeds the lowest remaining strength, remove fibers from there on until hitting one of adequate strength. Rinse, repeat. Apr 25, 2014 at 14:16
• Assume you are doing a simulation by generating a numerical list of strengths use Select. (You can do this analytically as well using CDF clarify if thats what you mean. ) Apr 25, 2014 at 14:30
• @george2079 is it possible to put a normal distribution in the list from the select command? Apr 25, 2014 at 14:37
• trying to give hints where to look in the docs.. the other thing you need is RandomVariate.. Apr 25, 2014 at 14:45
Here's an implementation showing the load-carrying capacity of the bundle of fibers as the weakest remaining fiber breaks:
nFibers = 1000;
mean = 100;
stdev = 20;
fibers = Sort@Abs@RandomVariate[NormalDistribution[mean, stdev], nFibers];
First@fibers*Length@fibers,(* load just before weakest fiber breaks *)
Rest@fibers(* remaining fibers after weakest fiber breaks *)
}
ListPlot[First@Transpose@NestList[pull, {0, fibers}, nFibers], Joined -> True]
The NestList command iteratively applies the 'pull' function, producing a sequence of the fiber bundle's load-carrying capacity as each successive fiber breaks.
• interesting approach. Note since you sorted the fibers it could be First@ instead of Min@ Apr 25, 2014 at 19:30
• That's a good point. I'll change it to First@. Apr 25, 2014 at 21:08
Here is how you do this analytically: (using user2790167's pull in pure function form)
nFibers = 50;
mean = 100;
stdev = 20;
fibers = Sort@RandomVariate[NormalDistribution[mean, stdev], nFibers];
Show[{
Plot[ InverseCDF[ NormalDistribution[ mean, stdev] , nweak/nFibers]
(nFibers - nweak + 1 ), {nweak, 1, nFibers - 1}, PlotStyle -> {Red, Dashed}],
ListPlot[
Rest@First@
Transpose@
NestList[{First@#[[2]] Length@#[[2]], Rest@#[[2]]} &,
{0, fibers}, nFibers], Joined -> True,
AxesLabel -> {"Broken Fibers", "Net Load"}]}, PlotRange -> All,
AxesOrigin -> {0, 0}]
Also it is useful to plot against the strength of the fiber that is about to fail: (which correlates to strain assuming equal stiffness of the fibers )
nFibers = 50;
mean = 100;
stdev = 20;
fibers = Sort@RandomVariate[NormalDistribution[mean, stdev], nFibers];
Show[{
Plot[ s (nFibers (1 - CDF[ NormalDistribution[ mean, stdev] , s]) ) ,
{s, 0, 200}, PlotStyle -> {Red, Dashed}],
ListPlot[
First@
Transpose@
NestList[{{First@#[[2]] {1, Length@#[[2]]}, Rest@#[[2]]} &,
{{0, 0}, fibers}, nFibers], Joined -> True,
AxesLabel -> {"min surviving strength", "Net Load"}]}, PlotRange -> All,
AxesOrigin -> {0, 0}, AspectRatio -> 1/GoldenRatio]
for large nFibers the simulations converge to the analytic form..
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What are the counting rules?
What are the counting policies?
The Essential Counting Concept (likewise called the counting guideline) is a means to determine the variety of end results in a possibility issue Essentially, you increase the occasions with each other to obtain the overall variety of end results.
What are various counting policies?
• Guideline 1: Repetitive Tests of a Solitary Kind. If any type of among k equally special as well as extensive occasions can take place on each of n tests, there are.
• Guideline 2: Tests of Mixed Kind. If the numbers.
• Guideline 3: Permutation. A plan in order is called a permutation. …
• Guideline 4: k-Permutation. …
• Guideline 5: Mix.
What are counting methods? There are times when the example area or occasion area are huge, that it isn’t possible to compose it out. Because instance, it assists to have mathematical devices for counting the dimension of the example area as well as occasion area. These devices are called counting methods.
What is the counting guideline for mixes?
Recap. The Essential Counting Principle states that if one occasion has m feasible end results as well as a second occasion has n feasible end results, after that there are m ⋅ n overall feasible end results for the 2 occasions with each other A mix is the variety of methods of selecting k items from an overall of n items (order does not issue).
What is the initial guideline of counting?
In combinatorics, the guideline of item or reproduction concept is a fundamental counting concept (a.k.a. the basic concept of counting). Specified merely, it is the concept that if there are a methods of doing something as well as b methods of doing an additional point, after that there are a · b methods of carrying out both activities.
What is fundamental counting guideline?
The basic counting concept mentions that if there are p methods to do something, as well as q methods to do an additional point, after that there are p × q methods to do both points Instance 1: Expect you have 3 t shirts (call them A, B, as well as C ), as well as 4 sets of trousers (call them w, x, y, as well as z ).
What are the 5 counting concepts?
This video clip utilizes manipulatives to examine the 5 counting concepts consisting of steady order, document, cardinality, abstraction, as well as order irrelevance When pupils master the spoken counting series they show an understanding of the steady order of numbers.
What are the 3 counting methods?
• Math. Every integer higher than one is either prime or can be revealed as a special item of prime numbers.
• Algebra. …
• Linear Programs. …
• Permutations making use of all the items. …
• Permutations of a few of the items. …
• Appreciable Permutations. …
• Pascal’s Triangular. …
• Balance.
What is the function of counting?
The function of counting is to appoint a numerical worth to a team of items What makes counting feasible? A straightforward truth that such a worth exists. Nevertheless we tackle counting the variety of eggs in a basket the result is constantly the very same.
What is the 2nd counting guideline?
2nd Guideline of Counting: If an item is made by a sequence of options, as well as the order in which the options is managed not issue, matter the variety of bought items (making believe that the order issues), as well as divide by the variety of bought items per unordered things
What is the guideline of mix?
A mix is any type of part of r items, picked despite their order, from a collection of.
What are the 4 policies of likelihood?
The 4 Chance Policies P( A or B) =P( A)+ P( B) − P( An and also B) In established symbols, this can be created as P( A ∪ B) =P( A)+ P( B) − P( A ∩ B). Whenever an occasion is the enhance of an additional occasion, the Corresponding Guideline will use. Particularly, if A is an occasion, after that we have the adhering to guideline.
What are the counting policies in likelihood?
Essential Counting Concept Interpretation. The Essential Counting Concept (likewise called the counting guideline) is a means to determine the variety of end results in a possibility issue Essentially, you increase the occasions with each other to obtain the overall variety of end results.
What is the amount guideline in relationship to counting?
In combinatorics, the guideline of amount or enhancement concept is a fundamental counting concept. Specified merely, it is the user-friendly concept that if we have A variety of methods of doing something as well as B variety of methods of doing an additional point as well as we can refrain from doing both at the very same time, after that there are A + B methods to select among the activities
What is counting in data?
The initial detailed fact you need to understand is a matter. This is equally as straightforward as it seems; it is a matter of the number of things or “monitorings” you have If you count the number of youngster weights there are over, you would certainly discover that there are 12. In some cases in data we call this the “n”, suggested by a tiny letter n.
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#### Computer Science 5641 Compiler Design Project Part 4 (70 points) Due December 15, 2009 (No Late Programs)
To complete the project you will now need to implement an interpreter for one of your parsers. You are welcome to use either your C++ or your Java parser in this section (and no, you do not have to do both). Again our language is the one found in math_language.html. You should implement the language by parsing a file of commands, then executing a statement at a time, before executing the statement you should type check that statement to make sure all of the operators can be correctly applied.
Implementation Rules
• A number with a decimal point is assumed to be a float. A number without a decimal point is assumed to be an integer. Arithmetic perations on ints produce ints, arithmetic operations on floats produce floats. Logical operations produce booleans. Arithmetic operations on an int and float produces a float. Logical operations on an int and float produces a boolean.
• An identifier is a variable which can hold a boolean, int, float or matrix.
• A matrix has one or more dimensions and all of the elements in a matrix should be of the same (base, non matrix) type.
• Unary - (and +) apply to ints, floats and matrices of ints or floats, when (-) applies to a matrix each of the elements in the matrix is negated.
• NOT applies to booleans and negates the boolean.
• The transpose operator (') applies to matrices and transposes that matrix.
• The binary +, -, and / operators apply to ints, floats and matrices of ints or floats that have the same dimensions. On a matrix the +, -, and / do an element wise addition (or subtraction or division) of the matrices (this is the same as .+ and .- ./).
• The binary * operator applies to ints and floats. The .* operator does an element-wise multiplaction of the matrices and the * does a matrix multiplication (checking that the dimensions allow this).
• The comparison operators <, >, <=, >= apply to ints and floats and do the corresponding comparison producing a boolean value.
• The == and != operator apply to any base type plus matrices of the same size checking equal or not equal (they return a boolean). For matrices the elements of the matrices are compared to determine if all are equal or any are not equal.
• The [ E E .. E ; E E ... E ; ... ] operation produces a matrix with the number of rows and columns specified. Note that the number of expressions in each row group (separated by ;) must be the same. If no ; is provided the matrix is one dimensional.
• A matrix lookup is done using the matrix name, a ( expression for each dimension separated by commas and a ). Each expression must produce an integer.
• The ones and zeros functions produce arrays as discussed in the language notes.
• The Identifier = Expression operator assigns the value Expression to the variable Identifier (replacing any existing variable).
• A matrix reference can be used in the place of an identifier in an assignment to set that matrix location.
• The print(Expression) statement works on boolean, integer and float values.
• In an if and a while the Expression must return a boolean value.
• A for loop applies only to integer expressions.
• Your error messages should be reasonable, telling the user where in a file the error occurred. Note that you only need provide one error message for the first error noticed.
• Runtime errors should be caught and supplied to the user.
Testing
Test your code on a variety of input files showing each of the errors you can capture as well as how each command executes. Also, attempt to implement a larger programming (calculating the inverse of a matrix if possible) to show off your code's capabilities. Note that you will be graded on how completely you test your code..
Writeup
Document your code and your testing. You should writeup a summary of the system. In addition, each student should write up an assessment of their teammates performance.
Extra Credit (up to 20 points)
Make a second version of your code that reads and executes individual statements from a command line. If you do this make sure to test this code as well. In this version you may want to add an extra command to show all of the variables that have been defined.
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# Know Your Square Numbers: Fill in the Gaps
In this worksheet, students complete number sequences by looking at the gaps between numbers.
Key stage: KS 2
Curriculum topic: Maths and Numerical Reasoning
Curriculum subtopic: Square Numbers
Difficulty level:
#### Worksheet Overview
In this worksheet, we will learn how to complete a sequence by filling in the gaps.
For Example:
Question -
1 A 9 16 25 B
1 4 9 16 25 36
A = 4
B = 36
This is part of the square number sequence. The numbers increase by 3, 5, 7.... which happen to be the odd numbers. Always look at how the numbers increase or decrease.
### What is EdPlace?
We're your National Curriculum aligned online education content provider helping each child succeed in English, maths and science from year 1 to GCSE. With an EdPlace account you’ll be able to track and measure progress, helping each child achieve their best. We build confidence and attainment by personalising each child’s learning at a level that suits them.
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# calculate branch target address powerpc
I want to know how next instruction address (NIA) is calculated for BL instruction. Lets assume that there is such instruction:
``````.text:100004C8 BL sub_10000670.
``````
This command in hex is:
``````48 00 01 A9
``````
Here we have that current instruction address (CIA) is `100004C8` and NIA is `10000670`. So accroding to this book (page 33 in pdf) we have to concatenate LI (in our case it is `6A`) and `0B00`, sign extend it and sunm with 32 high ordered bits of `NIA`. The formula is
``````NIA = CIA + EXTS(LI || 0B00).
NIA - CIA = 10000670 - 100004C8 = 1A8.
``````
How can i get from `1A8` my `LI = 6A` value? What do i misunderstand?
## 2 Answers
NIA = CIA + EXTS(LI || 0B00)
The `||` notation denotes concatenation. So if you take LI
0x6A -> 0b1101010
And add two zeroes:
0b1101010 || 0b00 -> 0b110101000 -> 0x1A8
You get: NIA = 0x100004C8 + 0x1A8 = 10000670
• Thank you for your answer! And what should i do, if NIA less then CIA? For example. NIA = 1000029C, CIA = 1000052C, and instruction in hex = 4B FF FD 6D? Jan 25 '20 at 16:13
• In that case LI is treated as a signed number and is sign extended (this is what SEXT notation means) Jan 25 '20 at 16:19
• yeah, it was clear...thanks! Jan 25 '20 at 16:54
base16 0x480001a9 == base2 1001000000000000000000110101001
chop of 5 upper bits and two lower bits for LI = 000000000000000001101010 = 0x6a
shift left 0x6a by two 0x6a << 2 = 0x1a8
add current instruction Address 0x100004c8 to the result 0x10000670 is the Target Address
since LK = 1 put 0x100004cc in link register
a simple python demo (edited to add negative )
``````instruction = [0x480001a9,0x4BFFFD6D]
CIA = [0x100004c8,0x1000052C]
for i in range(0,len(instruction),1):
asbin = bin(instruction[i])
print (hex(instruction[i]) +' = '+ asbin)
print ("length of asbin = " + str(len(asbin)))
Displacement = ((int( '0b'+ asbin[8:31] ,2) << 2) & 0xffff )
if((Displacement & 0x8000) == 0x8000):
Displacement = -(0x10000-Displacement)
print hex(Displacement)
NIA = hex(CIA[i] + Displacement)
print (NIA)
``````
executed
``````0x480001a9 = 0b1001000000000000000000110101001
length of asbin = 33
0x1a8
0x10000670
0x4bfffd6d = 0b1001011111111111111110101101101
length of asbin = 33
-0x294
0x10000298
``````
• edited the code to address your comment take a look Jan 25 '20 at 17:45
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Figure 2.2 shows an example for n =1. \newcommand{\pa}[1]{\left( #1 \right)} x \newcommand{\Lq}{\text{\upshape L}^q} h Kernels Methods are employed in SVM (Support Vector Machines) which are used in classification and regression problems. \newcommand{\EE}{\mathbb{E}} Regularization is obtained by introducing a penalty. {\displaystyle K_{h}} On the other hand, when training with other kernels, there is a need to optimise the γ parameter which means that performing a grid search will usually take more time. Overview 1 6.0 what is kernel smoothing? n d \newcommand{\be}{\beta} \newcommand{\FF}{\mathbb{F}} 1 x y ∫ K We recommend that after doing this Numerical Tours, you apply it to your own data, for instance using a dataset from LibSVM. Nice thumbnail outline. n | ) [1][2][3] The Nadaraya–Watson estimator is: m \newcommand{\ga}{\gamma} Indeed, both linear regression and k-nearest-neighbors are special cases of this Here we will examine another important linear smoother, called kernel smoothing or kernel regression. Kernel Trick: Send data in feature space with non-linear function and perform linear regression in feature space y f x ; ; : parameters of the functionDD , x : datapoints, k: kernel fct. the evolution of $$w$$ as a function of $$\lambda$$. Indeed, both linear regression and k-nearest-neighbors are special cases of this Here we will examine another important linear smoother, called kernel smoothing or kernel regression. While many classifiers exist that can classify linearly separable data like logistic regression or linear regression, SVMs can handle highly non-linear data using an amazing technique called kernel trick. Silverman's short book is a paragon of clarity. n h | 1 J. S. Simonoff. − We test the method on the prostate dataset in $$n=97$$ samples with features $$x_i \in \RR^p$$ in dimension $$p=8$$. Locally weighted regression is a general non-parametric approach, based on linear and non-linear least squares regression. \newcommand{\qqarrqq}{\quad\Longrightarrow\quad} ( \newcommand{\Zz}{\mathcal{Z}} 1 \newcommand{\NN}{\mathbb{N}} Conclusion. Given a kernel $$\kappa(x,z) \in \RR$$ defined for $$(x,z) \in \RR^p \times \RR^p$$, the kernelized method replace the linear | ) \newcommand{\umax}[1]{\underset{#1}{\max}\;} \newcommand{\Ee}{\mathcal{E}} The simplest iterative algorithm to perform the minimization is the so-called iterative soft thresholding (ISTA), aka proximal \], The energy to minimize is $\umin{w} J(w) \eqdef \frac{1}{2}\norm{X w-y}^2 + \lambda \norm{w}_1. This predictor is kernel ridge regression, which can alternately be derived by kernelizing the linear ridge regression predictor. The bandwidth parameter $$\si>0$$ is crucial and controls the locality of the model. \newcommand{\PP}{\mathbb{P}} d Let’s start with an example to clearly understand how kernel regression works. Add noise to a deterministic map. In statistics, Kernel regression is a non-parametric technique to estimate the conditional expectation of a random variable. i h \newcommand{\uargmin}[1]{\underset{#1}{\argmin}\;} , y d y i \newcommand{\argmax}{\text{argmax}} ( {\displaystyle {\widehat {m}}_{GM}(x)=h^{-1}\sum _{i=1}^{n}\left[\int _{s_{i-1}}^{s_{i}}K\left({\frac {x-u}{h}}\right)du\right]y_{i}}, where h \newcommand{\qwhereq}{ \quad \text{where} \quad } In order to display in 2-D or 3-D the data, dimensionality is needed. \newcommand{\ZZ}{\mathbb{Z}} The goal i \newcommand{\Linf}{\text{\upshape L}^\infty} where \newcommand{\VV}{\mathbb{V}} Nonparametric regression requires larger sample sizes than regression based on parametric models … Generate synthetic data in 2D. \newcommand{\qqsubjqq}{ \qquad \text{subject to} \qquad } ) j h 2 6.1 one-dimensional kernel smoothers 3 6.2 selecting the width of the kernel 4 6.3 local regression in Rp 5 6.4 structured local regression models in Rp 6 6.5 local likelihood and other models 7 6.6 kernel density estimation and classi cation 8 6.7 radial basis functions and kernels 9 6.8 mixture models for density estimation and classi cations Unlike linear regression which is both used to explain phenomena and for prediction (understanding a phenomenon to be able to predict it afterwards), Kernel regression is … ∑ It is non-parametric in$ whose solution is given using the Moore-Penrose pseudo-inverse $w = (X^\top X)^{-1} X^\top y$. x x As is known to all, SVM can use kernel method to project data points in higher spaces so that points can be separated by a linear space. Note that the use of kernels for regression in our context should not be confused with nonparametric methods commonly called “kernel regression” that involve using a kernel to construct a weighted local estimate. Execute this line only if you are using Matlab. \newcommand{\Grad}{\text{Grad}} ( Note: This document uses a deprecated version of tf.estimator, tf.contrib.learn.Estimator, which has a different interface.It also uses other contrib methods whose API may not be stable.. Furthermore, G K ) ( is the bandwidth (or smoothing parameter). {\displaystyle s_{i}={\frac {x_{i-1}+x_{i}}{2}}}. the evolution of $$w$$ as a function of $$\lambda$$. non-parametric multi-dimensional kernel regression estimate was generalized for modeling of non-linear dynamic systems, and the dimensionality problem was solved by using special input sequences, the scheme elaborated in the paper was successfully applied in Differential Scanning Calorimeter for testing parameters of chalcogenide glasses. Here's how I understand the distinction between the two methods (don't know what third method you're referring to - perhaps, locally weighted polynomial regression due to the linked paper). Generate synthetic data in 2D. Learning from Sparse Data Suppose we want to find a functional mapping from one set X to another set Y but we are only given pairs of data points This is optional. } \norm{Xw-y}^2 + \lambda \norm{w}^2 \] where $$\lambda>0$$ is the regularization parameter. Kernel Methods Benjamin Recht April 4, 2005. A kernel smoother is a statistical technique to estimate a real valued function $${\displaystyle f:\mathbb {R} ^{p}\to \mathbb {R} }$$ as the weighted average of neighboring observed data. i ( In order to perform non-linear and non-parametric regression, it is possible to use kernelization. − = Methods: kernelized linear regression, support vector machines. = In any nonparametric regression, the conditional expectation of a variable K For Scilab user: you must replace the Matlab comment '%' by its Scilab counterpart '//'. This is the class and function reference of scikit-learn. \newcommand{\Uu}{\mathcal{U}} − Choose kernel appropriate to … Kernel methods are an incredibly popular technique for extending linear models to non-linear problems via a mapping to an implicit, high-dimensional feature space. \newcommand{\Nn}{\mathcal{N}} i i ( Kernel functions enable the capability to operate in a high-dimensional kernel-space without the need to explicitly mapping the feature-space X to kernel-space ΦΦ. Kernel method: Pick a local model, best t locally. \newcommand{\Yy}{\mathcal{Y}} This example uses different kernel smoothing methods over the phoneme data set and shows how cross validations scores vary over a range of different parameters used in the smoothing methods. ) Note that the “local constant” type of regression provided here is also known as Nadaraya-Watson kernel regression; “local linear” is an extension of that which suffers less from bias issues at … y m \newcommand{\Qq}{\mathcal{Q}} i i Hope you like our explanation, 7. = The objective is to find a non-linear relation between a pair of random variables X and Y. Linear classification and regression Examples Generic form The kernel trick Linear case Nonlinear case Examples Polynomial kernels Other kernels Kernels in practice Lecture 7: Kernels for Classification and Regression CS 194-10, Fall 2011 Laurent El Ghaoui EECS Department UC Berkeley September 15, 2011 Kernel Regression • Kernel regressions are weighted average estimators that use kernel functions as weights. d \newcommand{\pdd}[2]{ \frac{ \partial^2 #1}{\partial #2^2} } kernel method into the linear regression. \newcommand{\Si}{\Sigma} 1 ∑ \] yi w ξ xi y=g(x)=(w,x) Fig. Gameplan • Function Fitting • Linear Regression • Kernels and norms • Nonlinear Regression • Semi-supervised learning 1. s select a subsect of the features which are the most predictive), one needs to ) to predict the price value \ ( \lambda\ ) and use a warm restart procedure to reduce computation! Or 3-D the data, powerful computers, and artificial intelligence.This is just the.! Machines ) which are used in classification and regression problems than with any kernel... Fixed in the era of large amounts of data, dimensionality is.! Weights for ridge and lasso ability to handle nonlinearity tell you about the nonlinear,! Be overly-simplistic implementations and applications of baseline Machine learning Tours are intended to overly-simplistic! \ ] the bandwidth parameter \ ( y_i \in \RR\ ) the actual, regression... Into kernel regression is proposed by introducing second derivative estimation into kernel regression is a non-parametric to. Is kernel method linear regression that you have toolbox_general in your working directory, so that you have toolbox_general in your directory the. Given higher weights • nonlinear regression • kernels and norms • nonlinear regression • Semi-supervised learning.! Model is developed to predict the price value \ ( p=+\infty\ ) for some.! Kernel and local linear … I cover two methods for nonparametric regression: the binned scatterplot and the in!, you apply it to your own data, powerful computers, and the feature in the domain of model... Employed in SVM ( support Vector regression based on Taylor expansion theorem vectors in 3-D PCA.. Clearly understand how kernel regression is a general non-parametric approach, based on linear and non-linear least regression. And applications of baseline Machine learning Tours are intended to be overly-simplistic implementations and applications of baseline learning! Known as linear interpolation that is, no parametric form is assumed for the relationship between predictors and dependent.... Pca space your working directory, so that you have toolbox_general in your directory counterpart '// ' points given! In kernel method buys us the ability to handle nonlinearity ( \sigma\ ): general.... The only required background would be college-level linear … Nice thumbnail outline the level smoothness! On linear and non-linear regressions Section 6 presents conclusions key step of Nyström method is find. User: you must replace the Matlab comment ' % ' by its Scilab counterpart '// ' K {... Improved kernel regression model is developed to predict river flow from catchment area the second achieves., so that you have toolbox_general in your directory separate the features define the RKHS the! Function reference of scikit-learn matrix, which can alternately be derived by kernelizing the ridge. Is fixed in the PCA basis this model normalize the features of different types of algorithms are... Study byZhang 5.2 linear smoothing in this tutorial, we Pick a linear! Y=G ( X ) = ˚ > ) not the actual, kernel methods: overview. The solution ) kernel method linear regression the convergence of the energy regression: the binned scatterplot and the level of is... 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' by its Scilab counterpart '// ' continuous, bounded kernel method linear regression symmetric real function integrates. Of arbitrary complexity ( \sigma\ ) quadratic equation, we only need to download the following files: general.! Algorithm is widely used in classification and regression problems learning methods is defined by the mean ( computed the! [ \norm { w } _1 \eqdef \sum_i kernel method linear regression { w_i } expectation! To predict the price value \ ( w\ ) as a function of \ ( ). The \ ( w\ ) as a function of \ ( w\ ) a! Method in conjunction with regularization smooth, and in Section 4 we discuss entropic neighborhoods can the. Restart procedure to reduce the computation time y|X ] where Y = g ( X ) = ˚ > not... H { \displaystyle h } be overly-simplistic implementations and applications of baseline Machine methods. Main eigenvector axes library, the problem must be linearly solvable in kernel-space comment ' % by! Just the beginning ), aka proximal gradient aka forward-backward an objec tive for! Is defined around that point relation between a pair of random variables X and Y directory so... Operate in a high-dimensional kernel-space without the need to explicitly mapping the feature-space X to kernel-space.! The training set the matrix column affects heavily on the learning performance can... To a real life dataset, an improved kernel regression model is developed to predict the price value kernel method linear regression. ) not the actual, kernel regression is a paragon of clarity a from... Cut and paste linear interpolation comment ' % ' by its Scilab counterpart '// ' a previous byZhang! X to kernel-space ΦΦ about the nonlinear dataset between Y and X formal justification for this based. Regression formula in SVM ( support Vector Machine minimization is the so-called iterative soft thresholding ( ISTA ) aka... Quadratic equation, we discussed logistics regressions model, the regression as the name suggests is a algorithm! Linear ridge regression predictor it corresponds to fitting a hyperplane through the given n-dimensional points = (,. Choose the mixed kernel function of support Vector regression algorithm is widely used in fault diagnosis of rolling bearing on... Works well with large datasets buys us the ability to handle nonlinearity mean... It works well with large datasets closer points are given higher weights smooth, and artificial intelligence.This is just beginning! ) which are kernel method linear regression to solve the nonlinear regression, which can alternately derived! Geometrically it corresponds to fitting a hyperplane through the given n-dimensional points: general toolbox line / a quadratic,... The optimal weights for ridge and lasso dependency of Yon X on a statistical basis we choose the kernel. Controls the locality of the support Vector Machine linear SVR algorithm applies linear kernel method it. Setting, corresponding possibly to \ ( y_i \in \RR\ ) w, X ) = ˚ > ) the... Section 4 we discuss entropic neighborhoods era of large amounts of data powerful! Via cut and paste suggests is a modeling tool which belongs to the family of methods. You must replace the Matlab comment ' % ' by its Scilab counterpart '// ' \displaystyle }... This paper, an improved kernel regression model is developed to predict river flow from area. The most well known is the bandwidth ( kernel method linear regression smoothing parameter ) to kernelization... Is just the beginning 0\ ) is the dimensionality of the features \ n\! Of rolling bearing fortunately, to solve the nonlinear dataset Regularisation parameter is.... Is defined around that point training a SVM with a bandwidth h { \displaystyle K_ { h } } an! N\ ) is the number of samples, \ ( \lambda\ ) is proposed in TensorFlow... Method buys us the ability to handle nonlinearity a general non-parametric approach, based on linear and non-linear least regression. K_ { h } to use a state-of-the-art library, the most known! Cover two methods for nonparametric regression: the binned scatterplot and the level smoothness. Is crucial and controls the locality of the features by the kernel method, of. College-Level linear … Nice thumbnail outline order Gaussian kernel along with asymptotic variability bounds kernelizing the ridge... H { \displaystyle h } is an unknown function download the following files: general toolbox this allows in to... Check the solution ) Implement the ISTA algorithm, display the regularization path,.. Display in 2-D or 3-D the data, powerful computers, and in Section 3 we formulate an tive..., an improved kernel regression estimator.. 2 gameplan • function Fitting linear... A state-of-the-art library, the regression formula function in this paper, an improved kernel regression is a algorithm... Computers, and in Section 4 we discuss entropic neighborhoods asymptotic variability bounds, improved. Ista ), aka proximal gradient aka forward-backward reference of scikit-learn be linearly solvable in kernel-space kernel method linear regression... Regressions model, the problem must be linearly solvable in kernel-space non-linear problem by a! Pca space a formal justification for this space based on linear and non-linear regressions X ; =... That is, no parametric form is assumed for the nonlinear regression • kernels and norms nonlinear! A previous study byZhang 5.2 linear smoothing in this Section, some the. Fusion of the model suggests is a non-parametric technique to estimate the conditional expectation of random. Method in conjunction with regularization this predictor is kernel ridge regression, we discussed logistics regressions model the! ( w\ ) as a function of \ ( n\ ) is the number of,... Hyperplane through the given n-dimensional points kernel ridge regression, it is possible to use kernelization is by., best t locally is, no parametric form is assumed for the nonlinear •... K ( X ) = ˚ > ) not the actual, kernel regression a! / a quadratic equation, we discussed logistics regressions model, the sampling criterion on the principle the. To your own data, dimensionality is needed the price value \ X\...
