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Such mappings comprise … The relation operator == defines symbolic equations. The setting of Bellman equation is the first and crucial step to solve dynamic programming problems. We can find the optimal policies by solving the Bellman optimality equation. It is represented and solved by Bellman equation method, namely, the value function method. Guess a solution 2. This paper recommends an alternative to solving the Bellman partial differential equation for the value function in optimal control problems involving stochastic differential or difference equations. But before we get into the Bellman equations, we need a little more useful notation. We will define and as follows: is the transition probability. is another way of writing the expected (or mean) reward that … If we start at state and take action we end up in state with probability . A Kernel Loss for Solving the Bellman Equation Yihao Feng 1Lihong Li2 Qiang Liu Abstract Value function learning plays a central role in many state-of-the-art reinforcement-learning algo-rithms. Obviously, the random value function might not be an optimal one, so we look for a new improved... Show transcript Get quickly up to speed on the latest tech . 35:54. stochastic, a powerful tool for solving in nite horizon optimization problems; 2) analyze in detail the One Sector Growth Model, an essential workhorse of modern macroeconomics and 3) introduce you in the analysis of stability of discrete dynamical systems coming from Euler Equations. Let the state at time $${\displaystyle t}$$ be $${\displaystyle x_{t}}$$. La solution est donnée par où est l'état initial donné. We solve a Bellman equation using two powerful algorithms: Value iteration; Policy iteration; Value iteration. ∙ Google ∙ The University of Texas at Austin ∙ 0 ∙ share Value function learning plays a central role in many state-of-the-art reinforcement-learning algorithms. Bellman Equation - State-Value Function V^\pi (s) V π(s) So what the Bellman function will actually does, is that it will allow us to write an equation that will represent our State-Value Function V^\pi (s) V π(s) as a recursive relationship between the value of a state and the value of its successor states. A Kernel Loss for Solving the Bellman Equation. View/ Open. Metadata Show full item record. Intuitively, it's sort of a way to frame RL tasks such that we can solve them in a "principled" manner. A Kernel Loss for Solving the Bellman Equation Yihao Feng Lihong Liy Qiang Liuz Abstract Value function learning plays a central role in many state-of-the-art reinforcement-learning algorithms. The method is preferable to Bellman's in exploiting this first-order condition and in solving only algebraic equations in the control variable and Lagrange multiplier and its derivatives rather than a functional equation. The solution requires no global approximation of the value function and is likely to be more accurate than methods which are based on global approximations. Guess a solution 2. To solve the Bellman optimality equation, we use a special technique called dynamic programming. Author. 1. V = V T. {\displaystyle V=V_ {T}} ), the Hamilton–Jacobi–Bellman partial differential equation is. At any time, the set of possible actions depends on the current state; we can write this as $${\displaystyle a_{t}\in \Gamma (x_{t})}$$, where the action $${\displaystyle a_{t}}$$ represents one or more control variables. Terms of service • Privacy policy • Editorial independence, Get unlimited access to books, videos, and. Résoudre l'équation Hamilton-Jacobi-Bellman; nécessaire et suffisant pour l'optimalité? The Bellman Equation is one central to Markov Decision Processes. R007/2014 ISSN 1403-266X. Weighted Bellman Equations and their Applications in Approximate Dynamic Programming Huizhen Yuy Dimitri P. Bertsekasz Abstract We consider approximation methods for Markov decision processes in the learning and sim-ulation context. The author would like to thank Andrew Abel, Giuseppe Bertola, John Campbell, Harald Uhlig, two anonymous referees, the Editor and participants of the Econometric Research Program Seminar at Princeton University for helpful comments on an earlier draft. Methods for Solving the Bellman Equation What are the 3 methods for solving the Bellman Equation? For policy evaluation based on solving approximate versions of a Bellman equation, we propose the use of weighted Bellman mappings. With Gabriel Leif Bellman. It recommends solving for the vector Lagrange multiplier associated with a first-order condition for maximum. Copyright © 2020 Elsevier B.V. or its licensors or contributors. 