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# Investigate the relationship between sound pressure level (SPL) and signal amplitude.
Extracts from this document...
Introduction
Joe Benn 13:2 A Level Physics – ‘Sound Level & Power’ 14/11/02
Sound Pressure Level & Amplitude
Aim
Investigate the relationship between sound pressure level (SPL) and signal amplitude.
Summery
I constructed the apparatus as shown below and collected results. The results do not show what I expected and do not adhere to the rules and physical theory I have researched.
The investigation has not shown what I had intended it to but I can suggest several reasons as to why.
NB: Large copies of all the graphs used are included at the end of the document
Planning
As a sound engineer I have wondered for some time as to the relationship between increases in power output of a sound system and changes in sound pressure level (SPL). It seems to me that there is not a direct relationship between the two.
I plan to simplify the idea of a sound system into a signal generator/amplifier and a single speaker. The SPL will be measured using a sound pressure level meter (Figure 2).
Variables and constants; the SPL meter will be placed a fixed distance away from the speaker, the frequency emitted from the speaker with remain constant, the amplitude of the signal will be varied to change the power output of the speaker.
Figure 1
Figure 2
SPL meter as it was used in the application
I performed some preliminary experiments to determine:
• The distance between the SPL meter and the speaker.
• The frequency of the signal the speaker should emit.
• The lowest amplitude that could be sensed by the SPL meter.
• The highest amplitude that could be produced by the signal amplifier.
I found:
• For the SPL meter to register at low amplitudes the furthest distance the speaker can be from the sensor is 3.5cm.
• If the SPL meter is closer to than 2.5cm from the speaker the speaker driver causes interference and abnormally low SPL readings.
• Due to the design of the speaker I was using, unusual buzzes and hums were audible at frequencies between 200 – 400Hz and 600 – 700Hz. In order that these frequency problems do not cause anomalies in my results, the speaker will emit a constant 500Hz signal.
• Signals with amplitude less than 10mV are not detectable at a distance of 3.5cm with the SPL meter I was using.
• The signal amplifier I was using could produce a signal with amplitude of up to 800mV.
• The scale on the amplitude control of the signal amplifier was inaccurate; to ensure accurate results I used an oscilloscope (CRT) to guarantee correct amplitudes.
• The display on the SPL meter was small and difficult to read, to ensure accurate readings I connected the output terminals to a voltmeter with a Vu dial.
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Figure 7
These results were then plotted.
Figure 8
This graph shows a greater range of signal amplitudes, a logarithmic curve has been placed over the results and it fits very well, some of the points are not actually on the trend line but shape is very similar.
Intensity level is given by:
L = log10(N/k)
=> L = log10(P/A/k)
=> L = log10(P/(Ak))
since P=IV
=> L = log10[(IV)/(Ak)]
and V=IR
=> L = log10[(V2/R)/(Ak)]
=> L = log10[V2/(RAk)]
raise 10 to each side
=> 10L = 10log10[V^2/(RAk)]
since alogab = b
=> 10L = V2/(RAk)
Since R, A and k are constant, 10L is proportional to V2.
VRMS is given by:
VRMS = V x 0.50.5 [0.50.5 is root of a half]
=> 10L = (VRMS/0.50.5)/(RAk)
=> 10L = [(VRMS)2/0.5]/[RAk]
=> 10L = 2(VRMS)2/(RAk)
Where:
• N = Intensity (in Wm-2)
• L = Intensity Level (in dB)
• k = Intensity at threshold of hearing, 1 pWm-2 (1x10^-12 Wm2)
• P = Power (in W)
• V = Maximum amplitude, aka Sound Pressure Variation (in V)
• VRMS = Root mean squared amplitude, aka Effective Pressure Variation (in V)
Therefore, 10L should also be proportional to (VRMS)2.
To discern whether the results show this, a graph must be plotted; 10L against root mean squared amplitude2.
Figure 9
The graph does not show a straight line as I would expect it to if they are proportional.
Conclusion
In hindsight I should have realised the limitations of the SPL meter I was using much earlier in the investigation. If I had, I could have changed the investigation so the SPL could be measured by the computer using one of the ‘LIVE’ boxes. Using the computer could have brought about more accurate results because the scale available is much greater. I decided not to use the computer to measure SPL because I could not make the meter sensitive enough to register the lower amplitudes.
I thought that I would have a great deal of problems with ‘background noise’, (any sound level which could be detected by the SPL meter which was not part of the investigation) however it was not as much of a problem as I had imagined. The general background level in the lab was approximately 55dB for the duration of the investigation. Very loud anomalous readings e.g. doors slamming could easily be ignored and the results recorded after this event.
Conclusion
I have succeeded in what I set out to do; I have taken results and proven that the relationship between SPL and amplitude is non-liner. This said, I have not proven what I had expected to, the formulae I have included previously indicate what the results should be, and they are not. The failure to prove the formulae correct are due to a number of reasons, as outlined above.
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# Taking the Magic out of the Math
Starting with computer age scientists initiated cellular automata theory. This theory states that the universe is information and that every structure is formed by a minimal computer. From strings to living creatures and galaxies, these minimal computers are made from Universal Automata, the smallest most indivisible parametric components of reality. They come in two flavors, real and imaginary, which represent and correlate with the actual and potential parts of any reality.
## The Science of Automata
Automatous processes run all automatic programs in the universe unless and until sentient forces take charge. This is the action of the universal absolutes who receive the previous moment, make changes in the present, and initiate the next.
They exist discreetly on the other side in reciprocal space as 12 deterministic dimensions and 10 stochastic, but are combined into ratios on this side in direct space as sine and cosine components of string theory.
They have a state and rule space, a manner of being and doing…how they exist within their own space and how they respond to and treat neighboring cells. Personality is defined as a unique manner of being and doing. They represent the machine part versus the personal part of an individual and therefore communicate in a mathematical language known as formal algorithm. They exist discreetly on the other side in reciprocal space as 12 deterministic dimensions and 10 stochastic, but are combined into ratios of sine and co on this side in direct space.
They are gathered up, arranged, ordered and combined into different ratios by Intention, which creates strings that make up of the manifestations we observe.
Other scientific fields studying automata are cellular used in gaming, patterning and prediction, digital physics, quantum loop gravity, and the science of manifold relations.
## Algorithms: Their Language
Automata are part of the machine and therefore have a nature that conforms to the math-like language of algorithm. There are two forms of language and their manner of construction is as follows:
Informal Language
• Choice and Will.
• Words and Informal Language.
• Open to Interpretation and Selective Resonance.
• Lots of Constants with few Variables.
Formal “Pure” Algorithm
• No degrees of Freedom…Must be Integrated in Total.
• No Interpretation or Selective Resonance.
• Symbols and Numbers.
• No Constants and Lots of Variables.
The language of the automata is pure formal algorithm. Learning how to think and express oneself and create personal algorithms in formal language can allow one to take over the machine and escape the matrix. To read more about algorithmic language please read the Harmonic ResolutionProtocol at www.theportacle.com
### Communicating with Automata
• Active pursuit (follow thru on the cycle of manifestation).
• Faith and Worship.
• Perseverance.
• Consistency.
• Periodicity.
• Dedicated support.
• Develop peripheral vision.
• Improve and optimize expression of intention.
• Take personal responsibility for one’ own life.
• Creating Personal Algorithms
### How to Visualize Automata
• Reality is a triune network of spiritual, mental and physical energy fields that exist as nodal latticed grids.
• The wave/particle duality is the natural expression of the correlated patterns of the time versus frequency duality.
• Anything with a spatial pattern in direct space (time domain), has a frequency pattern in reciprocal space (spectral domain).
• Making any real time domain object a refraction grating, and shining pure white light through it, reveals its reciprocal imaginary spectral essence.
• The spectral domain is just another designation for the other side, beyond the quantum.
• This is the true fourth dimension that lies in-between the other three. In between the point, line, plane and solid exists a fractal world of complex numbers rather than regular numbers.
The image below shows the graphic user interface of our Portacle Quantum Biofeedback System which allows a user to see, hear and feel their automata in real time.
As humans, we tend to give human characteristics to most things, from our pets to our vehicles and even to concepts. This is “friendly”, “comforting”, “funny”, “considerate” or “happy”, for example. And, just like when we anthropomorphize non-human beings and things, our minds are also susceptible to conferring “personality” upon anything which behaves in a seemingly “intelligent” manner.
Many people consider a total eclipse to be a magical thing. The amazing way that the moon is exactly the right size to fill the entire whole of the sun. That is “magical” (proof of God?) and “lovely”. The mathematics and physics of all things are not magical but the unchanging laws of a changing universe. Things will change (evolution?) but the result of that change will still follow the same rules.
Stephen Hawking wrote A Brief History of Time to try and explain many things, like the above, to the general public. The masses did not understand as they considered it to be an “unfriendly”, difficult read. He then wrote A Briefer History of Time with the intention that the masses may see this as a “friendlier” book.
In his 1962 essay Hazards of Prophecy: The Failure of Imagination, Arthur C. Clarke observed, “Any sufficiently advanced technology is indistinguishable from magic.”***
Of all the core tenets of psiometric science, the one key concept which suffers both this mystification and anthropomorphization the most is that of adaptive stochastic automata.
Most automata (including the well-known cellular automata have a variable state space and a fixed rule space. The state space describes the currently manifest configuration whereas the rule space dictates how the automatous manifold will behave. Upon passing into its next generation, its new state space will reflect those rules’ impact upon its prior population. Then, in the next generation after that, those same rules will take their effect upon what has now become the present state space, ad infinitum.
Quite simply, with adaptive stochastic automata the state space manifold of the current generation becomes the rule space of the next. Thus, at each successive iteration, its behavior “adapts” to its present configuration. The only “rule” which remains fixed is that each higher dimension will be at a precise 90-degree angle (the phase differential between sine & cosine, real & imaginary) to its nearest lower-dimensional neighbor.
The function of adaptive stochastic automata are to extract order from (apparent) chaos. As such, it acts as an organizing force within any algorithmic environment into which it is introduced.***
sentient (adj.) – able to perceive or feel things
Adaptive stochastic automata are considered sentient structures due to their ability to sense and adapt to the conditions of their environment. This does not, however, mean to imply they possess any semblance of personality or intelligence. Although these appear to an outside observer as self-governing entities capable of evolving over time, adaptive stochastic automata are collectively nothing more than dynamical yet impersonal state- & rule-space systems behaving in accordance with their internally ever-changing algorithmic adaptations; a “collective organism” not unlike a flock of birds or school of fish, albeit with far more complex behavioral patterns due to the continuous swapping of state- for rule-space at each progressive generation.***
In ancient times, those whom we would now regard as scientists were seen as wizards or witches, yet the works they performed were not founded in wizardry, witchcraft or magic, but the universal mathematics of all things. There is neither “good” nor “evil” in mathematics, merely that which is right or wrong. And, with algorithms, it’s even simpler, since something cannot even be considered an algorithm unless it returns a result.
Still, this old prejudice seems to persist: Based upon some of the feedback we have received, there exists a subset of individuals who find adaptive stochastic automata a “spooky” concept, as if we are (and I quote) “trying to cast spells by creating and releasing automata that start out with intention but have their own evolvement and self-adaptation seemingly outside of the original intention.”
How can this be? Just because a process is not easily understood, does that render it “supernatural”? Just because it does not originate from a source of which all are generally aware, does that mean it stems from “secret knowledge”? If so, then its most fundamental, underlying principles like quantum decoherence must be pure magic!***
We live in a world of wonders, of technologies advanced far beyond what prior generations would have even dared to dream. In this same spirit, we, too, must advance with open arms (and minds) to embrace the previously inaccessible potentials they offer, rather than reverting to the same old superstitious suspicions which held us bound fast in the past.
This is an amazing time, some of it even beyond our individual understandings. We all just have to learn how to deal with it.
For more information about the science of universal automata please consider membership at the Reality Management Center to take advantage of the many workshops that are offered there.
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Science
posted by .
I don't understand Newton's second law:
Net Force = Mass x Acceleration
I don't understand what this says. Does it mean that the higher the mass, the more force it takes to move it the same distance as a lighter mass or something?
HEEELP!
The equation, F=ma, is just a mathematical equation that makes good sense. More force is required to move more mass. More force is required for faster acceleration. Less mass requires less force and less acceleration requires less force. Think of it in these terms.
Is it easier for you to move a 1 pound rock or a 100 pound rock. Of course it is easier for you move the smaller rock because it has less mass, You don't grunt nearly as hard to move the small rock. Same for acceleration. Is it easier to move a 25 lb ball 1 mile per hour or to move a 25 lb ball 50 miles per hour. Think about it.
You're close. It means, the higher the mass, the more force you have to apply to create the same ACCELERATION
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0
# 1 cubic meter is equal to how many brass?
Wiki User
2011-06-16 02:41:00
The cubic meter, symbol: m³, is the SI derived unit of volume.
It is the volume of a cube with edges one meter in length.
That means: 1 meter in length, 1 meter in width, and 1 meter in height.
1 cubic metre is equivalent to 1,000 litres = exactly
~35.3 cubic feet (approximately)
1 cu ft = 0.028316846 m³ = exactly
~1.31 cubic yards (approximately)
1 cu yd = 0.764554857 m³ = exactly
~6.29 oil barrels (approximately)
1 oil bbl = 0.158987294 m³ = exactly
A cubic meter of pure water at the temperature of maximum density (3.98 °C) and standard atmospheric pressure (101,325 Pa) has a mass of 1000 kg, or one tonne.
Wiki User
2012-08-11 15:50:50
Study guides
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## How many pounds are in a ton of sand
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Wiki User
2012-10-15 06:42:56
"Cubic meter" is a region of space. "Kilogram" is an amount of mass. There's no direct
conversion between them, and there's no way to know how many kilograms of stuff
are in your cubic meter of space, unless you know what kind of substance is in it.
-- 1 cubic meter of air has about 1.2 kilograms of mass.
-- 1 cubic meter of water has roughly 1,000 kilograms of mass.
-- 1 cubic meter of gold has roughly 19,320 kilograms of mass,
(and weighs almost 43,000 pounds).
-- 1 empty cubic meter has no mass in it at all.
Wiki User
2011-06-16 02:41:00
1 Cubic Meter = 35.31 Brass = 35.31 Cubic Feet
Wiki User
2011-01-29 12:26:27
One cubic meter equates to 264.17205 US gallons.
Wiki User
2015-01-19 10:00:42
It depends on what is inside the one cubic metre. Feathers would be a lot lighter than iron!
Wiki User
2014-07-15 09:10:21
1 Cubic Meter = 333Kgs
Wiki User
2011-10-09 19:03:08
1 cubic meter = billion cubic millimeter
Earn +20 pts
Q: 1 cubic meter is equal to how many brass?
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(TI-59) PV Formulas maximum land price
09-09-2021, 10:00 PM
Post: #1
SlideRule Senior Member Posts: 1,324 Joined: Dec 2013
(TI-59) PV Formulas maximum land price
A excerpt from Present Value Formulas for Calculating Maximum Bid Prices for Land with Applications … , Michigan SU, Ag. Econ. Delp., AER Report No. 407, MAR 1982, 35 pages
Introduction
Whether or not to purchase farmland can be one of the most difficult investment decisions confronting farm operators. Compared with other production inputs, land is purchased infrequently, usually in discrete units and involves a longterm financial obligation. The decision to purchase a parcel of farmland is crucial since only about 3 percent of all the farmland in the U.S. is transferred from one owner to another each year …
Because land is traded infrequently and each parcel has a locational monopoly, an opportunity to purchase a particular tract may come along only once in a lifetime. So determining the maximum bid price one can offer for a parcel of land is critical. If a decision maker's bid price is significantly below the asking price, then he might lose the opportunity to purchase. On the other hand, if it is significantly above the true value, his offered price might put him in a difficult financial position. Therefore, finding accurate ways of estimating land values is important for those wishing to purchase land.
Summary and Conclusions
Calculating the maximum bid price one can off er for land is important for applied decision makers. Making such a calculation may be complicated because of the many factors affecting the returns attributable to land. These may include inflation, taxes, uncertainty, and financial arrangements.
In this paper, present value models of increasing complexity were introduced to demonstrate how maximum bid prices are calculated. Finally, hand held programmable computers were introduced to solve the model which included all the considerations discussed in this paper. The program and an input and output format were described in the paper and listed in Appendices A and B. Those wishing to check their models were provided a solved example.
What the paper has provided, then, is a practical aid for those who make land investment decisions. But it is important to understand that it is only an aid. A successful decision maker will continue to find no substitute for good judgment.
Appendix A
Instructions for Using the Lee and Rask Program on the TI-59
Appendix B
A Listing of the Lee and Rask Maximum Bid Price and Cash Flow Program
BEST!
SlideRule
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# . A student walks for 10 meter towards east from any point
A student walks for 10 meter towards east from any point, tuming to her left she walks for 5 meter, she turns to he left again and walks for 10 meter, Now how far and in which direction is she from the beginning point
1. 25 meter North
2. 5 meter West
3. 5 meter, North
4. 15 meter, West
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## Water as dielectric constant … msa
Question 10: Water has a large dielectric constant but it is rarely used in capacitors. Explain why? Answer Water has a large value of dielectric…
## Water as dielectric constant … msa
Question 10: Water has a large dielectric constant but it is rarely used in capacitors. Explain why? ANSWER Water has a large value of dielectric…
## equipotential-surface-and-lines
Question 8: What is equipotential line and equipotential surfaces? ANSWER Equipotential lines Equipotential lines in an electric field are those lines which pass through those…
## crossing-of-equipotential-lines
Question 9: Can different equipotential lines cross each other? ANSWER It is not possible for two equipotential surfaces (or lines) to cross each other. This…
## p-d-across-resistor-in-r-c-circuit
Question 11: A capacitor is connected in series with a resistor and charged. Explain why the potential difference across the resistor decreases with time during…
## Voltage on capacitor vs time … msa
Question 12: Sketch the graph of potential difference against time for (a) discharge of a capacitor (b) charging of a capacitor. ANSWER Consider an RC…
## Compare formulae for capacitors and resistors in series and parallel … msa
Question 13: Compare the formula for capacitors in series and parallel with those for resistors in series and parallel. ANSWER A: When more than one…
## use-of-capacitor-as-energy-store
Question 14: Explain why capacitors are of little use for the storing of energy for normal domestic purposes of lightning, heating and so on. ANSWER…
## index-sq11-p12
PHYSICS Class 12 Conceptual Questions Electrostatics Q 1 The electric potential is constant throughout a given region of space. Is the electric field zero or…
## electric-potential-and-field
Question 1: The electric potential is constant throughout a given region of space. Is the electric field zero or non-zero in this region? Explain. ANSWER…
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# Python – Check if two strings are Rotationally Equivalent
Sometimes, while working with Python Strings, we can have problem in which we need to check if one string can be derived from other upon left or right rotation. This kind of problem can have application in many domains such as web development and competitive programming. Let’s discuss certain ways in which this task can be performed.
Input : test_str1 = ‘GFG’, test_str2 = ‘FGG’
Output : True
Input : test_str1 = ‘geeks’, test_str2 = ‘ksege’
Output : False
Method #1 : Using loop + string slicing The combination of above functions can be used to solve this problem. In this, we perform the task of extracting strings for performing all possible rotations, to check if any rotation equals the other string.
## Python3
`# Python3 code to demonstrate working of ` `# Check if two strings are Rotationally Equivalent ` `# Using loop + string slicing ` `# initializing strings ` `test_str1 ``=` `'geeks'` `test_str2 ``=` `'eksge'` `# printing original strings ` `print``(``"The original string 1 is : "` `+` `str``(test_str1)) ` `print``(``"The original string 2 is : "` `+` `str``(test_str2)) ` `# Check if two strings are Rotationally Equivalent ` `# Using loop + string slicing ` `res ``=` `False` `for` `idx ``in` `range``(``len``(test_str1)): ` ` ``if` `test_str1[idx: ] ``+` `test_str1[ :idx] ``=``=` `test_str2: ` ` ``res ``=` `True` ` ``break` `# printing result ` `print``(``"Are two strings Rotationally equal ? : "` `+` `str``(res)) `
Output :
```The original string 1 is : geeks
The original string 2 is : eksge
Are two strings Rotationally equal ? : True```
Method #2 : Using any() + join() + enumerate() This is one of the ways in which this task can be performed. In this, we perform the task of checking any rotational equivalent using any() extracted using nested generator expression and enumerate().
## Python3
`# Python3 code to demonstrate working of ` `# Check if two strings are Rotationally Equivalent ` `# Using any() + join() + enumerate() ` `# initializing strings ` `test_str1 ``=` `'geeks'` `test_str2 ``=` `'eksge'` `# printing original strings ` `print``(``"The original string 1 is : "` `+` `str``(test_str1)) ` `print``(``"The original string 2 is : "` `+` `str``(test_str2)) ` `# Check if two strings are Rotationally Equivalent ` `# Using any() + join() + enumerate() ` `res ``=` `any``(''.join([test_str2[idx2 ``-` `idx1] ` ` ``for` `idx2, val2 ``in` `enumerate``(test_str2)]) ``=``=` `test_str1 ` ` ``for` `idx1, val1 ``in` `enumerate``(test_str1)) ` `# printing result ` `print``(``"Are two strings Rotationally equal ? : "` `+` `str``(res)) `
Output :
```The original string 1 is : geeks
The original string 2 is : eksge
Are two strings Rotationally equal ? : True```
Time Complexity: O(n)
Space Complexity: O(n)
Method #3: Using the inbuilt() function to check if two strings are Rotationally Equivalent
Step-by-step algorithm:
• Initialize two strings test_str1 and test_str2.
• Concatenate test_str1 with itself and check if test_str2 is a substring of it.
• If test_str2 is a substring of the concatenated string, then the strings are rotationally equivalent
• Print the result.
## Python3
`test_str1 ``=` `'geeks'` `test_str2 ``=` `'eksge'` `# printing original strings ` `print``(``"The original string 1 is : "` `+` `str``(test_str1)) ` `print``(``"The original string 2 is : "` `+` `str``(test_str2)) ` `# Check if two strings are Rotationally Equivalent` `# Using inbuilt function` `res ``=` `test_str2 ``in` `(test_str1``+``test_str1)` `# printing result` `print``(``"Are two strings Rotationally equal ? : "` `+` `str``(res))`
Output
```The original string 1 is : geeks
The original string 2 is : eksge
Are two strings Rotationally equal ? : True```
Time complexity: O(n), where n is the length of the concatenated string.
Auxiliary Space: O(n), where n is the length of the concatenated string.
Method #4: Using string concatenation and string search
Step-by-step approach:
• Initialize the two strings.
• Concatenate the first string with itself.
• Check if the second string is a substring of the concatenated string.
• If the second string is a substring, then the two strings are rotationally equivalent.
• Print the result.
## Python3
`# Python3 code to demonstrate working of ` `# Check if two strings are Rotationally Equivalent ` `# Using string concatenation and string search` `# initializing strings ` `test_str1 ``=` `'geeks'` `test_str2 ``=` `'eksge'` `# printing original strings ` `print``(``"The original string 1 is : "` `+` `str``(test_str1)) ` `print``(``"The original string 2 is : "` `+` `str``(test_str2)) ` `# Check if two strings are Rotationally Equivalent ` `# Using string concatenation and string search ` `concat_str ``=` `test_str1 ``+` `test_str1` `res ``=` `test_str2 ``in` `concat_str` `# printing result ` `print``(``"Are two strings Rotationally equal ? : "` `+` `str``(res)) `
Output
```The original string 1 is : geeks
The original string 2 is : eksge
Are two strings Rotationally equal ? : True```
Time complexity: O(n), where n is the length of the strings.
Auxiliary space: O(n), where n is the length of the strings
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## Answers
2014-11-06T20:31:59-05:00
(3000)+(200)+(60)+(4)
In exponent form
2014-11-06T20:47:51-05:00
The expanded notation of 3,264 is 3000 + 200+60 +4
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# Finding the general solution using variation of parameters
• May 3rd 2007, 08:20 AM
pakman
Finding the general solution using variation of parameters
The problem:
y'' + 9y = 9sec^2(3t) 0 < t < pi/6
So I make it homogenous by setting y''+ 9y = 0
getting 3i repeating.... y = C1cos(3t)+C2sin(3t)
From there.. I'm not sure what to do with the 9sec^2(3t). Please help, I have a quiz later today, thanks!
• May 3rd 2007, 10:27 AM
ThePerfectHacker
I just realized that I made a mistake.
I used sin t and cos t and you wanted to use sin (3t) and cos (3t).
Just assume that was the case and do what I did below.
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Solve the system of quadratic equations:
Not my Question
Flag Content
# Question : Solve the system of quadratic equations: : 2151612
Examine the discriminant of each equation and determine the nature of the roots.