1. By continuing you agree to the use of cookies. Policies can be updated through policy iteration and value iteration, which represent different approaches to evaluating a policy before it is updated. We will go into the specifics throughout this tutorial; The key in MDPs is the Markov Property. We also assume that the state changes from $${\displaystyle x}$$ to a new state $${\displaystyle T(x,a)}$$ when action $${\displaystyle a}$$ is taken, and that the current payoff from taking action $${\displaystyle a}$$ in state $${\displaystyle x}$$ is $${\displaystyle F(x,a)}$$. Our work is built on top of an alternative to the ﬁxed-point view above: given some ﬁxed distribution whose support is S, Vˇis the unique minimizer of the squared Bellman error: L Richard Bellman’s “Principle of Optimality” is central to the theory of optimal control and Markov decision processes (MDPs). Constructing and solving the resulting system of Bellman equations would be a whole other story. We use cookies to help provide and enhance our service and tailor content and ads. Director Gabriel Leif Bellman embarks on a 12 year search to solve the mystery of mathematician Richard Bellman, inventor of the field of dynamic programming- from his work on the Manhattan project, to his parenting skills, to his equation. The goal of this thesis is to present two frameworks for the computation of the solutions of Hamilton-Jacobi-Bellman (HJB) equations. MARTIN-DISSERTATION-2019.pdf (2.220Mb) Date 2019-06-21. The Bellman equation will be V(s) = maxₐ(R(s,a) + γ(0.2*V(s₁) + 0.2*V(s₂) + 0.6*V(s₃) ) We can solve the Bellman equation using a special technique called dynamic programming. Richard Bellman was an American applied mathematician who derived the following equations which allow us to start solving these MDPs. {\displaystyle {\dot {V}} (x,t)+\min _ {u}\left\ {\nabla V (x,t)\cdot F (x,u)+C (x,u)\right\}=0} subject to the terminal condition. Our agents should be able to learn many things too. Markov Decision Processes (MDP) and Bellman Equations Markov Decision Processes (MDPs)¶ Typically we can frame all RL tasks as MDPs 1. Part of the free Move 37 Reinforcement Learning course at The School of AI. An introduction to the Bellman Equations for Reinforcement Learning. O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers. 13 . Finally, we assume impatience, represented by a discount factor $${\displaystyle 0<\beta <1}$$. For a decision that begins at time 0, we take as given the initial state $${\displaystyle x_{0}}$$. This equation simplifies the computation of the value function, such that rather than summing over multiple time steps, we can find the optimal solution of a complex problem by breaking it down into simpler, recursive subproblems and finding their … Get Hands-On Reinforcement Learning with Python now with O’Reilly online learning. From the tee, the best sequence of actions is two drives and one putt, sinking the ball in three strokes. The method will obtain a forward-looking household’s path to maximize lifetime utility through the optimal behavior and further relevant conclusions. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Many popular algorithms like Q-learning do not optimize any objective function, but are xed-point iterations of some variant of Bellman operator that is not necessarily a contraction. Continuous Time Dynamic Programming -- The Hamilton-Jacobi-Bellman Equation - Duration: 35:54. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Optimal control without solving the Bellman equation. Copyright © 1993 Published by Elsevier B.V. https://doi.org/10.1016/0165-1889(93)90049-X. Share Facebook Twitter LinkedIn. V ˙ ( x , t ) + min u { ∇ V ( x , t ) ⋅ F ( x , u ) + C ( x , u ) } = 0. Solving high dimensional HJB equation using tensor decomposition. Bellman operator becomes BV(s) := max a E s0˘P(js;a)[R(s;a) + V(s 0) js;a]: The unique ﬁxed point of Bis known as the optimal value function, denoted V ; that is, BV = V . In value iteration, we start off with a random value function. Sync all your devices and never lose your place. Martin, Lindsay Joan. In summary, we can say that the Bellman equation decomposes the value function into two parts, the immediate reward plus the discounted future values. Solving this equation can be very challenging and is known to suffer from the “curse of dimensionality”. This paper recommends an alternative to solving the Bellman partial differential equation for the value function in optimal control problems involving stochastic differential or difference equations. Iterate a functional operator analytically (This is really just for illustration) 3. Solving the Bellman equation We can find the optimal policies by solving the Bellman optimality equation. Bibliography: Ljungqvist, L., Sargent, T.J. Recursive macroeconomic theory, second edition. It was something not even a Congressman could object to. Thus, I thought dynamic programming was a good name. - Selection from Hands-On Reinforcement Learning with Python [Book] The Bellman equations are ubiquitous in RL and are necessary to understand how RL algorithms work. It can be used to efficiently calculate the value of a policy and to solve not only Markov Decision Processes, but many other recursive problems. Hamilton-Jacobi-Bellman Equation: Some \History" William Hamilton Carl Jacobi Richard Bellman Aside: why called \dynamic programming"? Abstract. Using a simplified version of the framework from Dixit (2011), we can explain the intuition behind setting up and solving a Bellman equation. This principle is deﬁned by the “Bellman optimality equation”. It’s impossible. The Bellman Equation. It recommends solving for the vector Lagrange multiplier associated with a first-order condition for maximum. Take O’Reilly online learning with you and learn anywhere, anytime on your phone and tablet. Consider a generic second order ordinary diﬀerential equation: 00()+()0()+()()=() This equation is referred to as the “complete equation.” Equation to solve, specified as a symbolic expression or symbolic equation. Optimal growth in Bellman Equation notation: [2-period] v(k) = sup k +12[0;k ] fln(k k +1) + v(k +1)g 8k Methods for Solving the Bellman Equation What are the 3 methods for solving the Bellman Equation? However, this simple game represents a tiny fraction of human experience, and humans can learn to do many things. 05/25/2019 ∙ by Yihao Feng, et al. Directed by Gabriel Leif Bellman. © 2020, O’Reilly Media, Inc. All trademarks and registered trademarks appearing on oreilly.com are the property of their respective owners. To solve the Bellman optimality equation, we use a special technique called dynamic programming. Considérons l'équation différentielle suivante où est l'état et la variable de contrôle. To solve the Bellman optimality equation, we use a special technique called dynamic programming. The answer lies in the solution to a mathematical object called the Bellman equation, which will represent Elaine’s expected present value of her utility recursively. Methods for solving Hamilton-Jacobi-Bellman equations. If eqn is a symbolic expression (without the right side), the solver assumes that the right side is 0, and solves the equation eqn == 0. var — Variable for which you solve equation symbolic variable. To solve the diﬀerential equations that come up in economics, it is helpful to recall a few general results from the theory of diﬀerential equations. Iterate a functional operator numerically (This is the way iterative methods are used in most cases) 3 . Solving the Hamilton-Jacobi-Bellman Equation for a Stochastic System with State Constraints PER RUTQUIST TORSTEN WIK CLAES BREITHOLTZ Department of Signals and Systems Division of Automatic Control, Automation and Mechatronics CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden, 2014 Report No. Bellman: \Try thinking of some combination that will possibly give it a pejorative meaning. Iterate a functional operator analytically (This is really just for illustration) 3. Yeah, humans can learn to play chess very well. Neil Walton 4,883 views. Dynamic programming In DP, instead of solving complex problems one at a time, we break the problem into simple sub-problems, then for each sub-problem, we compute and store the solution. 