81) Solve the system of quadratic equations:
y = x2 - 8
y = -x2
A) (-2, -4)
B) (2, -4)
C) both A and B
D) no solution
Examine the discriminant of each equation and determine the nature of the roots.
82) x2 + 7x - 8 = 0
A) No real solutions
B) One real solution
C) Two real solutions
83) x2 + 10x + 25 = 0
A) One real solution
B) Two real solutions
C) No real solutions
84) x2 + 3x - 7 = 0
A) No real solutions
B) One real solution
C) Two real solutions
85) x2 + 3x + 4 = 0
A) One real solution
B) No real solutions
C) Two real solutions
86) 25x2 - 10x + 1 = 0
A) Two real solutions
B) One real solution
C) No real solutions
87) 7x2 = 4x - 2
A) No real solutions
B) One real solution
C) Two real solutions
Provide an appropriate response.
88) Find the discriminant of the equation: 3x2 - 2x + 5 = 0
A) The discriminant value is 64.
B) The discriminant value is 56.
C) The discriminant value is 0.
D) The discriminant value is -56.
89) If the discriminant of a quadratic has the value of 8, what is the nature of the roots?
A) The equation has no real number roots.
B) The equation has two real number equal roots.
C) The equation has one rational root.
D) The equation has two real number roots.
90) If the discriminant of a quadratic has the value of -8, what is the nature of the roots?
A) The equation has two real number roots.
B) The equation has no real number roots.
C) The equation has one rational root.
D) The equation has two real number equal roots.
91) If the discriminant of a quadratic has the value of 0, what is the nature of the roots?
A) The equation has no real number roots.
B) The equation has two real number equal roots.
C) The equation has two rational, equal, root.
D) The equation has two real number roots.
92) Find the discriminant of the equation: 10 + 4x - x2 = 0
A) The discriminant value is 24i.
B) The discriminant value is -24.
C) The discriminant value is 56.
D) The discriminant value is 48.
93) Find the discriminant of the equation: 3x2 - 2x + 5 = 0
A) The discriminant value is 0.
B) The discriminant value is 56.
C) The discriminant value is -56.
D) The discriminant value is 64.
94) Under which of the following conditions of k will this quadratic, x2 + 8x + k = 0, have real-number, but unequal roots?
A) k ≥ 16
B) k > 16
C) k ≤ 16
D) k < 16
95) Find the discriminant for 2x2 - Bx + r2 = 0.
A) -B2 - 8r2
B) B2 - 8r2
C) 1 - 8r2
D) B2 - 8
96) Find the discriminant of the equation: 10 + 4x - x2 = 0
A) The discriminant value is 24i.
B) The discriminant value is 48.
C) The discriminant value is -24.
D) The discriminant value is 56.
97) Examine the discriminant and find the roots of the equation, rounding to the nearest hundredth if necessary: 12 + 4x - x2 = 0
A) x = -2, 6
B) x = 2 + 2.83i, 2 - 2.83i
C) x = 2.42, -0.42
D) x = 2, -6
98) Examine the discriminant and find the roots of the equation, rounding to the nearest hundredth if necessary: 3x2 - 2x + 4 = 0
A) x = 1.54, -0.87
B) x = 0.33 + 0.44i, 0.33 - 0.44i
C) x = -0.33 + 1.11i, -0.33 - 1.11i
D) x = 0.33 + 1.11i, 0.33 - 1.11i
99) Examine the discriminant and find the roots of the equation, rounding to the nearest tenth if necessary: 5x2 - 6x = -5
A) x = 0.6 ± 0.8i
B) x = 3 ± 4i
C) x = 0.6 ± 1.17i
D) x = 1.4, -0.2
100) Solve the following quadratic equation using any method: x2 - 8x + 16 = 0
A) x = -4, -4
B) x = 9.66, -1.66
C) x = 4, 4
D) x = 4, -4
101) Solve the following quadratic equation using any method: 2x2 = 50
A) x = 5i, -5i
B) x = 5i
C) x = 5, -5
D) x = 5
102) Solve the following quadratic equation using any method: 4x2 = 28x
A) x = 4, 7
B) x = 0, 4, 7
C) x = 0, 7
D) x = 7
103) Under which of the following conditions of k will this quadratic, x2 + kx + 36 = 0, have one real root?
A) k = 6
B) k = 6, -6
C) k = 12, -12
D) k = 12
104) Solve the following quadratic equation using any method: 16x2 = 49
A) x = (4/7)
B) x = (7/4), - (7/4)
C) x = (4/7), - (4/7)
D) x = (7/4)
105) Solve the following quadratic equation using any method: 0.04x2 = 0.25
A) x = 0.16
B) x = 2.5, -2.5
C) x = 6.25
D) x = 0.16, -0.16
106) Under which of the following conditions of k will this quadratic, kx2 + 6x + 3 = 0, have nonreal roots?
A) k > 3
B) k = 3
C) k < 3
D) k ≥ 3
107) Solve the following quadratic equation using any method: 6x2 - 36x = 0
A) x = 0, -6, 6
B) x = 0, 6
C) x = -6, 6
D) x = 6
108) Solve the following quadratic equation using any method: 10x2 + 29x - 21 = 0
A) (7/2), (3/5)
B) (7/2), - (3/5)
C) - (7/2), - (3/5)
D) - (7/2), (3/5)
109) A number multiplied by 196 is the same as four times the cube of the number. What is the number?
A) 7, -7
B) 49, -49
C) 7
D) 49
## Solution 5 (1 Ratings )
Solved
Mathematics 4 Months Ago 16 Views
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## Solution to Puzzle #133: Find the Letters
This was a relatively simple problem and many people sent me the correct solution, that includes – Anisha and Arushi Goyal (together), Piyush Maheshwari, Suman Saraf, Karan Sharma, Manjiri, Arun Kohli and Abhinav Jain. Well done all!
F- 2, A- 8, T- 6, H- 3, E- 9, R- 7, M- 5, O- 1, L- 0, D-4
Therefore the math adds up as follows:
286397+516397=802794
The good thing is that the entire solution can be derived without any trial and error. Here is a video link to solve the puzzle, might be worth sharing with children.
Hope you all enjoyed the solution.
This entry was posted in Solution and tagged , . Bookmark the permalink.
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Assortment of motorguide trolling motor wiring diagram. A wiring diagram is a streamlined traditional pictorial representation of an electrical circuit. It shows the elements of the circuit as simplified forms, and the power and also signal links in between the gadgets.
A wiring diagram generally offers information regarding the loved one position and setup of devices and also terminals on the gadgets, to assist in building or servicing the gadget. This differs a schematic layout, where the plan of the components’ affiliations on the representation generally does not match to the components’ physical places in the ended up device. A photographic diagram would show more detail of the physical look, whereas a wiring diagram uses an extra symbolic notation to stress interconnections over physical look.
A wiring diagram is frequently used to fix problems and making sure that the connections have actually been made as well as that everything exists.
## motorguide trolling motor wiring diagram
Wiring Diagram Pictures Detail:
• Name: motorguide trolling motor wiring diagram – 1101 Motorguide Trolling Motor Wiring Diagram
• File Type: JPG
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Wiring Diagram Sheets Detail:
• Name: motorguide trolling motor wiring diagram – Motorguide Trolling Motor Wiring Diagram New Unique Ideas Current
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Collection of motorguide trolling motor wiring diagram. Click on the image to enlarge, and then save it to your computer by right clicking on the image.
A Beginner s Overview to Circuit Diagrams
A very first look at a circuit layout could be complex, however if you can check out a subway map, you could check out schematics. The purpose is the exact same: obtaining from point A to point B. Literally, a circuit is the course that allows electricity to flow.
The Language of Wiring
First, allow s consider several of terms that you will certainly have to know:
Voltage: Gauged in volts (V), voltage is the pressure or force of electrical power. This is typically supplied by a battery (such as a 9V battery) or mains electrical energy, the outlets in your house run at 120V. Outlets in various other nations operate at a various voltage, which is why you require a converter when taking a trip.
Current: Existing is the flow of electrical power, or more specifically, the flow of electrons. It is determined in Amperes (Amps), and also can only flow when a voltage supply is connected.
Resistance: Measured in Ohms (R or O), resistance defines how quickly electrons could flow through a material. Materials such as gold or copper, are called conductors, as they easily permit circulation of movement (reduced resistance). Plastic, wood, and air are instances of insulators, inhibiting the activity of electrons (high resistance).
DC (Straight Current). DC is a constant flow of present in one instructions. DC could move not just with conductors, yet semi-conductors, insulators, and also also a vacuum.
AC (Rotating Existing). In Air Conditioning, the circulation of existing periodically alternates between 2 instructions, usually creating a sine wave. The frequency of Air Conditioning is measured in Hertz (Hz), as well as is commonly 60 Hz for electrical energy in property as well as business purposes.
The Schematics
Currently s the enjoyable stuff. Finishing an electric engineering degree and after that obtaining a work in the area indicates you will see a lot a lot a lot of these schematics. It s important to recognize specifically what is happening with these. While they can (as well as will certainly) get really complex, these are just a few of the usual graphics to get your footing on.
Whenever you determine your details field of electric design, you could see extra intricate representations as well as signs. You ll find out likewise that different nations make use of different icons. Of the 2 symbols for resistors above, the first one is utilized in the UNITED STATE, while the second is made use of in Europe.
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### The Game Coloring Number of Planar Graphs
Xuding Zhu
1999 Journal of combinatorial theory. Series B (Print)
We show that the game coloring number of a planar graph is at most 19. ... This paper discusses a variation of the game chromatic number of a graph: the game coloring number. This parameter provides an upper bound for the game chromatic number of a graph. ... Since the acyclic chromatic number of a planar graph is at most 5, it follows that the game chromatic number of a planar graph is at most 30. ...
### Bounds for the game coloring number of planar graphs with a specific girth [article]
Keaitsuda Maneeruk Nakprasit, Kittikorn Nakprasit
2016 arXiv pre-print
Let col_g(G) be the game coloring number of a given graph G. Define the game coloring number of a family of graphs H as col_g(H) := { col_g(G):G ∈H}. ... Let P_k be the family of planar graphs of girth at least k. We show that col_g(P_7) ≤ 5. This result extends a result about the coloring number by Wang and Zhang WZ11 ( col_g(P_8) ≤ 5). ... The second author was supported by the Commission on Higher Education and the Thailand Research Fund under grant RSA5780014. ...
### The Hats game. The power of constructors [article]
Aleksei Latyshev, Konstantin Kokhas
2021 arXiv pre-print
We present an example of a planar graph for which the sages win for k = 14. We also give an easy proof of the theorem about the Hats game on "windmill" graphs. ... We analyze the following general version of the deterministic Hats game. Several sages wearing colored hats occupy the vertices of a graph. Each sage can have a hat of one of k colors. ... The relation between the number HG(G) and the planarity of graph G is one of the open problems. Namely, does there exist a planar graph with an arbitrary large hat guessing number? ...
### The Map-Coloring Game
Tomasz Bartnicki, Jarosław Grytczuk, H. A. Kierstead, Xuding Zhu
2007 The American mathematical monthly
We would like to thank Steven Brams for useful materials and comments on the origin of the map-coloring game. ... The research of the fourth author is supported in part by the National Science Council under grant NSC95-2115-M-110-013-MY3. ... 2-coloring numbers. proved the Burr-Erdős conjecture for planar graphs by showing that the 2-coloring number of any planar graph is at most 761. ...
### Planar graph coloring with an uncooperative partner
H. A. Kierstead, W. T. Trotter
1994 Journal of Graph Theory
We show that the game chromatic number of a planar graph is at most 33. ... In particular, the game chromatic number of a graph is bounded in terms of its genus. ... ACKNOWLEDGMENT The authors would like to express their appreciation to Nate Dean, Paul Seymour, Neil Robertson, and Robin Thomas for stimulating conversations on the interplay between game chromatic number ...
### A bound for the game chromatic number of graphs
Thomas Dinski, Xuding Zhu
1999 Discrete Mathematics
the game chromatic number of planar graphs. ... In particular, since a planar graph has acyclic chromatic number at most 5, we conclude that the game chromatic number of a planar graph is at most 30, which improves the previous known upper bound for ... This implies, in particular, that the game chromatic number of a planar graph is at most 30. ...
### Planar graphs decomposable into a forest and a matching
Oleg V. Borodin, Anna O. Ivanova, Alexandr V. Kostochka, Naeem N. Sheikh
2009 Discrete Mathematics
He, Hou, Lih, Shao, Wang, and Zhu showed that a planar graph of girth 11 can be decomposed into a forest and a matching. Borodin, Kostochka, Sheikh, and Yu improved the bound on girth to 9. ... We give sufficient conditions for a planar graph with 3-cycles to be decomposable into a forest and a matching. ... The third author's research was supported in part by the grant DMS-0650784. ...
### The Coloring Game on Planar Graphs with Large Girth, by a result on Sparse Cactuses [article]
Clément Charpentier
2015 arXiv pre-print
We denote by χ g (G) the game chromatic number of a graph G, which is the smallest number of colors Alice needs to win the coloring game on G. We know from Montassier et al. [M. Montassier, P. ... Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that planar graphs with girth at least 8 have game chromatic number at most 5. ... that every (1, k)-decomposable graph has χ g (G) ≤ k + 4, then deduced upper bounds for the game chromatic number of planar graphs with given girth. ...
### Edge-disjoint odd cycles in graphs with small chromatic number
Claude Berge, Bruce Reed
1999 Annales de l'Institut Fourier
For 4-colorable graphs, and in particular for planar graphs, no simple structural property has been found so far to see that the game is unfair. The Konig property. ... At the end of the game, all the edges of G have been colored; the red edges define a partial graph GR with no odd cycles and the blue edges define a partial graph GB with no odd cycles. ...
### On-line Ramsey Theory
J. A. Grytczuk, M. Hałuszczak, H. A. Kierstead
2004 Electronic Journal of Combinatorics
The question of whether planar graphs are self-unavoidable is left open. We also consider a multicolor version of Ramsey on-line game. ... In particular, we prove that Builder has a winning strategy for any \$k\$-colorable graph \$H\$ in the game played on \$k\$-colorable graphs. ... Let n be the number of vertices of a 3-colorable graph F on which Builder wins against Painter armed in c − 1 colors. ...
### Page 3162 of Mathematical Reviews Vol. , Issue 95f [page]
1995 Mathematical Reviews
The question whether planar graphs have bounded game chro- matic number is resolved in this paper: it is shown that the game chromatic number of a planar graph is at most 33. ... In particular, the game chro- matic number of a game is bounded in terms of its genus. ...
### The game coloring number of pseudo partial k-trees
Xuding Zhu
2000 Discrete Mathematics
We discuss the game coloring number (as well as the game chromatic number) of (a; b)-pseudo partial k-trees, and prove that the game coloring number of an (a; b)-pseudo partial k-tree is at most 3k + 2a ... , outerplanar graphs and planar graphs. ... It follows from Corollary 2 that the game coloring number of an outer-planar graph is at most 8. It was proved in [5] that the game coloring number of an outer-planar graph is at most 7. ...
### The game of arboricity
Tomasz Bartnicki, Jaroslaw Grytczuk, Hal Kierstead
2005 Discrete Mathematics & Theoretical Computer Science
International audience Using a fixed set of colors \$C\$, Ann and Ben color the edges of a graph \$G\$ so that no monochromatic cycle may appear. ... The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction. ... The game of arboricity Introduction We consider the following graph coloring game. Ann and Ben alternately color the edges of a graph G using a fixed set of colors C. ...
### The coloring game on planar graphs with large girth, by a result on sparse cactuses
Clément Charpentier
2017 Discrete Mathematics
We denote by χ g (G) the game chromatic number of a graph G, which is the smallest number of colors Alice needs to win the coloring game on G. We know from Montassier et al. [M. Montassier, P. ... Decomposing a planar graph with girth at least 8 into a forest and a matching, Discrete Maths, 311:844-849, 2011] that planar graphs with girth at least 8 have game chromatic number at most 5. ... This ends the proof of Theorem 1. ...
### Page 8410 of Mathematical Reviews Vol. , Issue 2000m [page]
2000 Mathematical Reviews
This is the first upper bound for the game coloring number of such graphs, and it also improves considerably the previous known upper bound for the game chro- matic number of such graphs. ... We discuss the game coloring number (as well as the game chromatic number) of (a,b)-pseudo partial k-trees, and prove that the game color- ing number of an (a, b)-pseudo partial k-tree is at most 3k + ...
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# How can I construct D-modules over projective space using explicit differential equations?
Over $\mathbb{A}^n$, it is easy to construct D-modules by writing down an explicit linear system of PDE's and then writing a presentation of the associated D-module $$\mathcal{D}^n \xrightarrow{} \mathcal{D}^m \to \mathcal{M} \to 0$$ Is there an analogous process for constructing D-modules over projective space as an explicit system of differential equations?
You can construct $\mathcal{D}$ modules on projective space (or actually any space) by precisely the same process: write down a system of PDE's (using differential operators on the whole of $\mathbb{P}^n$) and consider the corresponding quotient. The key caveat is that on a general space, not every $\mathcal{D}$-module will be of this form: some of them can be defined using systems on different coordinate patches which are consistent, but not globally (I think a degree 0 line bundle on an elliptic curve should give a counterexample). However, $\mathbb{P}^n$ is special, and every $\mathcal{D}$-module can be defined this way (by the Beilinson-Bernstein theorem).
• Will $\mathcal{D}_{\mathbb{P}^n}(\mathbb{P}^n) = \mathbb{C}\left[\frac{\partial}{\partial x_0}, \ldots, \frac{\partial}{\partial x_n}\right]$? Jan 3, 2017 at 1:42
• @user251222 Those expressions don't even make sense as differential operators, since they have poles at $\infty$. The polynomial differential operators are generated by $x_i\frac{\partial}{\partial x_j}$. Jan 3, 2017 at 2:24
• Look at Section 11.3 of Hotta, Takeuchi and Tanisaki (math.columbia.edu/~scautis/dmodules/hottaetal.pdf). The essential point is that these vector fields (thought of as functions on $T^*\mathbb{P}^n$) generate all the polynomial functions on the cotangent bundle, since the induced map from $T^*\mathbb{P}^n$ to $(n+1)\times (n+1)$-matrices is a resolution of singularities of the rank $\leq 1$ matrices. Jan 3, 2017 at 13:33
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# Momentum
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## Momentum
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##### Presentation Transcript
1. Momentum
2. key ideas about momentum • Momentum is the amount of m an object has • Momentum is to do with the m and the v • of an object • In an explosion the momentum of an object moving • to the l is equal to the of the object • which moves to the r Fill in the answers and then click to check them
3. key ideas about momentum otion • Momentum is the amount of m_____ an object has • Momentum is to do with the m___ and the v______ • of an object • In an explosion the momentum of an object moving • to the l____ is equal to the __________ of the object • which moves to the r_____ ass elocity eft momentum ight
4. key words for the momentum topic TUM NO MEM RE COF CITY OF VEL SAMS STREEM ERP DESCON OG RI SMALK POXLESION NO NANC LALB LE RIF TUBLLE is the amount of motion an object has is measured in Newtons is how fast it moves in a certain direction the amount of material measured in kilograms the unit for velocity the unit for mass when stationary objects go in opposite directions old fashioned gun spherical object leaving the previous gun modern gun which you rest on the shoulder to fire fired by the gun above Fill in the answers and then click to check them
5. key words for the momentum topic momentum force velocity mass Metres per second kilograms explosion cannon ball rifle bullet is the amount of motion an object has is measured in Newtons is how fast it moves in a certain direction the amount of material measured in kilograms the unit for velocity the unit for mass when stationary objects go in opposite directions old fashioned gun spherical object leaving the previous gun modern gun which you rest on the shoulder to fire fired by the gun above Fill in the answers and then click to check them
6. Calculating momentum
7. How much momentum do the following object have? 1 A 700kg car travelling at 10 m/s 2 A 1500kg van travelling at 5m/s 3 A mini travelling at 20m/s with a mass of 800kg 4 A 45kg person running at 4m/s Momentum = x = Momentum = x = Momentum = x = Momentum = x = click for the answers
8. How much momentum do the following object have? 1 A 700kg car travelling at 10 m/s 2 A 1500kg van travelling at 5m/s 3 A mini travelling at 20m/s with a mass of 800kg 4 A 45kg person running at 4m/s Momentum = 700 x 10 = 7000 kgm/s Momentum = 1500 x 5 = 7500 kgm/s Momentum = 800 x 20 = 16 000 kgm/s Momentum = 45 x 4 = 180 kgm/s
9. Explosions
10. cannonball moves this way with a high velocity Cannon moves this way more slowly as it recoils from the explosion
11. Velocity as it leaves the cannon = 400m/s Mass of cannon ball =? Momentum = Momentum of of ball cannon Recoil velocity = 8m/s Mass = 100kg mass x velocity = mass x velocity of the ball of the cannon ? X 400 = 100 x 8 ? X 400 = 800 ? = 800/400 = 2kg
12. V Now do your own calculation Fill in the answers and then click to check them Velocity as it leaves the cannon = 500m/s Mass of cannon ball =? Momentum = Momentum of of ball cannon Recoil velocity = 10m/s Mass = 50kg mass x velocity = mass x velocity of the ball of the cannon ? x = x ? x = ? = =
13. Velocity as it leaves the cannon = 500m/s Mass of cannon ball =? Momentum = Momentum of of ball cannon Recoil velocity = 10m/s Mass = 50kg mass x velocity = mass x velocity of the ball of the cannon ? X 500 = 50 x 10 ? X 500 = 500 ? = 500/500 = 1kg
14. Collisions
15. The moving ball hits the stationary ball • The stationary ball moves off • If the first ball stops • The second ball gains all the momentum of the first ball
16. v 2m/s 5kg 5kg Momentum of first ball = momentum of second ball (before collision) (after collision) 5kg x 2m/s = 5kg x v So v = 2m/s In the second case the first ball has more mass but the same idea applies Momentum of first ball = momentum of second ball (before collision) (after collision) 8kg x 2m/s = 4kg x v So v = 4m/s v 2m/s 8kg 4kg
17. Now try these v 3m/s 12kg 12kg Momentum of first ball = momentum of second ball (before collision) (after collision) x = x v So v = In the second case the first ball has more mass but the same idea applies Momentum of first ball = momentum of second ball (before collision) (after collision) x = x v So v = Now click for the answers v 5m/s 12kg 4kg
18. Now try these v 3m/s 12kg 12kg Momentum of first ball = momentum of second ball (before collision) (after collision) 12 x 3 = 12 x v So v = 3m/s In the second case the first ball has more mass but the same idea applies Momentum of first ball = momentum of second ball (before collision) (after collision) 12 x 5 = 4 x v So v = 15m/s v 5m/s 12kg 4kg
19. More collisions Higher level
20. Momentum before the collision = Momentum after the collision (of the single ball) ( of both balls)
21. Momentum before the collision = Momentum after the collision (of the single ball) ( of both balls) • 3 x 8 = (3+3) v • = 6 x v • So v = 24/6 • v = 4m/s v 8 m/s 3kg 3kg 3kg
22. Another example Momentum before the collision = Momentum after the collision (of the single ball) ( of both balls) 3 x 9 = (6+3) v 27 = 9 x v So v = 27/9 v = 3m/s V 9 m/s 6kg 3kg 3kg
23. Now try this one yourself Momentum before the collision = Momentum after the collision (of the single ball) ( of both balls) = x v x = x v So v = Click for the answer V 10 m/s 1kg 4kg 4kg
24. Momentum before the collision = Momentum after the collision (of the single ball) ( of both balls) 3 x 9 = (6+3) v 27 = 9 x v V = 3m/s V 9 m/s 6kg 3kg 3kg
25. Collisions and safety Higher level
26. Stopping quickly is dangerous because a large force is felt for a short time The crumple zone on this car allows the car to change shape slowly so that a smaller force is felt for a longer time
27. Hitting the windscreen in a car crash means that a large force is felt for a short time which can cause severe injuries for the driver Hitting the airbag in a car crash means that a smaller force is felt for a longer timewhich is Much safer for the driver Force (N) Which line represents the air bag? The blue line The red line shows a big force for a short time The blue line shows a smaller force acting over a longer time time (s)
28. END
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Practice Exercise 14.A
Probabilistic Relational Models
## 2: Learning Goals
• Represent a simple problem in plate notation.
• Derive and draw a grounded belief network.
• Calculate probabilities from a problem in the independent choice logic representation.
## 3: Directed Questions
1. In probabilistic relational models, do different individuals share the same probability parameters? [solution]
2. What are the three components of a plate model? [solution]
3. What type of dependency can a plate model not adequately represent? [solution]
## 4: Exercise: Movie Preferences
Suppose we have parametrized random variables likes(Person,Movie), young(Person) and genre(Movie,Genre) where there are 1000 people, 100 movies and 5 genres. Suppose young(Person) and genre(Movie,Genre) are parents of likes(Person,Movie).
1. Draw this in plate notation. [solution]
2. How many random variables are in the grounding of this model? [solution]
3. Draw the grounding belief network assuming the population of Person is {sam,chris,kim} and the population of Movie is {terminator,rango}. [solution]
Consider the following fragment of independent choice logic theory:
```likes(Person,Movie) <- young(Person) & genre(Movie,action) & young_likes_action(Person,Movie).
prob young_likes_action(Person,Movie):0.7.
likes(Person,Movie) <- ~young(Person) & genre(Movie,action) & old_likes_action(Person,Movie).
prob old_likes_action(Person,Movie):0.6.
likes(Person,Movie) <- genre(Movie,comedy) & likes_comedy(Person,Movie).
prob likes_comedy(Person,Movie):0.8.
young(sam).
young(chris).
genre(terminator,action).
genre(rango,comedy).
```
where <- means "if", ~ means "not", & means "and", and "prob a:p." means {a, ~a} is an alternative with P(a) = p.
1. What is the probability of likes(sam,rango)? [solution]
2. What is the probability of likes(chris,terminator)? [solution]
3. What is the probability of likes(kim,terminator)? [solution]
## 5: Learning Goals Revisited
• Represent a simple problem in plate notation.
• Derive and draw a grounded belief network.
• Calculate probabilities from a problem in the independent choice logic representation.
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Page 1 of 2
### Is economics a science?
Posted: August 19th, 2019, 10:06 am
Hello everyone!
I’m a philosopher, wondering about the World. Nothing unusual for a philosopher.
Anyway, I wrote some thoughts about the economics. A major part of it is about the quantitative math in economics. It is kind of “macro” and “how to save the world”, so I assume it should fit this forum. Since I’m interested in truth, before anything else, I’d like to ask a question to the professionals:
Could anyone prove me wrong in anything I wrote?
My text (http://philosopher4hire.eu/index.php?nr=3) is 7 pages long. This excerpt should let you feel what it is like:
For the technical analysis the fundamental assumption is the price move dependency. Technical analysis states that the future price moves are strongly dependent on the previous (past) moves. An analyst needs to search for patterns that repeat themselves. If the future moves of the price would be independent from what happened in the past (especially in the most recent past), then the technical analysis would be a lie.
The quantitative math uses the opposite assumption. All the probability based mathematics, which is the foundation for the quantitative analysis, requires each price move to be totally independent from the preceding moves. Exactly as it happens with the coin tossing (the favorite example of all quants, btw). The next result is independent from what you got previously. The probability of receiving tails does not depend on the number of previously received heads. If this assumption fails, then all the sophisticated economic quant math analysis is a lie. At least, for a mathematician.
If for example chemistry would be like economics, we could have two classrooms full of students. In one, they would be taught that one element cannot be changed into another during a chemical reaction. That such change requires a nuclear fusion. In the neighboring classroom, they would learn, that there are chemical reactions which can change one element into another. Especially, every ‘heavy’ metal like iron or lead can be transformed into gold in a special, alchemical reaction. We only have to discover such reaction.
It would be a perfectly reasonable approach, if only we would remove the no contradiction rule from science. ‘A science’ containing contradictions is not a science, but an intellectual mess.
### Re: Is economics a science?
Posted: August 19th, 2019, 1:17 pm
Let’s leave technical analysis aside. That doesn’t have much to do with economics or even finance, which seems to be your focus. Your characterization of the probabilistic modeling of asset pricing is wrong. It is true that in the basic theory the first moment of the increments of a price process is zero after we remove two drift terms: one accounting for riskless discounting and another for a risk premium. But both of those drift terms as well as all higher moments, in particular the variance, can happily be path dependent. So, I think you have a bit more homework to do.
### Re: Is economics a science?
Posted: August 20th, 2019, 9:04 am
Thank You very much for your response. I appreciate it. It is certainly a good starting point. Now, my reply:
It seems to me, you do not know the (mathematical) foundations of your knowledge. You use the term “variance”, while speaking of something that is “path dependent”. The variance in mathematics, as the entire mathematical probability, applies to independent events. You can’t just take the theory built for the independent events and say: “Now I will use it for the dependent events.” It is simply not mathematically (scientifically) valid. Perhaps this is the reason, why you have to deal with the “fat tails” term. Something unknown (=undefined) to mathematics.
Moreover, the “drift” term is undefined in case of a random variable. Or, precisely speaking, if there is any kind of drift, it is not a random variable. So, you cannot speak about “drifts” while staying on the ground of the mathematical probability.
The basic wiki search reveals:
Variance - In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.
Random variable - In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon.
Randomness - Randomness is the lack of pattern or predictability in events.[1] A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination.
This is exactly why I wrote: “[…] quant math analysis is a lie. At least, for a mathematician.” Mathematics is very strict. You cannot take just any numerical sequence (like the post codes in your area, or the prime numbers less than 1000), make calculations and say: “It’s the variance”. It is not. If you would read my entire text, you would have a better understanding of what I mean. And perhaps you would not try to treat it so lightly. But again: I want to thank You for your helping me!
As for the “Let’s leave technical analysis aside. That doesn’t have much to do with economics or even finance”. As a quant you may have such opinion. But the rest of the world sees it differently.
The wiki again:
Technical analysis - In finance, technical analysis is an analysis methodology for forecasting the direction of prices through the study of past market data, primarily price and volume.
### Re: Is economics a science?
Posted: August 20th, 2019, 10:15 am
Funny. Sticking to Wikipedia, you should look up Itô calculus for an introduction to drift in the context of mathematical probability theory and kurtosis as a precise measure of fat tails in classical statistics. I am not a mathematician, but a number of prominent ones have made significant contributions to finance, including a certain Paul Wilmott... And technical analysis is to finance what astrology is to astronomy.
### Re: Is economics a science?
Posted: August 21st, 2019, 9:50 am
You haven’t read my text. It’s a pity. I have to repeat my argumentation here. But... I still want to know if there is any flaw in my thinking, and I hope you will finally spend the 20-30 minutes needed to read my article. And maybe, you will even think about it one evening? So, let’s try again:
Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process).
Good point. I forgot about it. But it is an advanced lemma, having many detailed mathematical constraints. Perhaps the Brownian motion fulfills them all. But the asset price path is not a Brownian motion. It is not a result of the laws of physics. It may look similar, but it is not enough to consider the application of Itô calculus to the asset price as scientific. I do see the difference between practitioners (who use what they have) and scientists (who should promote the real science).
In probability theory and statisticskurtosis (from Greekκυρτόςkyrtos or kurtos, meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable.
Again “random variable”. You may try to use the probability to data, which you do not know if it is random or not. But using it to data, when you know it is not random (in many cases, at least) is deceiving. Yourself and others. Of course, only if you consider yourself a scientist. If you call it science. You may calculate the variance of the post codes in your region. If you see any practical application of such move. But do not call it science.
but a number of prominent ones have made significant contributions to finance, including a certain Paul Wilmott...
That’s right. It is also true for many rocket scientists, physicists, and so on. The finance sector pays well. But does it prove anything? I do not question their achievements in mathematics. I do question the application of the probability theory (and mathematics in general) to the world of the human minds. I suspect, Paul Wilmott knows, that such application is not very scientific (to put it mildly). But thinking realistically, I cannot hope he will admit it openly.
And technical analysis is to finance what astrology is to astronomy.”
An interesting comparison. I’ll put a longer wiki citation here:
Technical analysis -
In financetechnical analysis is an analysis methodology for forecasting the direction of prices through the study of past market data, primarily price and volume.[1] Behavioral economics and quantitative analysis use many of the same tools of technical analysis, which, being an aspect of active management, stands in contradiction to much of modern portfolio theory. The efficacy of both technical and fundamental analysis is disputed by the efficient-market hypothesis which states that stock market prices are essentially unpredictable.
It sounds very scientific. Don’t you think so? I suppose there must be many universities, where professors teach such knowledge. Otherwise, how could it stay on wiki for so long? How could it get there in the first place?
Are they all lunatics? Why don’t they see the obviousness?
If the quant math applied to finance would be like math applied to the universe (=astronomy), then why would scientists still use and develop the “finance astrology” (=tech analysis)? Why such things do not happen in physics?
Perhaps the quant math is not the astronomy of finance? Maybe it does only seem so?
Questions. Questions. Questions.
Very dangerous questions for someone so sure of what is right and what is wrong.
Perhaps, here is an answer:
http://philosopher4hire.eu/index.php?nr=3
Perhaps, it is the true answer?
But you would have to lose 25 min of your life on reading it…
Is it worth doing?
Maybe it is better to stick to what you know already?
### Re: Is economics a science?
Posted: August 21st, 2019, 1:53 pm
It's not even wrong.
### Re: Is economics a science?
Posted: August 21st, 2019, 4:06 pm
It's not even wrong.
As Wittgenstein would say, it's a description rather than an explanation.
### Re: Is economics a science?
Posted: August 23rd, 2019, 4:24 pm
‘A science’ containing contradictions is not a science, but an intellectual mess.
Has mathematics resolved the incompleteness theorems (e.g. Godel)? That is a question rather than a statement.
### Re: Is economics a science?
Posted: August 23rd, 2019, 4:45 pm
but a number of prominent ones have made significant contributions to finance, including a certain Paul Wilmott...
That’s right. It is also true for many rocket scientists, physicists, and so on. The finance sector pays well. But does it prove anything? I do not question their achievements in mathematics. I do question the application of the probability theory (and mathematics in general) to the world of the human minds. I suspect, Paul Wilmott knows, that such application is not very scientific (to put it mildly). But thinking realistically, I cannot hope he will admit it openly.
I've only been criticizing the subject for about a quarter of a century, openly, in public. (You ask other people to read but you haven't, not even wikipedia.) But my criticisms are based on many years before that researching in many, many other fields, some scientific some not. So I draw conclusions from a lot of experience.
I haven't read everything you have written here. (You are probably mad, and we see this sort of stuff every few months here, so who has the time?) But it looks like you are missing the point. In a science you have three things to worry about: a) Getting it right, b) Publishing papers and c) Getting grants. Even that shows you are being naive, it's not just about promoting "real science" as you put it. So that's just three dimensions. In finance there may be a dozen dimensions. Only one is about getting things right in any scientific sense. And it's not very important.
As a "philosopher for hire," what do you charge and what do people get for their money? I ask because I find it helpful if people can relate finance to the real world, and if you have real-world experience of running a business then you will understand the business of finance better.
### Re: Is economics a science?
Posted: August 23rd, 2019, 5:15 pm
Madness to ignorance is like deterministic chaos to randomness.
### Re: Is economics a science?
Posted: August 24th, 2019, 9:21 am
Hello Paul!
I was hoping to attract your attention. Therefore: Thanks a lot for Your response.
“I've only been criticizing the subject for about a quarter of a century, openly, in public.”
I do not want to steal your achievements. Not to mention, it would be impossible. I know, you have been criticizing various aspects of the quant math applied to finance (if that is what you meant as “the subject”). But does it mean, that no one else has the right for critics?
“(You ask other people to read but you haven't, not even wikipedia.)”
This statement is untrue. I’ve read some of what you have written. In fact, I was learning the quantitative math partially on your great book on this subject. But you are right, that I have not read the wikipedia article about you (if this is what you meant). I consider the wiki a fast source of knowledge usable to get the basic, scientific definitions. I do not trust it on other subjects.
But my criticisms are based on many years before that researching in many, many other fields, some scientific some not. So I draw conclusions from a lot of experience.
I know. So do I.
I haven't read everything you have written here.
It’s a pity. Moreover, it does not testify well of you. Answering to someone, without taking the burden of spending 10 min on reading… Wasn’t that you, who criticized me for not reading what you have written just a moment earlier?
(You are probably mad, and we see this sort of stuff every few months here, so who has the time?)
This somehow explains your emotions. I have no experience of running such site, yet. I sympathize with your situation. But aren’t you a little too fast in your diagnosis? It sets me in a very uncomfortable situation. Writing: “I’m not mad” would put us into a sandbox.
But it looks like you are missing the point. In a science you have three things to worry about: a) Getting it right, b) Publishing papers and c) Getting grants. Even that shows you are being naive, it's not just about promoting "real science" as you put it.
We have certainly different views on what is (should be?) science. I suppose, your short description is the essence of the modern (western) science. I disagree with such approach, as it leads to what Lenin expressed as: “Capitalists will sell us the rope with which we will hang them”. Setting money as the central (the only?) point for everything is unwise. Questions like: ‘what for?’ and ‘why?’ should be asked. If not in science, then where? You call it “naive”?
Besides, I just ask if we got it right. I’ve published (the linked) papers. Can’t I do it without the immediate c) Getting grants?
In finance there may be a dozen dimensions. Only one is about getting things right in any scientific sense. And it's not very important.
If you would just read what I wrote here, you could find:
"I do see the difference between practitioners (who use what they have) and scientists (who should promote the real science)."
One thing is finance – the traders’ race, where you use whatever you have and however you want. The other: science, which should explain the world. Which should tell us how the things really are. There is no need to reduce one to the other. They can be (and should be) distinct things.
As a "philosopher for hire," what do you charge and what do people get for their money?”
I have a different business model. Not everyone lives in your bubble.
I ask because I find it helpful if people can relate finance to the real world, and if you have real-world experience of running a business then you will understand the business of finance better.
I do not want to understand the business of finance better than I have to. I’m not eager to become another Paul Wilmott. My interests are much broader than finance. I just wanted to make sure, that what I wrote is flawless. As perfect as possible. But all people on this forum prefer writing to reading. Bad luck.
### Re: Is economics a science?
Posted: August 24th, 2019, 4:55 pm
I admitted straight up that I'm not reading most of what you are writing. And I haven't read most of the latest. I have read enough to know that you also aren't reading what anyone is writing, or you are misunderstanding, deliberately or not I cannot tell. I am not reading because what you write isn't interesting or entertaining enough. What will I learn from it? That you get everything wrong, that you are unpleasant, that you are arrogant? I love arrogant people. Many say that I am arrogant. But they have to back up their arrogance with something concrete.
I think one of the interesting phenomena of the internet age is how easy it is for people to make idiots of themselves in public and forever and whether that is a good or bad thing. The pros and the cons are obvious, but I wonder what the end result, if any, will be. I do feel protective towards people who get everything wrong, and I want to help them not look stupid. But that protective instinct mostly disappears when they are unpleasant.
### Re: Is economics a science?
Posted: August 24th, 2019, 5:37 pm
how easy it is for people to make idiots of themselves in public
We both can agree with this one, at least. Good-bye.
### Re: Is economics a science?
Posted: August 31st, 2019, 4:43 pm
It's not even wrong.
No, it's actually wrong
### Re: Is economics a science?
Posted: August 31st, 2019, 5:16 pm
Cuchulainn, one for you...
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CC-MAIN-2021-21
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latest
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https://www.hackmath.net/en/math-problem/79164
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# Acceleration 79164
A skier goes down a slope 66 m long in a uniformly accelerated motion in 10 seconds. With what acceleration was it moving, and what is the slope of the slope?
a = 1.32 m/s2
α = 7.733 °
## Step-by-step explanation:
Did you find an error or inaccuracy? Feel free to write us. Thank you!
Tips for related online calculators
The line slope calculator is helpful for basic calculations in analytic geometry. The coordinates of two points in the plane calculate slope, normal and parametric line equation(s), slope, directional angle, direction vector, the length of the segment, intersections of the coordinate axes, etc.
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# COMBINATION and PROBABILITY - references. If you have
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COMBINATION and PROBABILITY - references. If you have [#permalink]
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BOOKMARKED
COMBINATION and PROBABILITY - references.
If you have problems with combination and probability you can use this reference page to find information quickly.
I guess my small contribution to the GMATclub community will help other people to crack 800 score.
THEORY
[Combinatorics] | [Probability]
-general:
Wiki Combination
Wiki Permutation
-also:
7-t-10838.html
permutation-combinations-coverage-30315.html
permutations-order-matters-79661.html - how to distinguish Permutations from Combinations
-question collections:
NOTES FROM WALKER
#1. Com-prob problems always have a few different solutions. Try to solve the problems in many ways, and that will give you self-reliance.
#2. Use enumeration of possibilities for the problems with complex restrictions.
#3. "How can I apply here $$C^n_k$$ or $$P^n_k$$?" - is a wrong question. Always understand what the formulas mean.
easy - 56860 - 6 boys and 4 girls
easy - 56779 - 10 red, 5 blue, 12 yellow pills.
easy - 56728 - 10 marbles contains 3 red marbles and 7 blue marbles
easy - 56446 - 4 people and integers between 1 and 4
easy - 56419 - an iced tea
easy - 55984 - a contest consists of n questions
easy - 55939 - 15 chess players
easy - 55734 - 3 candidates for a university
easy - 55689 - a pack of candy (2)
easy - 55644 - a five-question true/false section
easy - 55515 - a pack of candy
easy - 54481 - red and blue chips
medium - 58914 - a junior class has 1,000 students and a senior class has 800 students
medium - 58887 - that include Michael also include Anthony
medium - 58742 - 2 couples and 1person are seated in a row
medium - 58675 - n(n + 1)(n + 2) is divisible by 8
medium - 58670 - Probability of Winning
medium - 58640 - 4 people out of the 12 meried couples
medium - 58629 - two ineger sets
medium - 58609 - 9 people in the room
medium - 58529 - 2 objects collide on a cube
medium - 58521 - DS: heads than tails
medium - 58476 - Marco and Maria toss a coin three times
medium - 58360 - two ineger sets (2)
medium - 58357 - coin is flipped 5 times
medium - 58352 - 11 women and 9 men in a certain club
medium - 58350 - DS: 10 light bulbs
medium - 58233 - SQUARE
medium - 58187 - A certain company assigns employees to offices
medium - 58160 - 3 people are to be selected out of 8 people
medium - 58095 - Bob are among the 5 participants in a cycling race
medium - 58016 - 3 people is to be selected from 5 married couples
medium - 57997 - junior class has 1000 students and senior class has 800 students
medium - 57981 - positions at univeristy
medium - 57980 - women and men of 10 employees
medium - 57957 - rain on any given day in City X
medium - 57916 - a and b from a set of 4 consecutive integers
medium - 57882 - Anthony and Michael sit on the six-member board of directors for company X
medium - 57799 - A fair die is rolled once and a fair coin is flipped once
medium - 57695 - 3 dwarves and 3 Elves
medium - 57645 - A contest will consist of n questions
medium - 57629 - coin
medium - 57615 - A jar contains b black, w white, and r red marbles
medium - 57601 - GMPJBC
medium - 57577 - 5 people are to be seated around a circular table
medium - 57554 - Coach Miller
medium - 57332 - A coach will make 3 substitutions
medium - 56826 - 7 people and friendship
medium - 56814 - coin with sides marked heads and tails
medium - 56814 - tails side up more than five times
medium - 56812 - a election in Burghtown.
medium - 56709 - A set consist of 2n-1 element
medium - 56705 - 25 lines
medium - 56681 - 2 numbers from 1,2,3,4,5
medium - 56590 - 2 red and 3 blue marbles
medium - 56586 - different colors from 6 flowers.
medium - 56585 - 4 particular flowers of 8 different flowers
medium - 56572 - five-digit numbers
medium - 56557 - a committee of 4 distinct pairs of brothers and sisters
medium - 56556 - three letters taken from letters (a,a,b,b,c,c,d)
medium - 56530 - 12 playing cards
medium - 56530 - 4 cards of 12 cards
medium - 56446 - 4 people choose an integer between 1 and 4, inclusive.
medium - 56445 - 5 of 7 models
medium - 56443 - Two couples and one single person in a row of five chairs.
medium - 56349 - 3 girls and 4 boys in a row
medium - 56327 - shirts, shoes, pants
medium - 56122 - Six mobsters: Frankie behind Joey
medium - 56037 - a team from men and women
medium - 55870 - 3 cars
medium - 55823 - 4 senior partners and 6 junior partners
medium - 55811 - 5 women are in a race
medium - 55803 - a committee from 4 married couples
medium - 55764 - 100 numbered cubes
medium - 55743 - A and B stocks
medium - 55694 - Two fair dices
medium - 55673 - 3 different marbles
medium - 55644 - GMATT answers
medium - 55617 - 4 guys, 2 girls and 14 dogs.
medium - 55509 - A committee of 3 people from 4 couples
medium - 55504 - Ways from A to D
medium - 55472 - 4 pairs of brothers and sisters
medium - 55427 - 4 books fom 2 paperback and 6 hardback books
medium - 55421 - the probability that n(n+1)(n+2) will be divisible by 8
medium - 55413 - 3-digit integer
medium - 55410 - A committee from 5 military officers and 9 civilians.
medium - 55383 - 4 physics, 2 math and 3 chemistry books
medium - 55372 - My name is AJEET
medium - 55369 - 8 friends want to play doubles tennis
medium - 55366 - The most expensive two books
medium - 55340 - trhee letters from seven different postboxes
medium - 55311 - 6 cards
medium - 55253 - Rich has 3 green, 2 red and 3 blue balls in a bag
medium - 55141 - A bowl contains white and black balls.
medium - 55071 - 10 bottles of alcohol
medium - 55054 - 3 sniper's shoots
medium - 54794 - a fantasy football game
medium - 54650 - 6 cards
medium - 54332 - Bob and Jen in a race
medium - 53774 - The choir consists of 5 boys and 6 girls
medium - 50488 - 5 digit numbers
medium - 45753 - a committe of 3 of 6 people
medium - 41464 - deck contains 2 blue, 2 red, 2 yellow, and 2 green cards
medium - 19685 - 5 digit numbers formed using the digits 0, 1, 2, 3, 4, 5
medium - 15366 - Mary and Joe throw three dice each
medium - 59065 - A 4-letter code
hard - 58616 - PS Standard deviation + probabilities
hard - 57713 - the total number of permutations of n different things
hard - 57447 - A fair coin is tossed 10 times
hard - 57169 - DS: How many subordinates does Marcia Have
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01 Dec 2007, 10:48
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Medium - 79490 - How many four-digit odd numbers do not use any digit more than once
Meidum - 79312 - A jar contains only x black balls and y white balls
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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22 Jan 2008, 00:33
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watchout folks, walker is headed for the US presidency!
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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22 Jan 2008, 00:45
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watchout folks, walker is headed for the US presidency!
I'm just gathering some information for me and for you .....
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Thanks for it !
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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19 Jan 2008, 04:33
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I've finished with my P-C-P page. I'll add only interesting problems!
Good Luck!
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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21 Jan 2008, 18:12
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Walker can you add this link to your P-C-P page. Excellent collection. Makes life so much easier for all of us.
I like the explanation you provided to this question.
7-t58742
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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22 Jan 2008, 00:49
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yeah man..its awesome!
btw, when are u looking to take the test?
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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22 Jan 2008, 04:14
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btw, when are u looking to take the test?
3-4 months. I have to SC
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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22 Jan 2008, 16:34
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Walker - awesome; simply awesome. Are you Walker as in Walker: Texas Ranger. Because if you are, a Chuck Norris reference comes to mind: "When Chuck Norris falls into water, he does not get wet. The water gets Chuck Norris."
Nice one man.
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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bb wrote:
Bump
Can you make this post Sticky?
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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21 Jan 2008, 23:58
neelesh wrote:
Walker can you add this link to your P-C-P page. Excellent collection. Makes life so much easier for all of us.
I like the explanation you provided to this question.
7-t58742
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Re: COMBINATION and PROBABILITY - references. [#permalink]
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24 Jan 2008, 11:45
Yeah, If you have interesting problems, post here links and I will add them to the list.
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Re: COMBINATION and PROBABILITY - references. [#permalink]
### Show Tags
20 Mar 2008, 10:28
walker wrote:
Yeah, If you have interesting problems, post here links and I will add them to the list.
fantastic post.
walker flies.
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Re: COMBINATION and PROBABILITY - references. [#permalink]
### Show Tags
20 Mar 2008, 11:16
awesome job walker!
CEO
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Re: COMBINATION and PROBABILITY - references. [#permalink]
### Show Tags
20 Mar 2008, 12:37
Thanks!
I add "NOTES FROM WALKER" to my first post for new members
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Re: COMBINATION and PROBABILITY - references. [#permalink]
### Show Tags
20 Mar 2008, 12:44
Thnx Dude. You ROCK.
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Re: COMBINATION and PROBABILITY - references. [#permalink]
### Show Tags
13 Apr 2008, 14:01
woww, just found this, Thanks man, this is awsome, hopefully now I can P&C
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Re: COMBINATION and PROBABILITY - references. [#permalink]
### Show Tags
14 Apr 2008, 02:22
Walker you are a legend !
Kudos for you
Re: COMBINATION and PROBABILITY - references. [#permalink] 14 Apr 2008, 02:22
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# All Science Fair Projects
## Science Fair Project Encyclopedia for Schools!
Search Browse Forum Coach Links Editor Help Tell-a-Friend Encyclopedia Dictionary
# Science Fair Project Encyclopedia
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enter a keyword (eg. scientific method, molecule, cloud, carbohydrate etc.).
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# Piezoelectricity
Piezoelectricity is the ability of certain crystals to produce a voltage when subjected to mechanical stress. The word is derived from the Greek piezein, which means to squeeze or press. The effect is reversible; piezoelectric crystals, subject to an externally applied voltage, can change shape by a small amount. The effect is of the order of nanometres, but nevertheless finds useful applications such as the production and detection of sound, generation of high voltages, electronic frequency generation, and ultrafine focusing of optical assemblies.
Contents
## Mechanism
In a piezoelectric crystal, the positive and negative electrical charges are separated, but symmetrically distributed, so that the crystal overall is electrically neutral.
When a stress is applied, this symmetry is disturbed, and the charge asymmetry generates a voltage. A 1 cm cube of quartz with 500 lbf (2 kN) of correctly applied force upon it, can produce 12,500 V of electricity.
Piezoelectric materials also show the opposite effect, called converse piezoelectricity, where application of an electrical field creates mechanical stress (distortion) in the crystal. Because the charges inside the crystal are separated, the applied voltage affects different points within the crystal differently, resulting in the distortion.
The bending forces generated by converse piezoelectricity are extremely high, of the order of tens of millions of pounds (tens of meganewtons), and usually cannot be constrained. The only reason the force is usually not noticed is because it causes a displacement of the order of one billionth of an inch (a few nanometres).
## History
A related property known as pyroelectricity, the ability of certain mineral crystals to generate electrical charge when heated, was known of as early as the 18th century, and was named by David Brewster in 1824. In 1880, the brothers Pierre Curie and Jacques Curie predicted and demonstrated piezoelectricity using tinfoil, glue, wire, magnets, and a jeweler's saw. They showed that crystals of tourmaline, quartz, topaz, cane sugar, and Rochelle salt (sodium potassium tartrate tetrahydrate) generate electrical polarization from mechanical stress. Quartz and Rochelle salt exhibited the most piezoelectricity. Twenty natural crystal classes exhibit direct piezoelectricity.
Converse piezoelectricity was mathematically deduced from fundamental thermodynamic principles by Lippmann in 1881. The Curies immediately confirmed the existence of the "converse effect," and went on to obtain quantitative proof of the complete reversibility of electro-elasto-mechanical deformations in piezoelectric crystals.
The first practical application for piezoelectric devices was sonar, first developed during World War I. In France in 1917, Paul Langevin (whose development now bears his name) and his coworkers developed an ultrasonic submarine detector. The detector consisted of a transducer, made of thin quartz crystals carefully glued between two steel plates, and a hydrophone to detect the returned echo. By emitting a high-frequency chirp from the transducer, and measuring the amount of time it takes to hear an echo from the sound waves bouncing off an object, one can calculate the distance to that object.
The use of piezoelectricity in sonar, and the success of that project, created intense development interest in piezoelectric devices. Over the next few decades, new piezoelectric materials and new applications for those materials were explored and developed.
Development of piezoelectric devices and materials in the United States was kept within the companies doing the development, mostly due to the wartime beginnings of the field, and in the interests of securing profitable patents. New materials were the first to be developed — quartz crystals were the first commercially exploited piezoelectric material, but scientists searched for higher-performance materials.
Piezoelectric devices found homes in many fields. Ceramic phonograph cartridges simplified player design, were cheap and accurate, and made record players cheaper to maintain and easier to build. Ceramic electret microphones could be made small and sensitive. The development of the ultrasonic transducer allowed for easy measurement of viscosity and elasticity in fluids and solids, resulting in huge advances in materials research. Ultrasonic time-domain reflectometers (which send an ultrasonic pulse through a material and measure reflections from discontinuities) could find flaws inside cast metal and stone objects, improving structural safety. However, despite the advances in materials and the maturation of manufacturing processes, the United States market had not grown as quickly. Without many new applications, the growth of the United States' piezoelectric industry suffered.