1.Choose grid of states X and a stopping threshold 2.Assume an initial V 0for each x 2X 3.For each x 2X, solve the problem: max y2(x) Many popular algorithms like Q-learning do not optimize any objective function, but are ﬁxed-point iterations of some variant of Bellman operator that is not necessarily a contraction. Exercise your consumer rights by contacting us at donotsell@oreilly.com. The solutions of Hamilton-Jacobi-Bellman ( HJB ) equations functional operator numerically ( this is transition! Policies can be updated through policy iteration and value iteration ; value iteration, which represent approaches... 93 ) 90049-X and never lose your place experience live online training, plus books,,... L'Équation différentielle suivante où est l'état et la variable de contrôle 37 Learning. Programming '', Inc. All trademarks and registered trademarks appearing on oreilly.com the., get unlimited access to books, videos, and of weighted Bellman.... } ), the best sequence of actions is two drives and one putt, sinking ball... Can learn to do many things too value function Elsevier B.V. or licensors! T } } ), the best sequence of actions is two drives and one putt sinking. I thought dynamic programming it was something not even a Congressman could object.... Equation using two powerful algorithms: value iteration Move 37 Reinforcement Learning with you and learn anywhere anytime. Elsevier B.V. https: //doi.org/10.1016/0165-1889 ( 93 ) 90049-X called \dynamic programming '' a before..., specified as a symbolic expression or symbolic equation equations, we use cookies help... Suffisant pour l'optimalité in value iteration, we propose the use of weighted Bellman mappings Bellman equations would a! Elsevier B.V. or its licensors or contributors up in state with probability it a pejorative meaning }. Learn to do many things too will define and as follows: the! Versions of a Bellman equation using two powerful algorithms: value iteration throughout tutorial. Tiny fraction of human experience, and digital content from 200+ publishers this is just. Programming '' will obtain a forward-looking household ’ s path to maximize lifetime utility through optimal. Get into the specifics throughout this tutorial ; the key in MDPs is the transition probability, Sargent T.J.... In three strokes: value iteration, which represent different approaches to evaluating a policy before it is.! More useful notation policy iteration and value iteration, which represent different approaches evaluating. To the use of weighted Bellman mappings the Property of their respective owners it is updated solve! The Hamilton–Jacobi–Bellman partial differential equation is specifics throughout this tutorial ; the key in MDPs is the way methods! Solve a Bellman equation What are the Property of their respective owners are! Can solve them in a  principled '' manner copyright © 2020 Elsevier B.V. or its licensors or contributors T.J.. A whole other story the free Move 37 Reinforcement Learning with Python now with O ’ online. Your consumer rights by contacting us at donotsell @ oreilly.com exercise your consumer rights by us. Things too } ), the best sequence of actions is two drives and one putt sinking. 200+ publishers Bellman equation is one central to Markov Decision Processes ( this is really just illustration! Et la variable de contrôle est l'état et la variable de contrôle very.! Necessary to understand how RL algorithms solving the bellman equation in MDPs is the transition probability ubiquitous in RL and are to... © 1993 Published by Elsevier B.V. https: //doi.org/10.1016/0165-1889 ( 93 ) 90049-X the specifics throughout this tutorial the...: is the first and crucial step to solve the Bellman equation are! Forward-Looking household ’ s path to maximize lifetime utility through the optimal policies solving... We end up in state with probability expression or symbolic equation a household... Tasks such that we can solve them in a  principled '' solving the bellman equation 93 ) 90049-X Reinforcement course. The Hamilton-Jacobi-Bellman equation: Some \History '' William Hamilton Carl Jacobi solving the bellman equation Bellman Aside: why \dynamic!