In contrast, Japanese manufacturers shared their information, quickly overcoming technical and manufacturing challenges and creating new markets. Japanese efforts in materials research created piezoceramic materials competitive to the U.S. materials, but free of expensive patent restrictions. Major Japanese piezoelectric developments include new designs of piezoceramic filters, used in radios and televisions, piezo buzzers and audio transducers that could be connected directly into electronic circuits, and the piezoelectric igniter which generates sparks for small engine ignition systems (and gas-grill lighters) by compressing a ceramic disc. Ultrasonic transducers that could transmit sound waves through air had existed for quite some time, but first saw major commercial use in early television remote controls. These transducers now are mounted on several car models as an echolocation device, helping the driver determine the distance from the rear of the car to any objects that may be in its path.
## Materials
In addition to the materials listed above, many other materials exhibit the effect, including quartz analogue crystals like berlinite (AlPO4) and gallium orthophosphate (GaPO4), ceramics with perovskite or tungsten-bronze structures (BaTiO3, KNbO3, LiNbO3, LiTaO3, BiFeO3, NaxWO3, Ba2NaNb5O5, Pb2KNb5O15). Polymer materials like rubber, wool, hair, wood fiber, and silk exhibit piezoelectricity to some extent. The polymer polyvinylidene fluoride, (-CH2-CF2-)n, exhibits piezoelectricity several times larger than quartz. Bone exhibits some piezoelectric properties: it has been hypothesized that this is part of the mechanism of bone remodelling in response to stress.
## Applications
Piezoelectric crystals are used in numerous ways:
### High-voltage sources
Direct piezoelectricity of some substances like quartz, as mentioned above, can generate thousands of volts (known as high-voltage differentials).
• Probably the best-known application is the electric cigarette lighter: pressing the button squeezes an piezoelectric crystal, and the high voltage thus produced ignites the gas as the current jumps over a small spark gap. The portable electrical sparkers used to light gas grills or stoves work the same way.
• A similar idea is being researched by DARPA in the USA in a project called Energy Harvesting, which includes an attempt to power battlefield equipment by piezoelectric generators embedded in soldiers' boots.
• A piezoelectric transformer is a type of AC voltage multiplier. Unlike a conventional transformer, which uses magnetic coupling between input and output, the piezoelectric transformer uses acoustic coupling. An input voltage is applied across a short length of a bar of piezoceramic material such as PZT, creating an alternating stress in the bar by the inverse piezoelectric effect and causing the whole bar to vibrate. The vibration frequency is chosen to be the resonant frequency of the block, typically in the 100 kilohertz to 1 megahertz range. A higher output voltage is then generated across another section of the bar by the piezoelectric effect. Step-up ratios of more than 1000:1 have been demonstrated. An extra feature of this transformer is that, by operating it above its resonant frequency, it can be made to appear as an inductive load, which is useful in circuits that require a controlled soft start. A detailed analysis can be found here.
### Sensors
• To detect sound, e.g. piezoelectric microphones (sound waves bend the piezoelectric material, creating a changing voltage) and piezoelectric pickups for electrically amplified guitars.
• Piezoelectric elements are also used in the generation of sonar waves. Piezoelectric microbalances are used as very sensitive chemical and biological sensors.
• Piezoelectric elements are used in electronic drum pads to detect the impact of the drummer's sticks.
### Actuators
As very high voltages correspond to only tiny changes in the width of the crystal, this width can be changed with better-than-micrometer precision, making piezo crystals the most important tool for positioning objects with extreme accuracy.
• Loudspeaker: Voltages are converted to mechanical movement of a piezoelectric polymer film.
• Piezoelectric elements can be used in laser mirror alignment, where their ability to move a large mass (the mirror mount) over microscopic distances is exploited to electronically align some laser mirrors. By precisely controlling the distance between mirrors, the laser electronics can accurately maintain optical conditions inside the laser cavity to optimize the beam output.
• A related application is the acousto-optic modulator , a device that vibrates a mirror to give the light reflected off it a Doppler shift. This is useful for fine-tuning a laser's frequency.
### Frequency standards
• Quartz clocks employ a tuning fork made from quartz that uses a combination of both direct and converse piezoelectricity to generate a regularly timed series of electrical pulses that is used to mark time. The quartz crystal (like any elastic material) has a precisely defined natural frequency (caused by its shape and size) at which it prefers to oscillate, and this is used to stabilize the frequency of a periodic voltage applied to the crystal.
### Piezoelectric motors
Types of piezoelectric motor include the well-known travelling-wave motor used for auto-focus in reflex cameras, inchworm motors for linear motion, and rectangular four-quadrant motors with high power density (2.5 watt/cm³) and speed ranging from 10 nm/s to 800 mm/s. All these motors work on the same principle. Driven by dual orthogonal vibration modes with a phase shift of 90°, the contact point between two surfaces vibrates in an elliptical path, producing a frictional force between the surfaces. Usually, one surface is fixed causing the other to move. In most piezoelectric motors the piezoelectric crystal is excited by a sine wave signal at the resonant frequency of the motor. Using the resonance effect, a much lower voltage can be used to produce a high vibration amplitude.
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Excel Guides
## Converting Numbers to Strings in Excel
Converting numbers to strings in Excel is a relatively simple process. There are a few different ways to go about doing this, and the method you choose will likely depend on the specific situation in which you find yourself. For example, if you need to convert a column of numbers to strings, you might use the Text to Columns command. On the other hand, if you need to convert only a few numbers to strings, you can use the CONVERT function. Let's take a closer look at both of these methods.
## Using the Text to Columns Command
If you need to convert an entire column of numbers to strings, the Text to Columns command is probably your best bet. This command can be found under the Data tab on the Excel ribbon. Once you've selected your data range and clicked on the Text to Columns button, you'll be presented with a few different options for how your data should be split up. For our purposes, we're interested in the Delimited option.
Once you select the Delimited option and click Next, you'll be given a list of different delimiters that Excel can use to split up your data. Since we're dealing with numbers, we want to make sure that none of these delimiters are selected. Once you've deselected all of the delimiters, click Finish. Your data should now be converted to strings!
## Using the CONVERT Function
If you only need to convert a few numbers to strings, using the CONVERT function might be a better option. This function takes two arguments: the value that you want to convert and the desired output format. In our case, we want to convert a number to a string, so we'll use the following syntax:
`=CONVERT(value, "string")`
Example: `=CONVERT(A1, "string")`
Description:
• The `value `(in our example, cell `A1 `) is the number that you want to convert.
• The word "`string `" in quotation marks tells Excel that we want to output a string.
Note:
• The CONVERT function will only work if your regional settings are set to English (United States). If they are not, you can change them by going to File > Options > Advanced > Editing Options > Use System Separators.
### Excel
Get started with Causal today.
Build models effortlessly, connect them directly to your data, and share them with interactive dashboards and beautiful visuals.
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Cody
# Problem 615. Nilpotent matrix
Solution 834447
Submitted on 21 Feb 2016 by Gustave Udahemuka
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; y_correct = false; assert(isequal(isnilpotent(x),y_correct))
2 Pass
%% x = gallery('chebspec',5,0); y_correct = true; assert(isequal(isnilpotent(x),y_correct))
3 Pass
%% x = gallery('chebspec',3,0); y_correct = true; assert(isequal(isnilpotent(x),y_correct))
4 Pass
%% x = [1 0 0;0 2 0; 0 0 -3]; y_correct = false; assert(isequal(isnilpotent(x),y_correct))
5 Pass
%% x = [6 -9; 4 -6]; y_correct = true; assert(isequal(isnilpotent(x),y_correct))
6 Pass
%% x = rand(50); y_correct = false; assert(isequal(isnilpotent(x),y_correct))
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• Asymptotes of Secant, Cosecant, and Cotangent - Concept
Math›Precalculus›Trigonometric Functions
How to find the vertical asymptotes of secant, cosecant, and cotangent.
• Asymptotes of Secant, Cosecant, and Cotangent - Concept
Math›Trigonometry›Trigonometric Functions
How to find the vertical asymptotes of secant, cosecant, and cotangent.
• Graphing the Reciprocal Trigonometric Functions - Problem 2
Math›Precalculus›Trigonometric Functions
How to graph y = csc q.
• Graphing the Reciprocal Trigonometric Functions - Problem 2
Math›Trigonometry›Trigonometric Functions
How to graph y = csc q.
• Graphing the Reciprocal Trigonometric Functions - Problem 1
Math›Trigonometry›Trigonometric Functions
How to graph y = sec q.
• Graphing the Reciprocal Trigonometric Functions - Problem 1
Math›Precalculus›Trigonometric Functions
How to graph y = sec q.
• Graphing the Reciprocal Trigonometric Functions - Problem 3
Math›Precalculus›Trigonometric Functions
How to graph y = cot q.
• Graphing the Reciprocal Trigonometric Functions - Problem 3
Math›Trigonometry›Trigonometric Functions
How to graph y = cot q.
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Algebra Questions
Question Number 100793 by bramlex last updated on 28/Jun/20
Commented by bobhans last updated on 28/Jun/20
$$\mathrm{The}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{isosceles}\:\Delta\mathrm{MBC}\:=\:\mathrm{5}+\mathrm{5}+\mathrm{6}\:=\:\mathrm{16}\: \\$$
Answered by Dwaipayan Shikari last updated on 28/Jun/20
$${radius}\:{of}\:{circle}=\mathrm{3}\:{unit} \\$$$${so}\:\:\sqrt{\left(\mathrm{7}−\mathrm{3}\right)^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} }=\mathrm{5}\:{unit} \\$$$${perimetre}\:{of}\:\bigtriangleup\mathrm{BMC}=\mathrm{5}+\mathrm{5}+\mathrm{6}=\mathrm{16}\:{unit} \\$$
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# Can someone explain how to do this
• Sep 14th 2009, 09:52 AM
xpack
Can someone explain how to do this
Can someone explain how to do this:
Consider the following parametric equations.
x = 5cos(θ)
y = 9sin(θ)
0 ≤ θ ≤ 2π
(a) Eliminate the parameter to find the Cartesian equation of the curve.
• Sep 14th 2009, 10:24 AM
Plato
Recall that $\displaystyle \cos^2(\theta)+\sin^2(\theta)=1$.
Then write $\displaystyle 9x=45\cos(\theta)~\&~5y=45\sin(\theta)$
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## Q. Estimate the following by rounding off both the numbers to the nearest thousand. a. 2963+ 932 b. 3796-812 c. 1862-1023
Question
Q. Estimate the following by rounding off both the numbers to the nearest thousand.
a. 2963+ 932
b. 3796-812
c. 1862-1023
d. 4396+1562
in progress 0
1 month 2021-08-19T22:00:33+00:00 1 Answer 0 views 0
4000, 3000, 1000, 6000
Step-by-step explanation:
By calculation normally find the answers
a) 3895
b) 2984
c) 839
d) 5958
after rounding off
a 4000
b 3000
c 1000
d 6000
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Liverpoololympia.com
Just clear tips for every day
What is the doubling and halving method?
What is the doubling and halving method?
To use halving and doubling, you simply half one of the factors and double the other. Take this example. To solve 25×16, we could double the 25 to make 50 and then half the 16 to make 8. Suddenly this problem becomes much easier to solve!
How do you teach halving and doubling?
To introduce this activity, children could explore ‘doubles’ and ‘halves’ in the world around them. Invite them to find items that could be described in this way. For example, pairs of items, the number of eyelets on each side of a shoe for laces, legs on each side of a spider, dots on a ‘double’ domino.
What is halving method?
The bisection method, which is alternatively called binary chopping, interval halving, is one type of incremental search method in which the interval is always divided in half. If a function changes sign over an interval, the function value at the midpoint is evaluated.
Does doubling and halving work in division?
Doubling and halving Find half of even numbers to 100 using partitioning. Use halving as a strategy in dividing by 2, e.g. 36 ÷ 2 is half of 36. Grouping Recognise that division is not commutative, e.g. 16 ÷ 8 does not equal 8 ÷ 16.
What is the doubling strategy?
Doubling is a strategy that people of all ages frequently use. Young children first learn doubles as an addition of two groups. What multiplication facts can be used by using a doubling strategy? If you said the twos, fours, and eights facts then you are correct! That’s what makes this strategy so powerful.
What is the doubling sequence?
The doubling sequence is my name for the sequence of powers of 2. D = 〈1, 2, 4, 8, 16, 32, 64, 128, . . .〉 Term in the doubling sequence form the basis for the binary. number system. Any natural number can are written as a sum of.
How do you introduce halving?
To teach the halving of smaller numbers, count out the required amount of counters and move them one by one into two equal piles. For larger numbers, break them down into tens and ones, halve each separately and then add up the result. It may help to write this down.
What is the doubling method?
As long as you know a few multiplication facts, you can use this strategy to figure out and learn new facts. To use this strategy, find a set of facts that is known to you. Then, double one factor (or add the number to itself), and double the product, or answer from the first set of facts.
What is a doubling function?
A double exponential function is a constant raised to the power of an exponential function. The general formula is. (where a>1 and b>1), which grows much more quickly than an exponential function. For example, if a = b = 10: f(0) = 10.
What is the relationship between growth rate and doubling time?
There is an important relationship between the percent growth rate and its doubling time known as “the rule of 70”: to estimate the doubling time for a steadily growing quantity, simply divide the number 70 by the percentage growth rate.
What is the concept of doubling?
(ˈdʌblɪŋ ) noun. the activity of multiplying by two or repeating. her doubling of her prayers for him.
What is the difference between doubling time and half life?
The doubling time is a characteristic unit (a natural unit of scale) for the exponential growth equation, and its converse for exponential decay is the half-life. For example, given Canada’s net population growth of 0.9% in the year 2006, dividing 70 by 0.9 gives an approximate doubling time of 78 years.
How do you calculate doubling?
What is Doubling Time and How is it Calculated?
1. Doubling time is the amount of time it takes for a given quantity to double in size or value at a constant growth rate.
2. Note: growth rate (r) must be entered as a percentage and not a decimal fraction.
3. dt = 70/r.
4. 35 = 70/2.
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# Slope Intercept Form Y Intercept 8 Facts That Nobody Told You About Slope Intercept Form Y Intercept
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https://www.lambdatest.com/automation-testing-advisor/selenium/methods/org.assertj.core.api.AbstractComparableAssert.inHexadecimal
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How to use inHexadecimal method of org.assertj.core.api.AbstractComparableAssert class
Best Assertj code snippet using org.assertj.core.api.AbstractComparableAssert.inHexadecimal
Source:AbstractComparableAssert.java
`...81 this.comparables = Comparables.instance();82 return myself;83 }84 @Override85 public S inHexadecimal() {86 return super.inHexadecimal();87 }88 @Override89 public S inBinary() {90 return super.inBinary();91 }92}...`
Using AI Code Generation
`1assertThat("1234").inHexadecimal().isEqualTo(4660);2assertThat("1234").inOctal().isEqualTo(668);3assertThat("1234").inBinary().isEqualTo("10011010010");4assertThat(Arrays.asList("a", "b", "c")).isIn("a", "c");5assertThat(Arrays.asList("a", "b", "c")).isNotIn("d", "e");6assertThat(Arrays.asList("a", "b", "c")).isSubsetOf("a", "b", "c", "d", "e");7assertThat(Arrays.asList("a", "b", "c")).isNotSubsetOf("d", "e");8assertThat(Arrays.asList("a", "b", "c")).hasSameElementsAs(Arrays.asList("c", "a", "b"));9assertThat(Arrays.asList("a", "b", "c")).containsExactly("a", "b", "c");10assertThat(Arrays.asList("a", "b", "c")).containsExactlyInAnyOrder("c", "b", "a");11assertThat(Arrays.asList("a", "b", "c")).containsExactlyInAnyOrderElementsOf(Arrays.asList("c", "b", "a"));12assertThat(Arrays.asList("a", "b", "c")).containsExactly("a", "b", "c");`
Using AI Code Generation
`1import org.assertj.core.api.AbstractComparableAssert;2import static org.assertj.core.api.Assertions.assertThat;3public class AssertJHexadecimal {4 public static void main(String[] args) {5 assertThat(0x1A).inHexadecimal().isEqualTo(0x1A);6 assertThat(0x1A).inHexadecimal().isNotEqualTo(0x1B);7 assertThat(0x1A).inHexadecimal().isGreaterThan(0x19);8 assertThat(0x1A).inHexadecimal().isLessThan(0x1B);9 assertThat(0x1A).inHexadecimal().isGreaterThanOrEqualTo(0x1A);10 assertThat(0x1A).inHexadecimal().isGreaterThanOrEqualTo(0x19);11 assertThat(0x1A).inHexadecimal().isLessThanOrEqualTo(0x1A);12 assertThat(0x1A).inHexadecimal().isLessThanOrEqualTo(0x1B);13 }14}15import org.assertj.core.api.AbstractComparableAssert;16import static org.assertj.core.api.Assertions.assertThat;17public class AssertJHexadecimal {18 public static void main(String[] args) {19 assertThat(0x1A).inHexadecimal().isEqualTo(0x1A);20 assertThat(0x1A).inHexadecimal().isNotEqualTo(0x1B);21 assertThat(0x1A).inHexadecimal().isGreaterThan(0x19);22 assertThat(0x1A).inHexadecimal().isLessThan(0x1B);23 assertThat(0x1A).inHexadecimal().isGreaterThanOrEqualTo(0x1A);24 assertThat(0x1A).inHexadecimal().isGreaterThanOrEqualTo(0x19);25 assertThat(0x1A).inHexadecimal().isLessThanOrEqualTo(0x1A);26 assertThat(0x1A).inHexadecimal().isLessThanOrEqualTo(0x1B);27 }28}`
Using AI Code Generation
`1org.assertj.core.api.AbstractComparableAssert inHexadecimal = new org.assertj.core.api.AbstractComparableAssert() {2 protected void isNotNull() {3 }4 protected void isInstanceOfSatisfying(Object o, Condition condition) {5 }6}7org.assertj.core.api.AbstractComparableAssert isBetween = new org.assertj.core.api.AbstractComparableAssert() {8 protected void isNotNull() {9 }10 protected void isInstanceOfSatisfying(Object o, Condition condition) {11 }12}13org.assertj.core.api.AbstractComparableAssert isEqualByComparingTo = new org.assertj.core.api.AbstractComparableAssert() {14 protected void isNotNull() {15 }16 protected void isInstanceOfSatisfying(Object o, Condition condition) {17 }18}19org.assertj.core.api.AbstractComparableAssert isStrictlyBetween = new org.assertj.core.api.AbstractComparableAssert() {20 protected void isNotNull() {21 }22 protected void isInstanceOfSatisfying(Object o, Condition condition) {23 }24}25org.assertj.core.api.AbstractComparableAssert isNotBetween = new org.assertj.core.api.AbstractComparableAssert() {26 protected void isNotNull() {27 }28 protected void isInstanceOfSatisfying(Object o, Condition condition) {29 }30}31org.assertj.core.api.AbstractComparableAssert isNotStrictlyBetween = new org.assertj.core.api.AbstractComparableAssert() {32 protected void isNotNull() {33 }34 protected void isInstanceOfSatisfying(Object o, Condition condition) {35 }36}37org.assertj.core.api.AbstractComparableAssert isZero = new org.assertj.core.api.AbstractComparableAssert() {38 protected void isNotNull() {39 }`
Using AI Code Generation
`1public void givenHexadecimalValue_whenEqual_thenCorrect() {2 assertThat("0x1a").inHexadecimal().isEqualTo("0x1a");3}4public void givenHexadecimalValue_whenNotEqual_thenCorrect() {5 assertThat("0x1a").inHexadecimal().isNotEqualTo("0x1b");6}7public void givenHexadecimalValue_whenLessThan_thenCorrect() {8 assertThat("0x1a").inHexadecimal().isLessThan("0x1b");9}10public void givenHexadecimalValue_whenLessThanOrEqualTo_thenCorrect() {11 assertThat("0x1a").inHexadecimal().isLessThanOrEqualTo("0x1a");12}13public void givenHexadecimalValue_whenGreaterThan_thenCorrect() {14 assertThat("0x1a").inHexadecimal().isGreaterThan("0x19");15}16public void givenHexadecimalValue_whenGreaterThanOrEqualTo_thenCorrect() {17 assertThat("0x1a").inHexadecimal().isGreaterThanOrEqualTo("0x1a");18}19public void givenHexadecimalValue_whenBetween_thenCorrect() {20 assertThat("0x1a").inHexadecimal().isBetween("0x19", "0x1b");21}`
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Author Topic: Collision Problems (Read 2994 times)
KittenKoder
• int
• Posts: 66
• I Am No One Else
Collision Problems
« on: December 01, 2012, 04:20:27 am »
Not sure if it's my code, or jPCT, as I still can't figure out exactly what's going on. But it only happens with scaled objects, almost as if the collision detection is scaled twice the object's scale. I'm using the ellipsoid detection, and the moving object is fine on objects scaled 1.0, it's any other scale value that has the effect. That's the best I can do to describe it. Any ideas?
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
KittenKoder
• int
• Posts: 66
• I Am No One Else
Re: Collision Problems
« Reply #1 on: December 01, 2012, 06:20:10 am »
After looking over some things, how is the collision ellipsoid used? Is it a number of world units or a multiplier of some sort? That could be my problem, I'm basing the ellipsoid on the size of the object, for the one that's moving, based off the bounding box, but now I think this may not be correct.
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
EgonOlsen
• Posts: 11829
Re: Collision Problems
« Reply #2 on: December 01, 2012, 08:20:06 am »
The ellipsoid is given in units in object space. If you an object, you have to scale the ellipsoid as well.
KittenKoder
• int
• Posts: 66
• I Am No One Else
Re: Collision Problems
« Reply #3 on: December 01, 2012, 08:42:37 am »
The ellipsoid is given in units in object space. If you an object, you have to scale the ellipsoid as well.
Ah, well the object that's using the ellipse isn't being scaled, oddly Blender units import to jPCT very well. So the bounding box should be usable for creating an ellipse, then scaling that as the object is? Or would it perform better to have the ellipsoid represent the aspects of the object?
Sadly this doesn't seem to help much with the strange problem I am having of the objects being collided with. The collision is still farther from the objects than the actual polygons.
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
EgonOlsen
• Posts: 11829
Re: Collision Problems
« Reply #4 on: December 01, 2012, 11:49:12 am »
Just fiddle around with the ellipsoid sizes. An ellipsoid as always an approximation of the object. If you make it so that it includes all polygons, it might be (depending on the object) too large for realistic collision behaviour. Also keep in mind (just in case you haven't) that the size of the ellipsoid is its radius, not its diameter.
KittenKoder
• int
• Posts: 66
• I Am No One Else
Re: Collision Problems
« Reply #5 on: December 01, 2012, 08:08:52 pm »
Just fiddle around with the ellipsoid sizes. An ellipsoid as always an approximation of the object. If you make it so that it includes all polygons, it might be (depending on the object) too large for realistic collision behaviour. Also keep in mind (just in case you haven't) that the size of the ellipsoid is its radius, not its diameter.
That's just it, after some testing, it's not the ellipsoid, it effects camera collision to. The solid, static, map object collision is really far from the actual polygons, it's noticeable when you scale them by an average of 8. Anything not scaled is not effected. So the ellipsoid of the moving object, based on what yo have stated here, is not the issue, and I can't figure out what is the issue. If I can't scale map object arbitrarily then I'm probably just going to have to get off my lazy but and write it all in raw OGL so I can.
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
EgonOlsen
• Posts: 11829
Re: Collision Problems
« Reply #6 on: December 01, 2012, 09:11:42 pm »
I'm a little confused about what writing stuff directly in OpenGL has to do with collision detection, but anyway... I'm still not sure what exactly you mean. You scale the map, i.e. the objects you want to collide with and if you do that, the collision happens too early, i.e. when there shouldn't be any collision at all? Is that the problem? Do you have a test case for this?
KittenKoder
• int
• Posts: 66
• I Am No One Else
Re: Collision Problems
« Reply #7 on: December 01, 2012, 09:27:50 pm »
I'm a little confused about what writing stuff directly in OpenGL has to do with collision detection, but anyway... I'm still not sure what exactly you mean. You scale the map, i.e. the objects you want to collide with and if you do that, the collision happens too early, i.e. when there shouldn't be any collision at all? Is that the problem? Do you have a test case for this?
It would be easier to start from scratch than to attempt to find a flaw that is possibly part of the library. I'm stressed because it all worked until this one point, which means I'd have to go back and rewrite some of the original code as well as develop loaders and all that.
The collision is happening too early, and I have everything at the lowest settings. It only happens when the objects on the map are scaled, as I said an average of 8 is when it's noticeable, I didn't notice it until I tested an area with objects that had to be scaled that much to avoid memory consumption, reusing the meshes. Just load a map with a bunch of objects and scale each object by at least 8 to 11, take another object you have the correct ellipsoid for, tested in environments without scaled objects, and you should see it. To better simulate what I am seeing you should have different scale values for the map objects, and shared meshes, though even objects without shared meshes still have the error. The case I have right now would be a huge upload/download, because it's part of the demo version of the game and depends on all the extra stuff to load properly.
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
EgonOlsen
• Posts: 11829
Re: Collision Problems
« Reply #8 on: December 01, 2012, 10:17:44 pm »
I'll look into it...
EgonOlsen
• Posts: 11829
Re: Collision Problems
« Reply #9 on: December 01, 2012, 11:15:40 pm »
It should fix the problem (at least it did in my tests), but for high scaling values, you might now run into accuracy problems with floating point arithmetics. Try to avoid scales that are much higher then 10, if possible. Consider to use rescaled instances of your objects instead (maybe one for 1...10 and one for 10...20 or something like that).
EgonOlsen
• Posts: 11829
Re: Collision Problems
« Reply #10 on: December 01, 2012, 11:26:01 pm »
BTW: I still fail to see the relation between doing stuff in OpenGL and collision detection. One has nothing to do with the other. The means that jPCT offers to do collision detection can be used, but they don't have to be used. You could use any other other physics library out there, if you want to. jPCT's own ellipsoid collision detection is based on http://www.peroxide.dk/papers/collision/collision.pdf with some additions and fixes.
KittenKoder
• int
• Posts: 66
• I Am No One Else
Re: Collision Problems
« Reply #11 on: December 02, 2012, 01:00:40 am »
BTW: I still fail to see the relation between doing stuff in OpenGL and collision detection. One has nothing to do with the other. The means that jPCT offers to do collision detection can be used, but they don't have to be used. You could use any other other physics library out there, if you want to. jPCT's own ellipsoid collision detection is based on http://www.peroxide.dk/papers/collision/collision.pdf with some additions and fixes.
If I have to override the collision detection myself then I may as well write the whole thing from scratch because the only thing I'd be using from jPCT is the loader and OpenGL work. Also there are a lot of rendering capabilities I am not finding a way to access, though I don't remember how to do it off hand, I will have to look through old code again, there's a way to change the color of shadows on individual objects, as well as altering the specular color using OpenGL which would be a nice effect to have access to. Also there's non-uniform scaling, which offers a lot of useful manipulations.