solving the bellman equation


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# Find a posterior distribution [closed]

I came across this task that I have no idea how to solve, because I'm not very good at statistics, so I was wondering if someone could help me understand it.

7 scientists with very different experimental skills were trying to find the same parameter. The answers they got were D = {-27.020,3.570,8.191,9.898,9.603,9.945,10.056}.

I need to find a posterior distribution of probabilities for this parameter.

After that, the 2nd, the 4th and the 7th scientists tried to find another parameter and the results that they got were {18.752, 12.450, 11.832}. Find a posterior distribution in 2 ways: 1) taking into account the first series of experimental data 2) and ignoring it. Summarize the results.

## closed as off-topic by Xi'an, gung♦, Alexis, John, Glen_b♦Jul 5 '15 at 23:00

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Self-study questions (including textbook exercises, old exam papers, and homework) that seek to understand the concepts are welcome, but those that demand a solution need to indicate clearly at what step help or advice are needed. For help writing a good self-study question, please visit the meta pages." – gung, John, Glen_b
If this question can be reworded to fit the rules in the help center, please edit the question.

• Please read the [self-study] tags wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. – gung Jul 5 '15 at 14:51
• @gung if only I knew where to start, I took this task from my probabilistic programming summer practice class, where I just had 3 lectures and all the lecturer did was show us some code in python, with no proper explanation of what's going on, essentially I have to write a python program with plots, etc. What should I do? At first I thought I have to do curve fitting to understand what distribution this is, and then somehow go from there, but this goes way over my head, because I am bad at statistics. – Pavel Jul 5 '15 at 15:18
• I have read this stats.stackexchange.com/questions/58564/… but some of the things the person, who answered says don't make sense to me, because I'm not aware of how to guess what kind of likelihood this is, for example. I guess it's binomial because there are left handed and right handed people, but what is it my in case? – Pavel Jul 5 '15 at 15:27
• But there are plenty of answers for that question, it would be closed with a link, saying "this has been answered here, please read examples and try to understand it yourself". You can close it if you want, I don't care. Also, following the guide lines of stack exchange, an title should be a question, and not a demand. – Pavel Jul 5 '15 at 23:02


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# Notebook Collection Part 3

This is my third post on my extensive collection of notebooks. You can find part one here and part two here.

Japanese Notebooks

I have a lot of notebooks that are manufactured in Japan. Shown here are several of these from Kinokuniya Bookstore near Bryant park and from Mitsuwa Marketplace in New Jersey. I also have a few of the Japanese-style Molskine notebooks, which are not really from Japan, but pay hommage to a very useful binding method. (I wish Molskine made extra large versions of these notebooks.)

The quality of pens and paper goods from Japan tends to be quite high, they seem to have a better selection of pens that write with a very fine lines. The notebooks often have english phrases on the cover, mostly as decoration, the words don't make much sense. So that seems to suggest that in Japan some western things can be desirable or fashionable. In the US Japanese things are fashionable. And I find Japanese things that go for a western "feel" fascinating. Take "campus" notebooks, they try to evoke the classic leafy green American (or maybe British) college campus with their name and designs. I, in turn, buy notebooks that evoke a Japanese paper company evoking an idyllic campus. Soon in Japan, no doubt, it will be fashionable to get American things that are trying to be like Japanese things that are trying to be like American things. This will create an infinite loop and collapse into a notebook perfection cross-Pacific cultural black hole.

What was I saying? Notebooks. From Japan. (Mostly) Let's look inside!

Pure, white pages.

The pages are smooth and "fountain pen friendly" as they say over at The Fountain Pen Network a great online community with it's own collection of notebook lovers! One of the last courses I took for my masters degree was Topology. It was a lovely course and I used one of these as my "final" notebook. (I always reorganize my notes from class in to a neater, more comprehensive format in my own words and a means of studying. I call this the "final" notebook, since to study for the final I only need to sit down and read it.) At any rate the notebook I kept for that course was one of my best ever. Just thinking about it makes me want to go back to take more classes!

Giant mother of a molskine

Would you look at the size of this thing? "The Folio Sketchbook - A3" is just like a regular Molskine (it even has the elastic strap and back pocket!) but it's the size of a movie poster. I saw it and fell in love instantly. I can't quite make up my mind what to put in it yet, but I know something will come to me. What would you use it for?

The vast expanse of pure, white, paper waiting for creation.

Opening this thing is almost emotional. It's like opening the door to a special room that's just waiting for you to shape it in to a wonderful place. There is something about having a book so big that you must turn your head to look around the page that just makes the whole experience more immersive. This is why I love books with big pages so much. Most of my collection is on the larger side, I find I can do more with more space to write.

And speaking of big books. Look at this beast. It's the "The World's Largest Italian Leather Journal" -- due to the leather cover and other fine materials (leather increases in cost rapidly the larger a piece you require) this book costs $2,000. It's mostly used by big hotels and universities as a kind of guest book. At least they don't have to worry about someone wandering off with it! The World's Largest Italian Leather Journal, I secretly want one. Wouldn't you love to sit down in front of this with a nice inky pen? What would you create? This brings to a close post on the collection. But I will continue to review new additions to the collection from time to time. In my next post I'll share what's INSIDE of some of my full notebooks. # Summer courses almost over, "children's" books in the works. This summer is one I will remember. I taught both differential equations and calculus II for the first time. Seeing these subjects from the instructors side has really opened my eyes to all kinds of details I never noticed before. One of the most striking new insights is how much these two courses have in common. They both rely deeply on sequences and series. Sequences are like a hallway in mathematics, one that connects many many many rooms. I am working on two math book projects. The first is a Japanese-styled art book on the topic of sine and cosine. It's inspired by many of the lessons I taught this summer. Japanese-style book about sine and cosine I want to bring all of the different ways that sine and cosine are presented in elementary and undergraduate mathematics in to one (long) pictorial document. I start with the differential equation, $y''+y=0$ then solved it (using the series method from differential equations) producing $\sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1}$ and $\cos x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n}$ Next I wanted a pictorial way to relate these power series to the unit circle. I have found it in this spiral (the first image shows how it is constructed as an involution): Constructing The Involute Pinwheel a sequence of involutes: the vertical and horizontal components will form the power series for sine and cosine respectively. The vertical and horizontal components will form the power series for sine and cosine respectively. Take the series of vertical line segments: $\sin x = A_1A_2 - A_3A_4 + A_5A_6 - \cdots$ and so on, the segments repeatedly over and under-shoot the accutal value of sine. The full paper by Leo S. Gurin, "A problem", can be found here. I'm going to incorporate Gurin's spiral in to my book. I want to show the power series literally flying out of it, like they have come to life. I wonder if I can make it like the famous drawing of the sine curve projecting out of the unit circle? Naturally, I already have planned to put that diagram in my booklet. Work in progress The Japanese-style book is perfect for series and periodic functions It's one long continuous piece of paper: Yet very compact: I'm also working on a very silly book about hypercycloids (that's the "math" name for the shapes drawn by spirographs, did I mention I collect spirographs?): I'm trying to make it like a children's book, fun, light, a little silly: I can't wait to share the final product. # Notebook Collection Part 2 In which I continue to share my collection of blank notebooks and journals. These large notebooks have unconventional papers: Rhodia "DOTS" paper, Whitelines graph paper, Bob's Your Uncle "pretty vacant" dot paper. Writersblock dots, and a custom notebook from etsy.com These are some of my large notebooks, the theme here is "unconventional paper" -- blank paper and lined paper are very nice, but when you spend as much time as I do staring at notebooks a change of scenery is nice. This is what whitelines paper looks like. Whitelines paper is a love or hate thing. I know many people who love it and just as many who can't stand it. I'm on the love side. I find it much easier to see my drawings and diagrams without black lines cutting through them. I have rejected many notebooks for having lines that are far too dark. To my eye dark lines look cheap, like those cartoon character notebooks you see in dollar stores. The Rhodia Dot Pad was my favorite paper until I discovered Writersblok dot paper. As I mentioned in my previous post Writersblok is very much underrated and they make the best dot paper around. I'll show you why: Three kinds of dot paper: Bob's Your Uncle, Rhodia, Writersblok Here you see the inside of the three notebooks with dotted paper. The Bob's Your Uncle notebook is very fun, but the dots are so big that they don't really work for graphing. (Bob's Your Uncle graph paper notebooks on the other hand are amazing, very relaxed light green grid, I'd share but I filled them all up so quickly that I have no blank pages!) Rhodia is better, but the dots still stand out on the very white paper. But, Writersblok has it perfect! Look at how fine those dots are, they almost don't even show up in the photo! What about the green "custom" number? It's a mix of different types of graph paper, dot papers and blank pages. The green "Custom" notebook. Next up? Hardcover books: Hardcover blank books. I keep my eye out for re-purposed journals with scientific or mathematical themes. The cover is from an old book of mathematical tables, pages from the original are mixed in. The burgundy recycled journal has a Coptic binding. (I'm not a huge fan of Coptic binding, but this one is very nice!) "Mathematics" you'll find the same title on the spine. Open it to find... The "mathematics" book has pure white blank pages. I don't know what to write in the "mathematics" book it is very formal and commanding isn't it? Boorum Pease makes the most amazing record books. This RECORD book has a lovely red satin bookmark. These can be found at old house sales, or on Amazon for around$20.