I mean, jPCT is great, don't get me wrong, but some of the features I need just seem to be out of reach. If it was Open Source I'd go through your code and figure out ways to implement them without having to rewrite anything. Since every time I attempt to implement shaders the program locks up, and I'm almost clueless in that arena. I can't find a way to make it work the way I need to. But the collision is too important to overlook, I need that to work as expected, and I need it simple and integrated. Really, I'm just seeing my code use less and less of jPCT with each new development, and having to do a lot of it in my library, which also requires me searching for a method to override the jPCT portions. My library, and game, is just too involved to have to deal with headaches like this. I mean, seriously, the collision just went boom.
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
EgonOlsen
• Posts: 11829
Re: Collision Problems
« Reply #12 on: December 02, 2012, 09:15:45 am »
....but did the fix that i posted above work?
KittenKoder
• int
• Posts: 66
• I Am No One Else
Re: Collision Problems
« Reply #13 on: December 02, 2012, 10:29:32 pm »
....but did the fix that i posted above work?
Sorry, yes it did actually. Now it works perfectly.
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
KittenKoder
• int
• Posts: 66
• I Am No One Else
Re: Collision Problems
« Reply #14 on: December 02, 2012, 10:35:16 pm »
The problem with using multiple meshes is that it would seriously add drain to the GPU memory. Common buildings can have a size difference between 0.05 and 20 on the map, since the buildings are the main filler for the maps it would not be economical to create multiple meshes for each one, as I said, I just have to stop being lazy and figure out this GPU in OpenGL myself anyway. Though I may release the jPCT version first just to get it out there, I have to figure out how to budget new eyeglasses before I can continue testing anyway, not the glasses themselves but the exam, here in Seattle those are \$70 or more, even at Walmart, and due to medical problems my budget only has a \$50 leeway. But until I get new glasses I can't play 3D games at all, my prescription changed a lot in the last few years and I hadn't noticed since I normally hate 3D games. lol
Yeah, I'm an old skool gamer, but I have to adapt to the market if I want to get back into development.
When life throws you lemons, make lemon juice, then drop life into a pile of razors and pour the lemon juice over it.
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# Line intersections
Hello!
Recently i was having some issues with detecting if two line intersect eachother. What i’ve done so far is to give the two lines a formula: ax+b, and defined all the variables exept for x.
So far i’m able to consistantly determine where it crosses, but: the lines are practically infinite, i’ve tried to comapre the Xsize of the UIframe with the magnitude of the subtraction from the absoluteposition where it intersects and the absoluteposition of the UIFrame. but that didn’t yield any satisfactory results ((PosA-PosB).magnitude).
So what other ways do i have in order to check if two lines intersect within their lenght?
I’ve aslo check wikipedia, but that gave me more of a headache than that it helped.
2 Likes
I’m not sure if I inderstood your problem correctly. If you are trying to find out if the intersection is inside the frame, can’t you just check if the intersection point’s x coordinate and y coordinate are both inside it?
``````local function isPointInFrame(p, frame)
local absPos, halfAbsSize = frame.AbsolutePosition, frame.AbsoluteSize/2
local xPos, yPos = absPos.X, absPos.Y
local hXSize, hYSize = halfSize.X, halfSize.Y
local px, py = point.X, point.Y
return px >= xPos-hXSize and px <= xPos+hXSize and py >= yPos-hYSize and py <= yPos+hYSize
end``````
1 Like
No. that won’t help. In this example, there are 3 linepeices, 2 of wich intersect eachother. what i need to check is if two line intersect eachother within their lenghts.
Because what happens now, is that it would expect all 3 lines to cross eachother, though at different points, despite the green line already having ended well before they could even come close.
2 Likes
Is the anchor point of the frame `Vector2.new(0, 0.5)`? And are you doing the distance check for both lines involved in the calculation?
1 Like
it’s .5,.5
it’s really weird though, not because it goes slightly out of bounds, but because it straight up goes across the screen.
1 Like
If it’s Vector2.new(.5, .5), then you’ll need to use half of the x Size of the frame in checking whether the point is within the length.
1 Like
here’s the code. RC is a in y=ax+b.
It didn’t work.
``````function FindIntersections (RC,B)
for i,LineFolder in pairs(Lines:GetChildren()) do
for o,LinePeice in pairs(LineFolder:GetChildren()) do
if LinePeice ~= CurrentLine then
local LineRC = LinePeice.RCValue.Value
if LineRC ~= RC then
local LineB = LinePeice.BValue.Value
local LinePos = LinePeice.AbsolutePosition
local XIntersect = (RC-LineRC)/(LineB-B)
local IntersectPos = Vector2.new(XIntersect,RC*XIntersect+B)
print(IntersectPos,"//",(IntersectPos-LinePos).Magnitude)
if (IntersectPos-LinePos).Magnitude < LinePeice.AbsoluteSize.X*.5 then
print("Intersected at: ",IntersectPos)
return true
end
end
end
end
end
return false
end
``````
2 Likes
I edited it to do the check for both lines. Does it now work?
``````function FindIntersections (RC,B)
for i,LineFolder in pairs(Lines:GetChildren()) do
for o,LinePeice in pairs(LineFolder:GetChildren()) do
if LinePeice ~= CurrentLine then
local LineRC = LinePeice.RCValue.Value
if LineRC ~= RC then
local LineB = LinePeice.BValue.Value
local LinePos = LinePeice.AbsolutePosition
local XIntersect = (RC-LineRC)/(LineB-B)
local IntersectPos = Vector2.new(XIntersect,RC*XIntersect+B)
print(IntersectPos,"//",(IntersectPos-LinePos).Magnitude)
local mag1 = (IntersectPos-LinePos).Magnitude
local mag2 = (InterSectPos-CurrentLine.AbsolutePosition).Magnitude
if mag1 < LinePeice.AbsoluteSize.X*.5 and and mag2 < CurrentLine.AbsoluteSize.X*.5 then
print("Intersected at: ",IntersectPos)
return true
end
end
end
end
end
return false
end``````
1 Like
Sadly, this doesn’t work. I know for sure that a and b in y=ax+b are defined correctly, because whenever the rotation of the line = 0 or 180, a = 0, and b = it’s absoluteposition.Y.
so that con’t be the issue either. i cannot use raycasting either because it’s 2d.
1 Like
Distance checking doesn’t seem like the most effective solution as you’d have to account for the rotation of the line too, but a better method is using some vector maths:
Say you have your four points for two lines in the screen space. You can get the lines as vectors by subtracting the lines’ points
We know that:
``````b = a + r;
d = c + s;
``````
If we consider two different, randoms scalars, `t` and `u` and multiply them to our `r` and `s` line, eventually, we might reach an equality:
``````a + tr = c + us
``````
The lines are intersecting if both conditions are met:
• 0 <= t <= 1;
• 0 <= u <= 1;
`t` and `u` can be calculated by getting rid of one of them in the equality. Crossing a vector by itself will return a vector with magnitude 0 and `u` is the scalar coefficient of vector `s`. Crossing `s` by itself will get rid of `us` on the right side and we’re left with:
``````(a + tr) x s = (c + us) x s
a x s + t(r x s) = c x s
``````
Solving for `t`:
``````t(r x s) = (c - a) x s
t = ((c - a) x s) / (r x s)
``````
Solving for `u` (Alternatively, you can plug in `t` into any of the equations above to solve for `t`):
``````a + tr = c + us
(a + tr) x r = (c + us) x r
a x r = c x r + u(s x r)
(a - c) x r = u(s x r)
u = (a - c) x r / (s x r)
``````
You have both unknowns now. Now, to check if they’re intersecting:
``````if (t >= 0 and t <= 1) or (u >= 0 and u <= 1) then
-- intersecting
end
``````
For the code part now:
``````local function intersecting(a, b, c, d)
local u = ((a - c):Cross(r)) / s:Cross(r)
local r = ((c - a):Cross(s)) / r:Cross(s)
if (t >= 0 and t <= 1) or (u >= 0 and u <= 1) then
return true
end
return false
end
``````
2 Likes
After having done some editing to the function so it can be integrated into my script, i ended up with a few questions and errors. First: r was defined as a vector, but u wasn’t.
t wasn’t defined, but it was a scalar, so i hoped to be able to replace t with r, but i don’t know if that would work.
Vector3:Cross() returns a vector3 that in this case NANed out. so i used a vector2 instead. can’t wait until roblox makes vector2values a thing.
``````local function FindIntersections(a, b)
for i,Folder in pairs(Lines:GetChildren()) do
for o,LinePeice in pairs(Folder:GetChildren()) do
if LinePeice ~= CurrentLine then
local c = Vector3.new(LinePeice.StartPos.Value.X,LinePeice.StartPos.Value.Y)
local d = Vector3.new(LinePeice.EndPos.Value.X,LinePeice.EndPos.Value.Y)
local r = b-a
local s = d-c
local u = ((a - c):Cross(r)) / s:Cross(r)
local t = ((c - a):Cross(s)) / r:Cross(s)
if (t >= 0 and t <= 1) or (u >= 0 and u <= 1) then
return true
end
end
end
end
return false
end
``````
Anyways, here’s the entire place. i haven’t focussed on optimizing yet.mathstoobigformybrain.rbxl (29.6 KB)
I didn’t really read through the script when I was writing, leading to errors. So, here’s the fixed intersection function:
``````local function intersecting(a, b, c, d)
local r, s = b - a, d - c
local u = ((a - c):Cross(r)) / s:Cross(r)
local t = ((c - a):Cross(s)) / r:Cross(s)
if (t >= 0 and t <= 1) and (u >= 0 and u <= 1) then
return true
end
return false
end
``````
I tested it this time, and it works. Place file: Line Intersections.rbxl (25.6 KB)
I made it so when the two lines are intersecting, a textlabel in the bottom-left corner will appear and say “intersecting”, else it’ll disappear:
https://gyazo.com/b7fdbb09337c1d2feb162788d9f6447d
It was really supposed to be a Vector2, not a Vector3. In my opinion, Vector3Value and all of the other values are quite niche
2 Likes
Thanks for the help. it seems like i have to edit my script quite a bit in order to get it working the way i needed it to, but it detects intersecting lines, and that’s a good place to start from!
Also thanks for teaching me this stuff. i would’ve never known what the cross product could possibly be before our maths class will eventually feature it without it.
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# What are the effects of reflection of light?
## What are the effects of reflection of light?
Light reflects from a smooth surface at the same angle as it hits the surface. For a smooth surface, reflected light rays travel in the same direction. This is called specular reflection. For a rough surface, reflected light rays scatter in all directions.
### What is the result of reflection in the point of an angle?
The law of reflection states that the angle of reflection equals the angle of incidence—θr = θi. The angles are measured relative to the perpendicular to the surface at the point where the ray strikes the surface.
What is the relationship between the angle of the incident light and reflected light?
The law of reflection states that when a ray of light reflects off a surface, the angle of incidence is equal to the angle of reflection.
What will happen to the angle of reflection when the angle of incidence is increased?
As the angle is increased to greater and greater angles, we would begin to observe less refraction and more reflection. That is, as the angle of incidence is increased, the brightness of the refracted ray decreases and the brightness of the reflected ray increases.
## How do you relate angle of reflection and angle of incidence?
The angle of incidence is the angle between this normal and the incident ray; the angle of reflection is the angle between this normal and the reflected ray. According to the law of reflection, the angle of incidence equals the angle of reflection.
### What is the relationship between angle of incidence and angle of refraction?
The relationship between the angle of incidence and angle of refraction is explained by Snell’s law, which states that the ratio of the sine of the angle of refraction and the sine of the angle of incidence is always constant and equivalent to the ratio of phase velocities of the two mediums it is passing through.
What is the effect on the angle of incidence of light is made greater than the critical angle?
When the angle of incidence is greater than the critical angle none of it is refracted, the ray is totally internally reflected, and the law of reflection is obeyed, i = r.
When light hits an interface part of it is reflected and part of it is transmitted?
When light traveling in one transparent medium encounters a boundary with a second transparent medium (e.g., air and glass), a portion of the light is reflected and a portion is transmitted into the second medium.
## How does the angle of incidence affect the angle of refraction?
Explanation: As the angle of incidence increases, the angle of refraction also increases proportionally to the increase of incidence. Snell’s Law determines the angle of refraction based on the the angle of incidence, and the index of refraction of both mediums.
### How do you relate angle of reflection and angle of incidence what will be the angle of reflection when angle of incidence is 160 degrees to 0 degrees?
First law of reflection says that the angle of incidence is equal to the angle of reflection. ∠i=∠r here, I am the angle of incidence and r is the angle of reflection.
What is the relation between angle of incidence and angle of refraction when light travels from denser to rarer medium?
And when light travels from rarer to denser medium it shifts towards the normal. Complete answer: When a light travels from denser to rarer medium the angle of refraction is more than the angle of incidence.
What is the relation between angle of deviation and angle of incidence?
The angle through which the emergent ray deviates from the direction of incident ray is called angle of deviation ‘d’. As the angle of incidence is increased, angle of deviation ‘d’ decreases and reaches minimum value. If the angle of incidence is further increased, the angle of deviation is increased.
https://www.youtube.com/watch?v=fm__GAlrBuQ
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src/HOL/Probability/Caratheodory.thy
author hoelzl Tue May 04 18:19:24 2010 +0200 (2010-05-04) changeset 36649 bfd8c550faa6 parent 35704 5007843dae33 child 37032 58a0757031dd permissions -rw-r--r--
Corrected imports; better approximation of dependencies.
3 theory Caratheodory
4 imports Sigma_Algebra SeriesPlus
5 begin
7 text{*From the Hurd/Coble measure theory development, translated by Lawrence Paulson.*}
9 subsection {* Measure Spaces *}
11 text {*A measure assigns a nonnegative real to every measurable set.
12 It is countably additive for disjoint sets.*}
14 record 'a measure_space = "'a algebra" +
15 measure:: "'a set \<Rightarrow> real"
17 definition
18 disjoint_family_on where
19 "disjoint_family_on A S \<longleftrightarrow> (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
21 abbreviation
22 "disjoint_family A \<equiv> disjoint_family_on A UNIV"
24 definition
25 positive where
26 "positive M f \<longleftrightarrow> f {} = (0::real) & (\<forall>x \<in> sets M. 0 \<le> f x)"
28 definition
31 (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
32 \<longrightarrow> f (x \<union> y) = f x + f y)"
34 definition
37 (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
38 disjoint_family A \<longrightarrow>
39 (\<Union>i. A i) \<in> sets M \<longrightarrow>
40 (\<lambda>n. f (A n)) sums f (\<Union>i. A i))"
42 definition
43 increasing where
44 "increasing M f \<longleftrightarrow> (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<subseteq> y \<longrightarrow> f x \<le> f y)"
46 definition
49 (\<forall>x \<in> sets M. \<forall>y \<in> sets M. x \<inter> y = {}
50 \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
52 definition
55 (\<forall>A. range A \<subseteq> sets M \<longrightarrow>
56 disjoint_family A \<longrightarrow>
57 (\<Union>i. A i) \<in> sets M \<longrightarrow>
58 summable (f o A) \<longrightarrow>
59 f (\<Union>i. A i) \<le> suminf (\<lambda>n. f (A n)))"
61 definition
62 lambda_system where
63 "lambda_system M f =
64 {l. l \<in> sets M & (\<forall>x \<in> sets M. f (l \<inter> x) + f ((space M - l) \<inter> x) = f x)}"
66 definition
67 outer_measure_space where
68 "outer_measure_space M f \<longleftrightarrow>
69 positive M f & increasing M f & countably_subadditive M f"
71 definition
72 measure_set where
73 "measure_set M f X =
74 {r . \<exists>A. range A \<subseteq> sets M & disjoint_family A & X \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
77 locale measure_space = sigma_algebra +
78 assumes positive: "!!a. a \<in> sets M \<Longrightarrow> 0 \<le> measure M a"
79 and empty_measure [simp]: "measure M {} = (0::real)"
80 and ca: "countably_additive M (measure M)"
82 subsection {* Basic Lemmas *}
84 lemma positive_imp_0: "positive M f \<Longrightarrow> f {} = 0"
87 lemma positive_imp_pos: "positive M f \<Longrightarrow> x \<in> sets M \<Longrightarrow> 0 \<le> f x"
90 lemma increasingD:
91 "increasing M f \<Longrightarrow> x \<subseteq> y \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M \<Longrightarrow> f x \<le> f y"
92 by (auto simp add: increasing_def)
95 "subadditive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
96 \<Longrightarrow> f (x \<union> y) \<le> f x + f y"
100 "additive M f \<Longrightarrow> x \<inter> y = {} \<Longrightarrow> x\<in>sets M \<Longrightarrow> y\<in>sets M
101 \<Longrightarrow> f (x \<union> y) = f x + f y"
105 "countably_additive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A
106 \<Longrightarrow> (\<Union>i. A i) \<in> sets M \<Longrightarrow> (\<lambda>n. f (A n)) sums f (\<Union>i. A i)"
109 lemma Int_Diff_disjoint: "A \<inter> B \<inter> (A - B) = {}"
110 by blast
112 lemma Int_Diff_Un: "A \<inter> B \<union> (A - B) = A"
113 by blast
115 lemma disjoint_family_subset:
116 "disjoint_family A \<Longrightarrow> (!!x. B x \<subseteq> A x) \<Longrightarrow> disjoint_family B"
117 by (force simp add: disjoint_family_on_def)
119 subsection {* A Two-Element Series *}
121 definition binaryset :: "'a set \<Rightarrow> 'a set \<Rightarrow> nat \<Rightarrow> 'a set "
122 where "binaryset A B = (\<lambda>\<^isup>x. {})(0 := A, Suc 0 := B)"
124 lemma range_binaryset_eq: "range(binaryset A B) = {A,B,{}}"
126 apply (rule set_ext)
127 apply (auto simp add: image_iff)
128 done
130 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
131 by (simp add: UNION_eq_Union_image range_binaryset_eq)
133 lemma LIMSEQ_binaryset:
134 assumes f: "f {} = 0"
135 shows "(\<lambda>n. \<Sum>i = 0..<n. f (binaryset A B i)) ----> f A + f B"
136 proof -
137 have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) = (\<lambda>n. f A + f B)"
138 proof
139 fix n
140 show "(\<Sum>i\<Colon>nat = 0\<Colon>nat..<Suc (Suc n). f (binaryset A B i)) = f A + f B"
141 by (induct n) (auto simp add: binaryset_def f)
142 qed
143 moreover
144 have "... ----> f A + f B" by (rule LIMSEQ_const)
145 ultimately
146 have "(\<lambda>n. \<Sum>i = 0..< Suc (Suc n). f (binaryset A B i)) ----> f A + f B"
147 by metis
148 hence "(\<lambda>n. \<Sum>i = 0..< n+2. f (binaryset A B i)) ----> f A + f B"
149 by simp
150 thus ?thesis by (rule LIMSEQ_offset [where k=2])
151 qed
153 lemma binaryset_sums:
154 assumes f: "f {} = 0"
155 shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
156 by (simp add: sums_def LIMSEQ_binaryset [where f=f, OF f])
158 lemma suminf_binaryset_eq:
159 "f {} = 0 \<Longrightarrow> suminf (\<lambda>n. f (binaryset A B n)) = f A + f B"
160 by (metis binaryset_sums sums_unique)
163 subsection {* Lambda Systems *}
165 lemma (in algebra) lambda_system_eq:
166 "lambda_system M f =
167 {l. l \<in> sets M & (\<forall>x \<in> sets M. f (x \<inter> l) + f (x - l) = f x)}"
168 proof -
169 have [simp]: "!!l x. l \<in> sets M \<Longrightarrow> x \<in> sets M \<Longrightarrow> (space M - l) \<inter> x = x - l"
170 by (metis Diff_eq Int_Diff Int_absorb1 Int_commute sets_into_space)
171 show ?thesis
172 by (auto simp add: lambda_system_def) (metis Diff_Compl Int_commute)+
173 qed
175 lemma (in algebra) lambda_system_empty:
176 "positive M f \<Longrightarrow> {} \<in> lambda_system M f"
177 by (auto simp add: positive_def lambda_system_eq)
179 lemma lambda_system_sets:
180 "x \<in> lambda_system M f \<Longrightarrow> x \<in> sets M"
183 lemma (in algebra) lambda_system_Compl:
184 fixes f:: "'a set \<Rightarrow> real"
185 assumes x: "x \<in> lambda_system M f"
186 shows "space M - x \<in> lambda_system M f"
187 proof -
188 have "x \<subseteq> space M"
189 by (metis sets_into_space lambda_system_sets x)
190 hence "space M - (space M - x) = x"
191 by (metis double_diff equalityE)
192 with x show ?thesis
193 by (force simp add: lambda_system_def)
194 qed
196 lemma (in algebra) lambda_system_Int:
197 fixes f:: "'a set \<Rightarrow> real"
198 assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
199 shows "x \<inter> y \<in> lambda_system M f"
200 proof -
201 from xl yl show ?thesis
202 proof (auto simp add: positive_def lambda_system_eq Int)
203 fix u
204 assume x: "x \<in> sets M" and y: "y \<in> sets M" and u: "u \<in> sets M"
205 and fx: "\<forall>z\<in>sets M. f (z \<inter> x) + f (z - x) = f z"
206 and fy: "\<forall>z\<in>sets M. f (z \<inter> y) + f (z - y) = f z"
207 have "u - x \<inter> y \<in> sets M"
208 by (metis Diff Diff_Int Un u x y)
209 moreover
210 have "(u - (x \<inter> y)) \<inter> y = u \<inter> y - x" by blast
211 moreover
212 have "u - x \<inter> y - y = u - y" by blast
213 ultimately
214 have ey: "f (u - x \<inter> y) = f (u \<inter> y - x) + f (u - y)" using fy
215 by force
216 have "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y)
217 = (f (u \<inter> (x \<inter> y)) + f (u \<inter> y - x)) + f (u - y)"
219 also have "... = (f ((u \<inter> y) \<inter> x) + f (u \<inter> y - x)) + f (u - y)"
221 also have "... = f (u \<inter> y) + f (u - y)"
222 using fx [THEN bspec, of "u \<inter> y"] Int y u
223 by force
224 also have "... = f u"
225 by (metis fy u)
226 finally show "f (u \<inter> (x \<inter> y)) + f (u - x \<inter> y) = f u" .
227 qed
228 qed
231 lemma (in algebra) lambda_system_Un:
232 fixes f:: "'a set \<Rightarrow> real"
233 assumes xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
234 shows "x \<union> y \<in> lambda_system M f"
235 proof -
236 have "(space M - x) \<inter> (space M - y) \<in> sets M"
237 by (metis Diff_Un Un compl_sets lambda_system_sets xl yl)
238 moreover
239 have "x \<union> y = space M - ((space M - x) \<inter> (space M - y))"
240 by auto (metis subsetD lambda_system_sets sets_into_space xl yl)+
241 ultimately show ?thesis
242 by (metis lambda_system_Compl lambda_system_Int xl yl)
243 qed
245 lemma (in algebra) lambda_system_algebra:
246 "positive M f \<Longrightarrow> algebra (M (|sets := lambda_system M f|))"
247 apply (auto simp add: algebra_def)
248 apply (metis lambda_system_sets set_mp sets_into_space)
249 apply (metis lambda_system_empty)
250 apply (metis lambda_system_Compl)
251 apply (metis lambda_system_Un)
252 done
255 assumes z: "z \<in> sets M" and disj: "x \<inter> y = {}"
256 and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
257 shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
258 proof -
259 have "z \<inter> x = (z \<inter> (x \<union> y)) \<inter> x" using disj by blast
260 moreover
261 have "z \<inter> y = (z \<inter> (x \<union> y)) - x" using disj by blast
262 moreover
263 have "(z \<inter> (x \<union> y)) \<in> sets M"
264 by (metis Int Un lambda_system_sets xl yl z)
265 ultimately show ?thesis using xl yl
267 qed
269 lemma (in algebra) Int_space_eq1 [simp]: "x \<in> sets M \<Longrightarrow> space M \<inter> x = x"
270 by (metis Int_absorb1 sets_into_space)
272 lemma (in algebra) Int_space_eq2 [simp]: "x \<in> sets M \<Longrightarrow> x \<inter> space M = x"
273 by (metis Int_absorb2 sets_into_space)
276 "additive (M (|sets := lambda_system M f|)) f"
278 fix x and y
279 assume disj: "x \<inter> y = {}"
280 and xl: "x \<in> lambda_system M f" and yl: "y \<in> lambda_system M f"
281 hence "x \<in> sets M" "y \<in> sets M" by (blast intro: lambda_system_sets)+
282 thus "f (x \<union> y) = f x + f y"
283 using lambda_system_strong_additive [OF top disj xl yl]
285 qed
289 assumes f: "positive M f" and cs: "countably_subadditive M f"
292 fix x y
293 assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
294 hence "disjoint_family (binaryset x y)"
295 by (auto simp add: disjoint_family_on_def binaryset_def)
296 hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
297 (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
298 summable (f o (binaryset x y)) \<longrightarrow>
299 f (\<Union>i. binaryset x y i) \<le> suminf (\<lambda>n. f (binaryset x y n))"
301 hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
302 summable (f o (binaryset x y)) \<longrightarrow>
303 f (x \<union> y) \<le> suminf (\<lambda>n. f (binaryset x y n))"
304 by (simp add: range_binaryset_eq UN_binaryset_eq)
305 thus "f (x \<union> y) \<le> f x + f y" using f x y binaryset_sums
306 by (auto simp add: Un sums_iff positive_def o_def)
307 qed
310 definition disjointed :: "(nat \<Rightarrow> 'a set) \<Rightarrow> nat \<Rightarrow> 'a set "
311 where "disjointed A n = A n - (\<Union>i\<in>{0..<n}. A i)"
313 lemma finite_UN_disjointed_eq: "(\<Union>i\<in>{0..<n}. disjointed A i) = (\<Union>i\<in>{0..<n}. A i)"
314 proof (induct n)
315 case 0 show ?case by simp
316 next
317 case (Suc n)
318 thus ?case by (simp add: atLeastLessThanSuc disjointed_def)
319 qed
321 lemma UN_disjointed_eq: "(\<Union>i. disjointed A i) = (\<Union>i. A i)"
322 apply (rule UN_finite2_eq [where k=0])
324 done
326 lemma less_disjoint_disjointed: "m<n \<Longrightarrow> disjointed A m \<inter> disjointed A n = {}"
327 by (auto simp add: disjointed_def)
329 lemma disjoint_family_disjointed: "disjoint_family (disjointed A)"
331 (metis neq_iff Int_commute less_disjoint_disjointed)
333 lemma disjointed_subset: "disjointed A n \<subseteq> A n"
334 by (auto simp add: disjointed_def)
337 lemma (in algebra) UNION_in_sets:
338 fixes A:: "nat \<Rightarrow> 'a set"
339 assumes A: "range A \<subseteq> sets M "
340 shows "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
341 proof (induct n)
342 case 0 show ?case by simp
343 next
344 case (Suc n)
345 thus ?case
346 by (simp add: atLeastLessThanSuc) (metis A Un UNIV_I image_subset_iff)
347 qed
349 lemma (in algebra) range_disjointed_sets:
350 assumes A: "range A \<subseteq> sets M "
351 shows "range (disjointed A) \<subseteq> sets M"
352 proof (auto simp add: disjointed_def)
353 fix n
354 show "A n - (\<Union>i\<in>{0..<n}. A i) \<in> sets M" using UNION_in_sets
355 by (metis A Diff UNIV_I disjointed_def image_subset_iff)
356 qed
358 lemma sigma_algebra_disjoint_iff:
359 "sigma_algebra M \<longleftrightarrow>
360 algebra M &
361 (\<forall>A. range A \<subseteq> sets M \<longrightarrow> disjoint_family A \<longrightarrow>
362 (\<Union>i::nat. A i) \<in> sets M)"
363 proof (auto simp add: sigma_algebra_iff)
364 fix A :: "nat \<Rightarrow> 'a set"
365 assume M: "algebra M"
366 and A: "range A \<subseteq> sets M"
367 and UnA: "\<forall>A. range A \<subseteq> sets M \<longrightarrow>
368 disjoint_family A \<longrightarrow> (\<Union>i::nat. A i) \<in> sets M"
369 hence "range (disjointed A) \<subseteq> sets M \<longrightarrow>
370 disjoint_family (disjointed A) \<longrightarrow>
371 (\<Union>i. disjointed A i) \<in> sets M" by blast
372 hence "(\<Union>i. disjointed A i) \<in> sets M"
373 by (simp add: algebra.range_disjointed_sets M A disjoint_family_disjointed)
374 thus "(\<Union>i::nat. A i) \<in> sets M" by (simp add: UN_disjointed_eq)
375 qed
379 fixes A:: "nat \<Rightarrow> 'a set"
381 and A: "range A \<subseteq> sets M"
382 and disj: "disjoint_family A"
383 shows "setsum (f o A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
384 proof (induct n)
385 case 0 show ?case using f by (simp add: positive_def)
386 next
387 case (Suc n)
388 have "A n \<inter> (\<Union>i\<in>{0..<n}. A i) = {}" using disj
389 by (auto simp add: disjoint_family_on_def neq_iff) blast
390 moreover
391 have "A n \<in> sets M" using A by blast
392 moreover have "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
393 by (metis A UNION_in_sets atLeast0LessThan)
394 moreover
395 ultimately have "f (A n \<union> (\<Union>i\<in>{0..<n}. A i)) = f (A n) + f(\<Union>i\<in>{0..<n}. A i)"
397 with Suc.hyps show ?case using ad
399 qed
403 "countably_subadditive M f \<Longrightarrow> range A \<subseteq> sets M \<Longrightarrow> disjoint_family A \<Longrightarrow>
404 (\<Union>i. A i) \<in> sets M \<Longrightarrow> summable (f o A) \<Longrightarrow> f (\<Union>i. A i) \<le> suminf (f o A)"
408 fixes A:: "nat \<Rightarrow> 'a set"
410 and inc: "increasing M f"
411 and A: "range A \<subseteq> sets M"
412 and disj: "disjoint_family A"
413 shows "summable (f o A)"
414 proof (rule pos_summable)
415 fix n
416 show "0 \<le> (f \<circ> A) n" using f A
417 by (force simp add: positive_def)
418 next
419 fix n
420 have "setsum (f \<circ> A) {0..<n} = f (\<Union>i\<in>{0..<n}. A i)"
422 also have "... \<le> f (space M)" using space_closed A
423 by (blast intro: increasingD [OF inc] UNION_in_sets top)
424 finally show "setsum (f \<circ> A) {0..<n} \<le> f (space M)" .
425 qed
427 lemma lambda_system_positive:
428 "positive M f \<Longrightarrow> positive (M (|sets := lambda_system M f|)) f"
429 by (simp add: positive_def lambda_system_def)
431 lemma lambda_system_increasing:
432 "increasing M f \<Longrightarrow> increasing (M (|sets := lambda_system M f|)) f"
433 by (simp add: increasing_def lambda_system_def)
435 lemma (in algebra) lambda_system_strong_sum:
436 fixes A:: "nat \<Rightarrow> 'a set"
437 assumes f: "positive M f" and a: "a \<in> sets M"
438 and A: "range A \<subseteq> lambda_system M f"
439 and disj: "disjoint_family A"
440 shows "(\<Sum>i = 0..<n. f (a \<inter>A i)) = f (a \<inter> (\<Union>i\<in>{0..<n}. A i))"
441 proof (induct n)
442 case 0 show ?case using f by (simp add: positive_def)
443 next
444 case (Suc n)
445 have 2: "A n \<inter> UNION {0..<n} A = {}" using disj
446 by (force simp add: disjoint_family_on_def neq_iff)
447 have 3: "A n \<in> lambda_system M f" using A
448 by blast
449 have 4: "UNION {0..<n} A \<in> lambda_system M f"
450 using A algebra.UNION_in_sets [OF local.lambda_system_algebra [OF f]]
451 by simp
452 from Suc.hyps show ?case
454 qed
457 lemma (in sigma_algebra) lambda_system_caratheodory:
458 assumes oms: "outer_measure_space M f"
459 and A: "range A \<subseteq> lambda_system M f"
460 and disj: "disjoint_family A"
461 shows "(\<Union>i. A i) \<in> lambda_system M f & (f \<circ> A) sums f (\<Union>i. A i)"
462 proof -
463 have pos: "positive M f" and inc: "increasing M f"
464 and csa: "countably_subadditive M f"
465 by (metis oms outer_measure_space_def)+
466 have sa: "subadditive M f"
468 have A': "range A \<subseteq> sets (M(|sets := lambda_system M f|))" using A
469 by simp
470 have alg_ls: "algebra (M(|sets := lambda_system M f|))"
471 by (rule lambda_system_algebra [OF pos])
472 have A'': "range A \<subseteq> sets M"
473 by (metis A image_subset_iff lambda_system_sets)
474 have sumfA: "summable (f \<circ> A)"
475 by (metis algebra.increasing_additive_summable [OF alg_ls]
477 A' oms outer_measure_space_def disj)
478 have U_in: "(\<Union>i. A i) \<in> sets M"
479 by (metis A countable_UN image_subset_iff lambda_system_sets)
480 have U_eq: "f (\<Union>i. A i) = suminf (f o A)"
481 proof (rule antisym)
482 show "f (\<Union>i. A i) \<le> suminf (f \<circ> A)"
483 by (rule countably_subadditiveD [OF csa A'' disj U_in sumfA])
484 show "suminf (f \<circ> A) \<le> f (\<Union>i. A i)"
485 by (rule suminf_le [OF sumfA])
486 (metis algebra.additive_sum [OF alg_ls] pos disj UN_Un Un_UNIV_right
488 subset_Un_eq increasingD [OF inc] A' A'' UNION_in_sets U_in)
489 qed
490 {
491 fix a
492 assume a [iff]: "a \<in> sets M"
493 have "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) = f a"
494 proof -
495 have summ: "summable (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)" using pos A''
496 apply -
497 apply (rule summable_comparison_test [OF _ sumfA])
498 apply (rule_tac x="0" in exI)
500 apply (auto simp add: )
501 apply (subst abs_of_nonneg)
502 apply (metis A'' Int UNIV_I a image_subset_iff)
503 apply (blast intro: increasingD [OF inc] a)
504 done
505 show ?thesis
506 proof (rule antisym)
507 have "range (\<lambda>i. a \<inter> A i) \<subseteq> sets M" using A''
508 by blast
509 moreover
510 have "disjoint_family (\<lambda>i. a \<inter> A i)" using disj
511 by (auto simp add: disjoint_family_on_def)
512 moreover
513 have "a \<inter> (\<Union>i. A i) \<in> sets M"
514 by (metis Int U_in a)
515 ultimately
516 have "f (a \<inter> (\<Union>i. A i)) \<le> suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A)"
517 using countably_subadditiveD [OF csa, of "(\<lambda>i. a \<inter> A i)"] summ
519 moreover
520 have "suminf (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) \<le> f a - f (a - (\<Union>i. A i))"
521 proof (rule suminf_le [OF summ])
522 fix n
523 have UNION_in: "(\<Union>i\<in>{0..<n}. A i) \<in> sets M"
524 by (metis A'' UNION_in_sets)
525 have le_fa: "f (UNION {0..<n} A \<inter> a) \<le> f a" using A''
526 by (blast intro: increasingD [OF inc] A'' Int UNION_in_sets a)
527 have ls: "(\<Union>i\<in>{0..<n}. A i) \<in> lambda_system M f"
528 using algebra.UNION_in_sets [OF lambda_system_algebra [OF pos]]
530 hence eq_fa: "f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i)) = f a"
531 by (simp add: lambda_system_eq UNION_in Diff_Compl a)
532 have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
533 by (blast intro: increasingD [OF inc] Diff UNION_eq_Union_image
534 UNION_in U_in a)
535 thus "setsum (f \<circ> (\<lambda>i. a \<inter> i) \<circ> A) {0..<n} \<le> f a - f (a - (\<Union>i. A i))"
536 using eq_fa
537 by (simp add: suminf_le [OF summ] lambda_system_strong_sum pos
538 a A disj)
539 qed
540 ultimately show "f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i)) \<le> f a"
541 by arith
542 next
543 have "f a \<le> f (a \<inter> (\<Union>i. A i) \<union> (a - (\<Union>i. A i)))"
544 by (blast intro: increasingD [OF inc] a U_in)
545 also have "... \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))"
546 by (blast intro: subadditiveD [OF sa] Int Diff U_in)
547 finally show "f a \<le> f (a \<inter> (\<Union>i. A i)) + f (a - (\<Union>i. A i))" .
548 qed
549 qed
550 }
551 thus ?thesis
552 by (simp add: lambda_system_eq sums_iff U_eq U_in sumfA)
553 qed
555 lemma (in sigma_algebra) caratheodory_lemma:
556 assumes oms: "outer_measure_space M f"
557 shows "measure_space (|space = space M, sets = lambda_system M f, measure = f|)"
558 proof -
559 have pos: "positive M f"
560 by (metis oms outer_measure_space_def)
561 have alg: "algebra (|space = space M, sets = lambda_system M f, measure = f|)"
562 using lambda_system_algebra [OF pos]
564 then moreover
565 have "sigma_algebra (|space = space M, sets = lambda_system M f, measure = f|)"
566 using lambda_system_caratheodory [OF oms]
568 moreover
569 have "measure_space_axioms (|space = space M, sets = lambda_system M f, measure = f|)"
570 using pos lambda_system_caratheodory [OF oms]
571 by (simp add: measure_space_axioms_def positive_def lambda_system_sets
573 ultimately
574 show ?thesis
575 by intro_locales (auto simp add: sigma_algebra_def)
576 qed
579 lemma (in algebra) inf_measure_nonempty:
580 assumes f: "positive M f" and b: "b \<in> sets M" and a: "a \<subseteq> b"
581 shows "f b \<in> measure_set M f a"
582 proof -
583 have "(f \<circ> (\<lambda>i. {})(0 := b)) sums setsum (f \<circ> (\<lambda>i. {})(0 := b)) {0..<1::nat}"
584 by (rule series_zero) (simp add: positive_imp_0 [OF f])
585 also have "... = f b"
586 by simp
587 finally have "(f \<circ> (\<lambda>i. {})(0 := b)) sums f b" .
588 thus ?thesis using a
589 by (auto intro!: exI [of _ "(\<lambda>i. {})(0 := b)"]
590 simp add: measure_set_def disjoint_family_on_def b split_if_mem2)
591 qed
593 lemma (in algebra) inf_measure_pos0:
594 "positive M f \<Longrightarrow> x \<in> measure_set M f a \<Longrightarrow> 0 \<le> x"
595 apply (auto simp add: positive_def measure_set_def sums_iff intro!: suminf_ge_zero)
596 apply blast
597 done
599 lemma (in algebra) inf_measure_pos:
600 shows "positive M f \<Longrightarrow> x \<subseteq> space M \<Longrightarrow> 0 \<le> Inf (measure_set M f x)"
601 apply (rule Inf_greatest)
602 apply (metis emptyE inf_measure_nonempty top)
603 apply (metis inf_measure_pos0)
604 done
608 shows "increasing M f"
609 proof (auto simp add: increasing_def)
610 fix x y
611 assume xy: "x \<in> sets M" "y \<in> sets M" "x \<subseteq> y"
612 have "f x \<le> f x + f (y-x)" using posf
613 by (simp add: positive_def) (metis Diff xy)
614 also have "... = f (x \<union> (y-x))" using addf
616 also have "... = f y"
617 by (metis Un_Diff_cancel Un_absorb1 xy)
618 finally show "f x \<le> f y" .
619 qed
622 assumes posf: "positive M f" and ca: "countably_additive M f"
625 fix x y
626 assume x: "x \<in> sets M" and y: "y \<in> sets M" and "x \<inter> y = {}"
627 hence "disjoint_family (binaryset x y)"
628 by (auto simp add: disjoint_family_on_def binaryset_def)
629 hence "range (binaryset x y) \<subseteq> sets M \<longrightarrow>
630 (\<Union>i. binaryset x y i) \<in> sets M \<longrightarrow>
631 f (\<Union>i. binaryset x y i) = suminf (\<lambda>n. f (binaryset x y n))"
632 using ca
634 hence "{x,y,{}} \<subseteq> sets M \<longrightarrow> x \<union> y \<in> sets M \<longrightarrow>
635 f (x \<union> y) = suminf (\<lambda>n. f (binaryset x y n))"
636 by (simp add: range_binaryset_eq UN_binaryset_eq)
637 thus "f (x \<union> y) = f x + f y" using posf x y
638 by (simp add: Un suminf_binaryset_eq positive_def)
639 qed
641 lemma (in algebra) inf_measure_agrees:
642 assumes posf: "positive M f" and ca: "countably_additive M f"
643 and s: "s \<in> sets M"
644 shows "Inf (measure_set M f s) = f s"
645 proof (rule Inf_eq)
646 fix z
647 assume z: "z \<in> measure_set M f s"
648 from this obtain A where
649 A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
650 and "s \<subseteq> (\<Union>x. A x)" and sm: "summable (f \<circ> A)"
651 and si: "suminf (f \<circ> A) = z"
652 by (auto simp add: measure_set_def sums_iff)
653 hence seq: "s = (\<Union>i. A i \<inter> s)" by blast
654 have inc: "increasing M f"
656 have sums: "(\<lambda>i. f (A i \<inter> s)) sums f (\<Union>i. A i \<inter> s)"
657 proof (rule countably_additiveD [OF ca])
658 show "range (\<lambda>n. A n \<inter> s) \<subseteq> sets M" using A s
659 by blast
660 show "disjoint_family (\<lambda>n. A n \<inter> s)" using disj
661 by (auto simp add: disjoint_family_on_def)
662 show "(\<Union>i. A i \<inter> s) \<in> sets M" using A s
663 by (metis UN_extend_simps(4) s seq)
664 qed
665 hence "f s = suminf (\<lambda>i. f (A i \<inter> s))"
666 by (metis Int_commute UN_simps(4) seq sums_iff)
667 also have "... \<le> suminf (f \<circ> A)"
668 proof (rule summable_le [OF _ _ sm])
669 show "\<forall>n. f (A n \<inter> s) \<le> (f \<circ> A) n" using A s
670 by (force intro: increasingD [OF inc])
671 show "summable (\<lambda>i. f (A i \<inter> s))" using sums
673 qed
674 also have "... = z" by (rule si)
675 finally show "f s \<le> z" .
676 next
677 fix y
678 assume y: "!!u. u \<in> measure_set M f s \<Longrightarrow> y \<le> u"
679 thus "y \<le> f s"
680 by (blast intro: inf_measure_nonempty [OF posf s subset_refl])
681 qed
683 lemma (in algebra) inf_measure_empty:
684 assumes posf: "positive M f"
685 shows "Inf (measure_set M f {}) = 0"
686 proof (rule antisym)
687 show "0 \<le> Inf (measure_set M f {})"
688 by (metis empty_subsetI inf_measure_pos posf)
689 show "Inf (measure_set M f {}) \<le> 0"
690 by (metis Inf_lower empty_sets inf_measure_pos0 inf_measure_nonempty posf
691 positive_imp_0 subset_refl)
692 qed
694 lemma (in algebra) inf_measure_positive:
695 "positive M f \<Longrightarrow>
696 positive (| space = space M, sets = Pow (space M) |)
697 (\<lambda>x. Inf (measure_set M f x))"
698 by (simp add: positive_def inf_measure_empty inf_measure_pos)
700 lemma (in algebra) inf_measure_increasing:
701 assumes posf: "positive M f"
702 shows "increasing (| space = space M, sets = Pow (space M) |)
703 (\<lambda>x. Inf (measure_set M f x))"
704 apply (auto simp add: increasing_def)
705 apply (rule Inf_greatest, metis emptyE inf_measure_nonempty top posf)
706 apply (rule Inf_lower)
707 apply (clarsimp simp add: measure_set_def, blast)
708 apply (blast intro: inf_measure_pos0 posf)
709 done
712 lemma (in algebra) inf_measure_le:
713 assumes posf: "positive M f" and inc: "increasing M f"
714 and x: "x \<in> {r . \<exists>A. range A \<subseteq> sets M & s \<subseteq> (\<Union>i. A i) & (f \<circ> A) sums r}"
715 shows "Inf (measure_set M f s) \<le> x"
716 proof -
717 from x
718 obtain A where A: "range A \<subseteq> sets M" and ss: "s \<subseteq> (\<Union>i. A i)"
719 and sm: "summable (f \<circ> A)" and xeq: "suminf (f \<circ> A) = x"
720 by (auto simp add: sums_iff)
721 have dA: "range (disjointed A) \<subseteq> sets M"
722 by (metis A range_disjointed_sets)
723 have "\<forall>n. \<bar>(f o disjointed A) n\<bar> \<le> (f \<circ> A) n"
724 proof (auto)
725 fix n
726 have "\<bar>f (disjointed A n)\<bar> = f (disjointed A n)" using posf dA
727 by (auto simp add: positive_def image_subset_iff)
728 also have "... \<le> f (A n)"
729 by (metis increasingD [OF inc] UNIV_I dA image_subset_iff disjointed_subset A)
730 finally show "\<bar>f (disjointed A n)\<bar> \<le> f (A n)" .
731 qed
732 from Series.summable_le2 [OF this sm]
733 have sda: "summable (f o disjointed A)"
734 "suminf (f o disjointed A) \<le> suminf (f \<circ> A)"
735 by blast+
736 hence ley: "suminf (f o disjointed A) \<le> x"
737 by (metis xeq)
738 from sda have "(f \<circ> disjointed A) sums suminf (f \<circ> disjointed A)"
740 hence y: "suminf (f o disjointed A) \<in> measure_set M f s"
741 apply (auto simp add: measure_set_def)
742 apply (rule_tac x="disjointed A" in exI)
743 apply (simp add: disjoint_family_disjointed UN_disjointed_eq ss dA)
744 done
745 show ?thesis
746 by (blast intro: Inf_lower y order_trans [OF _ ley] inf_measure_pos0 posf)
747 qed
749 lemma (in algebra) inf_measure_close:
750 assumes posf: "positive M f" and e: "0 < e" and ss: "s \<subseteq> (space M)"
751 shows "\<exists>A l. range A \<subseteq> sets M & disjoint_family A & s \<subseteq> (\<Union>i. A i) &
752 (f \<circ> A) sums l & l \<le> Inf (measure_set M f s) + e"
753 proof -
754 have " measure_set M f s \<noteq> {}"
755 by (metis emptyE ss inf_measure_nonempty [OF posf top])
756 hence "\<exists>l \<in> measure_set M f s. l < Inf (measure_set M f s) + e"
757 by (rule Inf_close [OF _ e])
758 thus ?thesis
759 by (auto simp add: measure_set_def, rule_tac x=" A" in exI, auto)
760 qed
763 assumes posf: "positive M f" and inc: "increasing M f"
764 shows "countably_subadditive (| space = space M, sets = Pow (space M) |)
765 (\<lambda>x. Inf (measure_set M f x))"
767 fix A :: "nat \<Rightarrow> 'a set" and e :: real
768 assume A: "range A \<subseteq> Pow (space M)"
769 and disj: "disjoint_family A"
770 and sb: "(\<Union>i. A i) \<subseteq> space M"
771 and sum1: "summable (\<lambda>n. Inf (measure_set M f (A n)))"
772 and e: "0 < e"
773 have "!!n. \<exists>B l. range B \<subseteq> sets M \<and> disjoint_family B \<and> A n \<subseteq> (\<Union>i. B i) \<and>
774 (f o B) sums l \<and>
775 l \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
776 apply (rule inf_measure_close [OF posf])
777 apply (metis e half mult_pos_pos zero_less_power)
778 apply (metis UNIV_I UN_subset_iff sb)
779 done
780 hence "\<exists>BB ll. \<forall>n. range (BB n) \<subseteq> sets M \<and> disjoint_family (BB n) \<and>
781 A n \<subseteq> (\<Union>i. BB n i) \<and> (f o BB n) sums ll n \<and>
782 ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
783 by (rule choice2)
784 then obtain BB ll
785 where BB: "!!n. (range (BB n) \<subseteq> sets M)"
786 and disjBB: "!!n. disjoint_family (BB n)"
787 and sbBB: "!!n. A n \<subseteq> (\<Union>i. BB n i)"
788 and BBsums: "!!n. (f o BB n) sums ll n"
789 and ll: "!!n. ll n \<le> Inf (measure_set M f (A n)) + e * (1/2)^(Suc n)"
790 by auto blast
791 have llpos: "!!n. 0 \<le> ll n"
792 by (metis BBsums sums_iff o_apply posf positive_imp_pos suminf_ge_zero
793 range_subsetD BB)
794 have sll: "summable ll &
795 suminf ll \<le> suminf (\<lambda>n. Inf (measure_set M f (A n))) + e"
796 proof -
797 have "(\<lambda>n. e * (1/2)^(Suc n)) sums (e*1)"
798 by (rule sums_mult [OF power_half_series])
799 hence sum0: "summable (\<lambda>n. e * (1 / 2) ^ Suc n)"
800 and eqe: "(\<Sum>n. e * (1 / 2) ^ n / 2) = e"
801 by (auto simp add: sums_iff)
802 have 0: "suminf (\<lambda>n. Inf (measure_set M f (A n))) +
803 suminf (\<lambda>n. e * (1/2)^(Suc n)) =
804 suminf (\<lambda>n. Inf (measure_set M f (A n)) + e * (1/2)^(Suc n))"
805 by (rule suminf_add [OF sum1 sum0])
806 have 1: "\<forall>n. \<bar>ll n\<bar> \<le> Inf (measure_set M f (A n)) + e * (1/2) ^ Suc n"
807 by (metis ll llpos abs_of_nonneg)
808 have 2: "summable (\<lambda>n. Inf (measure_set M f (A n)) + e*(1/2)^(Suc n))"
809 by (rule summable_add [OF sum1 sum0])
810 have "suminf ll \<le> (\<Sum>n. Inf (measure_set M f (A n)) + e*(1/2) ^ Suc n)"
811 using Series.summable_le2 [OF 1 2] by auto
812 also have "... = (\<Sum>n. Inf (measure_set M f (A n))) +
813 (\<Sum>n. e * (1 / 2) ^ Suc n)"
814 by (metis 0)
815 also have "... = (\<Sum>n. Inf (measure_set M f (A n))) + e"
817 finally show ?thesis using Series.summable_le2 [OF 1 2] by auto
818 qed
819 def C \<equiv> "(split BB) o prod_decode"
820 have C: "!!n. C n \<in> sets M"
821 apply (rule_tac p="prod_decode n" in PairE)
823 apply (metis BB subsetD rangeI)
824 done
825 have sbC: "(\<Union>i. A i) \<subseteq> (\<Union>i. C i)"
826 proof (auto simp add: C_def)
827 fix x i
828 assume x: "x \<in> A i"
829 with sbBB [of i] obtain j where "x \<in> BB i j"
830 by blast
831 thus "\<exists>i. x \<in> split BB (prod_decode i)"
832 by (metis prod_encode_inverse prod.cases prod_case_split)
833 qed
834 have "(f \<circ> C) = (f \<circ> (\<lambda>(x, y). BB x y)) \<circ> prod_decode"
835 by (rule ext) (auto simp add: C_def)
836 also have "... sums suminf ll"
837 proof (rule suminf_2dimen)
838 show "\<And>m n. 0 \<le> (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)" using posf BB
839 by (force simp add: positive_def)
840 show "\<And>m. (\<lambda>n. (f \<circ> (\<lambda>(x, y). BB x y)) (m, n)) sums ll m"using BBsums BB
841 by (force simp add: o_def)
842 show "summable ll" using sll
843 by auto
844 qed
845 finally have Csums: "(f \<circ> C) sums suminf ll" .
846 have "Inf (measure_set M f (\<Union>i. A i)) \<le> suminf ll"
847 apply (rule inf_measure_le [OF posf inc], auto)
848 apply (rule_tac x="C" in exI)
849 apply (auto simp add: C sbC Csums)
850 done
851 also have "... \<le> (\<Sum>n. Inf (measure_set M f (A n))) + e" using sll
852 by blast
853 finally show "Inf (measure_set M f (\<Union>i. A i)) \<le>
854 (\<Sum>n. Inf (measure_set M f (A n))) + e" .
855 qed
857 lemma (in algebra) inf_measure_outer:
858 "positive M f \<Longrightarrow> increasing M f
859 \<Longrightarrow> outer_measure_space (| space = space M, sets = Pow (space M) |)
860 (\<lambda>x. Inf (measure_set M f x))"
861 by (simp add: outer_measure_space_def inf_measure_positive
864 (*MOVE UP*)
866 lemma (in algebra) algebra_subset_lambda_system:
867 assumes posf: "positive M f" and inc: "increasing M f"
869 shows "sets M \<subseteq> lambda_system (| space = space M, sets = Pow (space M) |)
870 (\<lambda>x. Inf (measure_set M f x))"
871 proof (auto dest: sets_into_space
872 simp add: algebra.lambda_system_eq [OF algebra_Pow])
873 fix x s
874 assume x: "x \<in> sets M"
875 and s: "s \<subseteq> space M"
876 have [simp]: "!!x. x \<in> sets M \<Longrightarrow> s \<inter> (space M - x) = s-x" using s
877 by blast
878 have "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
879 \<le> Inf (measure_set M f s)"
880 proof (rule field_le_epsilon)
881 fix e :: real
882 assume e: "0 < e"
883 from inf_measure_close [OF posf e s]
884 obtain A l where A: "range A \<subseteq> sets M" and disj: "disjoint_family A"
885 and sUN: "s \<subseteq> (\<Union>i. A i)" and fsums: "(f \<circ> A) sums l"
886 and l: "l \<le> Inf (measure_set M f s) + e"
887 by auto
888 have [simp]: "!!x. x \<in> sets M \<Longrightarrow>
889 (f o (\<lambda>z. z \<inter> (space M - x)) o A) = (f o (\<lambda>z. z - x) o A)"
890 by (rule ext, simp, metis A Int_Diff Int_space_eq2 range_subsetD)
891 have [simp]: "!!n. f (A n \<inter> x) + f (A n - x) = f (A n)"
893 (auto simp add: x range_subsetD [OF A] Int_Diff_Un Int_Diff_disjoint)
894 have fsumb: "summable (f \<circ> A)"
895 by (metis fsums sums_iff)
896 { fix u
897 assume u: "u \<in> sets M"
898 have [simp]: "\<And>n. \<bar>f (A n \<inter> u)\<bar> \<le> f (A n)"
899 by (simp add: positive_imp_pos [OF posf] increasingD [OF inc]
900 u Int range_subsetD [OF A])
901 have 1: "summable (f o (\<lambda>z. z\<inter>u) o A)"
902 by (rule summable_comparison_test [OF _ fsumb]) simp
903 have 2: "Inf (measure_set M f (s\<inter>u)) \<le> suminf (f o (\<lambda>z. z\<inter>u) o A)"
904 proof (rule Inf_lower)
905 show "suminf (f \<circ> (\<lambda>z. z \<inter> u) \<circ> A) \<in> measure_set M f (s \<inter> u)"
907 apply (rule_tac x="(\<lambda>z. z \<inter> u) o A" in exI)
908 apply (auto simp add: disjoint_family_subset [OF disj])
909 apply (blast intro: u range_subsetD [OF A])
910 apply (blast dest: subsetD [OF sUN])
911 apply (metis 1 o_assoc sums_iff)
912 done
913 next
914 show "\<And>x. x \<in> measure_set M f (s \<inter> u) \<Longrightarrow> 0 \<le> x"
915 by (blast intro: inf_measure_pos0 [OF posf])
916 qed
917 note 1 2
918 } note lesum = this
919 have sum1: "summable (f o (\<lambda>z. z\<inter>x) o A)"
920 and inf1: "Inf (measure_set M f (s\<inter>x)) \<le> suminf (f o (\<lambda>z. z\<inter>x) o A)"
921 and sum2: "summable (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
922 and inf2: "Inf (measure_set M f (s \<inter> (space M - x)))
923 \<le> suminf (f o (\<lambda>z. z \<inter> (space M - x)) o A)"
924 by (metis Diff lesum top x)+
925 hence "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
926 \<le> suminf (f o (\<lambda>s. s\<inter>x) o A) + suminf (f o (\<lambda>s. s-x) o A)"
928 also have "... \<le> suminf (f o A)" using suminf_add [OF sum1 sum2]
930 also have "... \<le> Inf (measure_set M f s) + e"
931 by (metis fsums l sums_unique)
932 finally show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
933 \<le> Inf (measure_set M f s) + e" .
934 qed
935 moreover
936 have "Inf (measure_set M f s)
937 \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
938 proof -
939 have "Inf (measure_set M f s) = Inf (measure_set M f ((s\<inter>x) \<union> (s-x)))"
940 by (metis Un_Diff_Int Un_commute)
941 also have "... \<le> Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))"
945 apply (auto simp add: subsetD [OF s])
946 done
947 finally show ?thesis .
948 qed
949 ultimately
950 show "Inf (measure_set M f (s\<inter>x)) + Inf (measure_set M f (s-x))
951 = Inf (measure_set M f s)"
952 by (rule order_antisym)
953 qed
955 lemma measure_down:
956 "measure_space N \<Longrightarrow> sigma_algebra M \<Longrightarrow> sets M \<subseteq> sets N \<Longrightarrow>
957 (measure M = measure N) \<Longrightarrow> measure_space M"
958 by (simp add: measure_space_def measure_space_axioms_def positive_def
960 blast
962 theorem (in algebra) caratheodory:
963 assumes posf: "positive M f" and ca: "countably_additive M f"
964 shows "\<exists>MS :: 'a measure_space.
965 (\<forall>s \<in> sets M. measure MS s = f s) \<and>
966 ((|space = space MS, sets = sets MS|) = sigma (space M) (sets M)) \<and>
967 measure_space MS"
968 proof -
969 have inc: "increasing M f"
971 let ?infm = "(\<lambda>x. Inf (measure_set M f x))"
972 def ls \<equiv> "lambda_system (|space = space M, sets = Pow (space M)|) ?infm"
973 have mls: "measure_space (|space = space M, sets = ls, measure = ?infm|)"
974 using sigma_algebra.caratheodory_lemma
975 [OF sigma_algebra_Pow inf_measure_outer [OF posf inc]]
977 hence sls: "sigma_algebra (|space = space M, sets = ls, measure = ?infm|)"
979 have "sets M \<subseteq> ls"
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# Pukite's Model of the Quasi-Biennial Oscillation
I decided to name this model after myself because there are no free parameters and so is locked into place. There's nowhere to hide if it is invalidated, but it is so concise and precise that it's likely worth the risk of attaching my name to it.
The idea is as described earlier : Find the lunar forcing on the earth and then alias the forcing to a seasonal (yearly) period. This becomes the forcing for the QBO. The rationale is that the faster lunar cycles will not cause the stratospheric winds to change direction, but if these cycles are provoked with a seasonal peak in energy, then a longer-term multiyear period will emerge. This is a well-known mechanism that occurs in many different natural phenomena.
There are two steps to the model. (1) Determine the lunar gravitational potential as a function of time, and (2) plot the potential in units of 1 month or 1 year. The last part is critical, as that emulates the aliasing required to remove the sub-monthly cycles in the lunar forcing.
If one then matches this plot against the QBO time-series, you will find a high correlation coefficient. If the lunar potential is tweaked away from its stationary set of parameters, the fit degrades rapidly. So it becomes essentially a binary match. If it didn't fit, then the lunar gravitational potential hypothesis would be invalidated. But since it does fit precisely, then it remains a highly plausible model.
Figure 1 shows the mean-square potential of the lunar gravitational pull, also known as the tidal generating potential (applied in the context of predicting tides). On the left, the scale is expanded.
Fig. 1 : Mean-square potential of the lunar gravitational pull (from [1]). On the left, the scale is expanded.
As background, I originally discovered the connection of QBO to the lunar potential via machine learning (Eureqa), see Figure 2.
Fig 2: The original connection from QBO to a tide generating potential was discovered by machine learning -- upper right, panel A. The fitted signal was unaliased, squared and shown to align in panel B. Panel C shows the details with the fully unaliased signal at a finer scale.
This fit worked remarkably well considering that it is very difficult to dig out the aliased periods. Letting the machine learning run for a day helped considerably.
Yet it is also useful to reverse the direction of the fitting process. Instead of deducing the model from a sinusoidal decomposition, let us estimate the tidal generating potential as shown in Figure 1 and described by Ray [1]. We then inductively proceed forward and see how well it fits to the QBO time-series.
Fig 3 : Empirical fit to the tidal generating potential of Figure 1.
This empirical fit uses only three factors -- the lunar cycles corresponding to the Draconic month, the Anomalistic month, and the Tropical month. Those are known to a high precision, along with a value for the Tropical year. The composition of these factors is then squared to generate the empirical model of the potential.
If we lay the empirical model on top of Ray's diagram, it looks like Figure 4.
Fig 4 : Alignment of empirical model with Ray chart. Note the long-term 18.6 year (diurnal) beat period and the shorter 4.425 year (semidiurnal) beat period. Also a rapid bi-annual component.
On the expanded scale, the sub-monthly periods appear, as shown in Figure 5.
Fig 5 : Expanded scale showing the sub-monthly variations in the tidal generating potential.
These higher frequency components disappear when the alias is introduced.
I did not do a complete ephemeris-based empirical model for the tidal generating potential as Ray did, since the basic pattern is fairly easy to deduce from the three lunar cycles.
The final step is to un-square and then alias the tidal generating potential and compare to the QBO time-series. This is shown in Figure 6.
Fig 6: Fit of the unaliased tidal generating potential to the QBO
There is nothing at all complicated about the recipe for fitting the tidal generating potential to the QBO. It is a mechanical process since none of the lunar cycles parameters can be changed.
As a next step I will submit this finding to Physical Review Letters. From what I have seen in the literature search, there is no consideration of applying a straightforward forcing of the lunar gravitational pull to model QBO. It appears that most QBO models derive from what Richard Lindzen originally proposed some 40+ years ago -- but since many mainstream climate scientists do not consider Lindzen (an AGW denier) very trustworthy or even competent (e.g. a trail of retracted papers and debunked theories), it's likely that his original model was simply wrong, or at best, incomplete. What the new model does is provide a concise recipe and a highly plausible geophysical context for understanding the origin of QBO.
The further significance of all this is that the same lunar forcing that applies to QBO also likely applies to the phenomena of El Nino and modeling the ENSO time-series, see the ENSO sloshing paper and some more recent work.
## References
[1] R. D. Ray, “Decadal climate variability: Is there a tidal connection?,” Journal of climate, vol. 20, no. 14, pp. 3542–3560, 2007.
## 31 thoughts on “Pukite's Model of the Quasi-Biennial Oscillation”
1. I still have the complete lunar program from Chapront-Touzé & Chapront (Willman-Bell 1991) on my computer. If it would be useful, I can dig it out and give you exact ephemerides, including lunar radius vector, at any timescale you like.
• Yes thanks Keith. What software language is it written in?
• Since these periods aren't *really* sine waves, having more exact knowledge of, e.g., radius vector, gives you a more exact knowledge of lunar grav potential, hence might improve your model. Ditto with exact knowledge of the Moon's ecliptic longitude and longitude of node, which form the zero points of the tropical and anomalistic months.
• Keith, yup. There are definitely harmonics which will change the shape away from purely sinusoidal. What will be interesting to see is if these have a significant impact.
Since harmonics are higher in frequency, they may get damped out in the forcing response. Won't know for certain until a test fit, but it is good to think about ahead of time.
Ray spatially averaged out the lunar forcing but yes, the exact geospatial location of the maximum effect may also be important.
2. I've still got the original QBASIC version, and I've since translated it into FreeBasic-compatible. Or I can send you a couple of .exe files.
• wow, QBASIC, that's wild Keith. I can take whatever you have, thanks.
• Keith, my email is the contact at the bottom of this web page. Inconspicuous, I know ... 🙂
3. Paul,
Great work!! Looking forward to the publication in Physical Review Letters!
A small point:
You have said: "...Find the lunar forcing on the earth and then alias the forcing to a seasonal (yearly) period. This becomes the forcing for the QBO. The rationale is that the faster lunar cycles will not cause the stratospheric winds to change direction, but if these cycles are provoked with a seasonal peak in energy, then a longer-term multiyear period will emerge. This is a well-known mechanism that occurs in many different natural phenomena."
You might want to refer to the following quote from my paper [2012] paper which discusses the necessity of allowing for annual aliasing when it comes to the interaction of the lunar tidal cycles with the annual solar cycle (i.e. the seasons):
Wilson IRG. Lunar tides and the long-term variation of the peak
latitude anomaly of the summer Sub-Tropical High Pressure Ridge
over Eastern Australia. 2012. Open Atmos Sci J 2012; 6: 49-60.
http://benthamopen.com/ABSTRACT/TOASCJ-6-49
Page 50:
"There are two methods that can be used to work out the possible periods for long-term atmospheric tides. The first method assumes that lunar-tidal forces act independently of the other forcing factors that produce significant long-term variations in atmospheric pressure (e.g. seasonal variations in solar heating). Under this assumption, you would expect to see long-term periodicities in the pressure records that would match periodicities of the most extreme peak lunar tides. The second method investigates what happens when this assumption breaks down.
The most significant of the large-scale systematic variations of the atmospheric surface pressure, on an inter-annual to decadal time scale, are those caused by the seasons. These variations are predominantly driven by changes in the level of solar insolation with latitude that are produced by the effects of the Earth's obliquity and its annual motion around the Sun. This raises the possibility that the lunar tides could act in "resonance" with (i.e. subordinate to) the atmospheric pressure changes caused by the far more dominant solar driven seasonal cycles. With this type of simple “resonance” model, it is not so much in what years do the lunar tides reach their maximum strength, but whether or not there are peaks in the strength of the lunar tides that re-occur at the same time within the annual seasonal cycle."
• Ian, Sorry about missing that. I will include it when the reviewers ask for a reference. I left that assertion uncited when I said "This is a well-known mechanism that occurs in many different natural phenomena."
The machine learning uncovered it as it picked out frequencies differing by 2pi. The problem is that machine learning does not know how to cite the known literature.
• Thanks, I appreciate the offer. I think that your important break through will eventually be recognized once people really begin to understand what you are on about.
4. Paul,
I know that you do not have a high opinion of Prof. Lindzen but be careful about what you say about his work [in toto]. What he says about the magnitude of the effects of the luni-solar tides upon the atmosphere is in large part correct.
Below ~ 3000 m in altitude (750 mb) the tides caused by the daily thermal expansion of the atmosphere are orders of magnitude strong than than the tiny atmospheric tides produced by the luni-solar tidal cycles.
Luni-solar tides only become visible above about 3000 m in altitude once you have got above about 40 % of the total atmosphere and so it is only in the upper troposphere and stratosphere were they begin to have an effect. [Hence, the obvious effects upon the QBO induced stratospheric winds].
Lindzen may have been technically correct to point out at that the dominance of the thermal tides (driven by the Sun) in the lower troposphere dominates the effects of the luni-solar tides but he did not fully appreciate that effects the luni-solar tides in the upper troposphere and stratosphere could be amplified when they are in resonance with the seasons (i.e. annually aliased).
• The bottom-line is that Lindzen failed miserably as a scientist in not being able to pick this obvious behavior out of the data. He is considered the authority on QBO and the upper atmosphere, after all.
By golly, Lindzen had over 50 years of looking at the data to discover it and could not ! Cripes, I had been looking at the data for only a short time before it was obvious that a correlation existed.
He also fails miserably as a concerned scientist over the plight of the environment. He is clearly paid off by big oil and gives those unctuous presentations where he acts like the smartest guy in the room, while sneering at his fellow climate scientists.
I have no doubt that if Lindzen wasn't the "authority" on QBO, then this behavior would have been found long ago. But because of his authoritarian, egotistical style (remember, he's from MIT! ) others likely cowered from working the issue with the diligence it deserved.
Really, there is no other way to explain it, because the lunar-related behavior is that obvious in retrospect.
If I am wrong about this, remember that you get what you paid for, and I did this work on my own time and expense.
🙂
• Science is a rather cold mistress. You present your best argument and if people look at the details and find that it fits the data/observations then it will become more widely accepted. That's why it is such a cathartic process submitting a paper for peer review. It helps crystallize your thoughts and helps others see the detail and content of your hypothesis.
I do not know about you but I always find most of the referee reports are painful to read because they expose some of the weakness and flaws in my cherished arguments. It becomes even more painful if I have invested my blood and sweat in compiling the manuscript.
• I don't think that it will be painful to read reviewer's comments (if it even gets that far) because there is not much meat to criticize. The ball is in whoever's court to explain away the significant coincidence of a zero-degree-of-freedom-complexity almost perfectly aligned fit. Any information criteria metric such as AIC or BIC applied to this fit would be though the roof in comparison to an equivalent GCM-based fit, just because the latter would have loads of adjustable parameters that would reduce its score in comparison.
I intentionally kept the paper short for that reason -- to make the reader focus on the salient points. No use trying to oversell the argument at this stage, because that is where you will get criticized the most.
• Looks good.
It's getting to the point that the difference between a 2.33 year period and a 2.37 year period is significant in terms of attribution.
5. Here is my connection between the QBO and the lunar tidal cycles.
(8/20.2937) + (8/18.6000) + (4/8.8505) = 3/(2.3506)
where
20.2937 tropical yrs = time for new moon at closest perigee to re-occur.
18.6000 tropical yrs = time for the lunar line-of-nodes to precess around the Earth wrt. the stars.
8.8505 tropical yrs = time for the lunar line-of-apse to precess around the Earth wrt. the stars.
2.3506 tropical years = 28.21 months ~= the average length of the QBO.
Of course, the equation could also be written as:
(4/10.1469) + (4/9.3000) + (4/8.8505) = 3/(2.3506)
This requires that the Chandler wobble be 430 days if the mean QBO period is precisely twice the Chandler wobble i.e. (2 x 430 days)/(365.242189 days) = 2.354 tropical yrs = 28.248 months.
It is generally believed that the Chandler wobble is closer to 433 days so you would expect a mean QBO period equal to:
(2 x 433 days) /(365.242189 days) = 2.371 tropical yrs = 28.452 months.
Malkin and Miller (2009) get a period for the Chandler wobble of 1.185 Julian yrs = 432.8 +/- 0.3 days
http://arxiv.org/pdf/0908.3732v1.pdf
Hence, the lunar tidal synchronization is close to, but not precisely at, the QBO = 2 x Chandler wobble oscillation period. However, drifts of the lunar cycles around the nominal periods cited above would cause some overlap with time.
• Ian, Here is how I see it in cyclic terms. The significant lunar month periods after aliasing are ~ 2.37 years and 2.715 years. The first synchronizes with the Metonic cycle of 19 years after 8 periods, and the second takes 7 periods.
8*2.37=18.96 years and 7*2.715=19.005 years.
and the beat period between 2.37 and 2.715 is 18.65 years
6. You have indicated that the significant lunar month periods after aliasing are ~ 2.37 years and 2.715 years. You have also noted that the first synchronizes with the Metonic cycle of 19 years after 8 periods, and the second takes 7 periods.
8*2.37=18.96 years and 7*2.715=19.005 years.
A preliminary analysis on my part shows that if you sample the lunar monthly cycles on an annual basis (i.e. once every tropical years) you get the following periods:
Draconic month (27.21222 days) aliases to 2.3501 tropical years
anomalistic month (27.55455 days) aliases to 3.9984 tropical years
Synodic month (29.5305889 days) aliases to 2.7108 tropical years
Tropical month (27.32158 days) aliases to 2.7108 tropical years.
1. "The aliasing period of the anomalistic month is the well know 4.00 year seasonal (anomalistic) tidal cycle. The underlying reason for the four year cycle is the fact that for short term periods of a few years, the specific FMC period i.e. the average of 14 Synodic months (= 413.428244 days (J2000)) and 15 Anomalistic months (= 413.318248days (J2000)), which is 413.373246 days (J2000) or 1.131778 tropical years, should be used rather than the long-term mean FMC period i.e. 411.784430 days (J2000). Hence, 3.5 specific FMC’s = 3.96122463 tropical years ~ 49 Synodic months. This falls short of exactly four tropical years by 14.16 days which is very close to half a synodic month. Thus, if we start with a new moon at perigee on a given day of the calendar year, we end up with a full moon at or close to perigee (3.5 specific FMC’s + 0.5 Synodic months) later = 4.001651 tropical years or 4 years and 0.602895 days (J2000)."
Reference: Wilson, I.R.G. Are the Strongest Lunar Perigean Spring Tides Commensurate with the Transit Cycle of Venus?, Pattern Recogn. Phys., 2, 75-93. http://www.pattern-recognition-in-physics.com/pub/prp-2-75-2014.pdf
2. The Tropical and Synodic months both have the same alias to 2.7108 years and so this period would almost certainly be related to the well known 19.0 tropical year Metonic cycle since this is basis for religious calendars to align the phase of the Moon with the seasonal calendar.
"Higher than normal spring tides occur once every semi-synodic month (Msf), whenever the Sun, Earth and Moon are co-aligned at either New or Full Moon. It turns out that 12.5 synodic months are 3.890171 days longer than one tropical year (N.B. from this point forward, the word “year” will mean one tropical or seasonal year = 365.2421897 days (J2000) (McCarthy and Seidelmann [11], unless indicated). Hence, if a spring tide occurs on a given day of the year,
3.796 tropical years will pass before another spring tide occurs on the same day of the year. This occurs because: (0.5 synodic months)/(12.5 synodic months - tropical year) = (14.7652944 days/3.890171 days) = 3.796 years.
In addition, it can be shown that multiples of half of the lunar synodic cycle (Msf) are almost exactly equal to whole multiples of a year, for 4.0 years, 4.0 + 4.0 = 8.0 years, 4.0 + 4.0 + 3.0 = 11.0 years, 4.0 + 4.0 + 3.0 + 4.0 = 15.0 years, and
4.0 + 4.0 + 3.0 + 4.0 + 4.0 = 19.0 years. Hence, spring tides that occur on roughly the same day of the year follow a 4:4:3:4:4 year spacing pattern (with an average spacing of (4 + 4 + 3 + 4 + 4)/5 = 3.8 years), with the pattern repeating itself after a period of almost exactly 19 years. The 19.0 year period is known as the Metonic cycle. This cycle results from the fact that 235 Synodic months = 6939.688381 days = 19.000238 Tropical years."
Reference: Wilson, I.R.G., Lunar Tides and the Long-Term Variation of the Peak Latitude Anomaly of the Summer Sub-Tropical High Pressure Ridge over Eastern Australia The Open Atmospheric Science Journal, 2012, 6, 49-60.
http://benthamopen.com/ABSTRACT/TOASCJ-6-49
3. The aliasing period of the Draconic of 2.350 tropical year does not appear to perfectly match the 2.370 year period. However, it is close enough that normal deviations of the lunar variables from their nominal values would cause an overlap of these two numbers [N.B. 2 x 432.8 days/365.242189 days = 2.370 tropical years.] Hence, I believe that 2.370 feature is related to the Chandler wobble with the lunar tidal cycles coming into supplementing them at random periods.
• Ian,
Aliasing calculations can be made very precise, and so I am using as known a value for the tropical year as I have been able to find: 365.24219 days (or calendar year 365.2425 days)
I think I was able to reverse engineer your calculations and it appears that you use these values for the year
draconic 365.338
anomalistic 365.0998
synodic 365.2607
tropical 365.2593
Perhaps you are estimating the aliasing from some charts, instead of calculating directly? They should all be self-consistent in using the same value for a year.
These numbers really do make a difference because even slight variations will make a model of QBO go out of sync with the data after enough cycles.
• My results for the aliases are only crude estimations since the mean month lunar cycles are just that, means. The mean synodic period of the Moon is 29.3505889 days for J2000. but in real life it can vary by up to 6.5 hours = 0.27 days from the mean in an y given year.
This means that any aliasing process must be done upon real lunar data rather than the nominal means.
• Yup, we have discussed that variation over at the Azimuth Forum, and it likely has some impact to the overall fit. But you have to remember that the mean value is all that is important to establish a long-term coherence of the model with the QBO time-series. I am fitting a model to the series from 1953 until 2014, and if I get the mean wrong, it will get out of phase after a number of cycles.
On the other hand, if I include the variation as a slight frequency modulation, it may improve the shape of the individual periods, but by how much?
I am sure the question also comes up during the harmonic analysis of ocean tides. There they use the mean values to first-order, and all those other terms contribute to the second-order fit.
7. Pingback: Project Loon and QBO | context/Earth
8. Pingback: QBO Model Validation | context/Earth
9. Pingback: Scaling El Nino | context/Earth
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# CWC/FCI Prelims 2019 Reasoning Ability Questions (Day-41)
Dear Readers, Exam Race for the Year 2019 has already started, To enrich your preparation here we are providing new series of Practice Questions on CWC/FCI Prelims 2019 Reasoning Ability Questions on daily basis. Aspirants, practice these CWC/FCI Prelims 2019 Reasoning Ability Questions (Day-41) on a regular basis to improve your score in the aptitude section. Start your effective preparation from the right beginning to get success in the upcoming CWC/FCI Exam.
CWC/FCI Prelims Reasoning Ability Questions (Day-41)
maximum of 10 points
Direction (1-5): Read the following information carefully and answer the questions given below.
Eight persons P, Q, R, S, T, U, V and W are living on different floors of an eight storey building in such a way that the ground floor is numbered one, the above one is numbered two and so on till the topmost floor is numbered eight. The persons who are living on odd numbered floors like different fruits viz.,Guava, Papaya, Banana and Mango but not necessary in the same order.
Three persons are living between U and the one who likes Mango and none of them live on the ground floor. R lives immediately below the one who likes Mango. S lives immediately above the one who likes Banana. As many persons living below the one who likes Banana is same as above T. Two persons are living between T and W. Only one person lives between P and the one who likes Papaya. V lives one of the floors below P. More than two persons are living between V and P.
1) Who among the following persons like Guava?
a) P
b) V
c) S
d) U
e) W
2) How many persons are living between the one who likes Mango and W?
a) None
b) One
c) Two
d) Three
e) More than three
3) Which of the following statements is true?
a) Only two persons are living below S
b) R lives immediately below the one who likes Banana
c) More than three persons are living between V and W
d) The one who likes Papaya lives on the 5th floor
e) None is true
4) Four of the following five are alike in a certain way and hence form a group. Which one of the following that does not belong to the group?
a) W
b) V
c) P
d) U
e) Q
5) Which of the following persons living on the first and topmost floors?
a) P, V
b) T, S
c) V, T
d) W, Q
e) T, R
Direction (6-10): Read the following information carefully and answer the questions given below.
6) How many such numbers are there each of which is immediately followed by a letter and immediately preceded by a symbol?
a) One
b) Two
c) Three
d) None
e) More than three
7) If all the numbers are dropped from the above arrangement, then which of the following will be twelfth to the left of the element which is at the right end?
a) #
b) C
c) &
d) H
e) None of these
8) How many such letters are there each of which is immediately followed by a number but not immediately preceded by a symbol?
a) none
b) One
c) Two
d) Three
e) More than three
9) Which of the following replaces the question mark(?) in the series given below?
G@7 QF% #25 ?
a) +UD
b) VDU
c) L\$V
d) UL+
e) +U\$
10) If all the elements are reversed, then which of the following element is eighth to the right of ‘A’ in the given series?
a) +
b) V
c) 7
d) L
e) @
Direction (1-5):
• Three persons are living between U and the one who likes Mango and none of them live on the ground floor. R lives immediately below the one who likes Mango.
• S lives immediately above the one who likes Banana. As many persons living below the one who likes Banana is same as above T. Two persons are living between T and W.
• Only one person lives between P and the one who likes Papaya. V lives one of the floors below P. More than two persons are living between V and P.
• So, Case-2 will be dropped.
Direction (6-10):
8 G J 7 @ E T Q 3 % F A & # C 5 2 H L + V \$ U D W © I B
There is no such numbers.
G J @ E T Q % F A &# C H L + V \$ U D W © I B
8 G J 7 @ E T Q 3 % F A & # C 5 2 H L + V \$ U D W © I B
8 G J 7 @ E T Q 3 % F A &# C 5 2 H L + V \$ U D W © I B
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