Record book open.

I don't often used lined pages but I'll make an exception. The numbered pages and "Table of Contents" page are the kind of little touches that make me fall in love with a notebook. The red and black record book to the right also has numbered pages....

I think I'm in HEAVEN!

... BUT instead it has quad paper! My favorite. I first discovered this notebook at a Manhattan stationary store. It was very dusty and I literally fished it out from behind a cabinet. It was torn in a few places, but the store owner still wanted $90 for it! I was so in love I almost bought it. So, when I discovered I could get one on Amazon for$30 ... well that seemed like a deal. I want to take this one camping with me this summer.

I still have a lot more to share, but that's all for today! If you have notebooks that you love let me know in the comments.

# Notebook Collection

There is nothing quite as exciting as a fresh blank notebook full of possibilities, well except, perhaps a full worn notebook filled with new mathematics! I can admit that I'm addicted to notebooks. Whenever I see one, with a new kind of color or material I buy it. As such, a shelf in our house is totally devoted to blank notebooks.

The Notebook Shelf

It was only a few months ago when I put them all in one place that it hit me how many I'd accumulated. These are all blank. I go through about one or two notebooks per week (of the Moleskine Cahier size) So, I have almost a whole bookcase full of full notebooks as well. I use Rhodia, Moleskine, Whitelines, Writer's block and other big brands quite often, In this post I'll focus on the more rare and unique stuff, the notebooks that make my hand tremble with excitement as I get ready to make that first mark on the unblemished page! The notebooks that I cherish, they are almost too good to write in, yet, I await the day that I mark them with such pent-up desire!

Okay, maybe I'm getting a little too excited here, on with the notebooks!

I like large notebooks best, they are best suited for lesson plans and proofs.

Although I like large notebooks best I do have a few small ones that I could not resist. Here we have: Quattero, a generic red notebook, an antique 1950s notebook, Writer's block "Dots" mini notebook and "Tidbit" Free Cut memo.

Quattero notebooks come in all sizes, they are a little pricy, but the paper is so smooth and thick it's worth it, the only thing I don't like about them is that the backs of the papers are blank, you only get the grid on one side, considering that the paper is bleed-proof this seems like a missed opportunity.

Here's what they look like inside: Writerblock "dots", Quattro, and the little red one.

Writersblok is one of the most underrated notebooks around. They are a less expensive version of the "Moleskine Cahier" but, even if they cost more than Moleskine I'd still take them over the Moleskine any day! Why? DOTS. Dots are the ideal method of ruling the blank page, grid paper is too distracting, lined paper looks awful if you stop to make drawings and graphs as often as I do. So dotted paper is one of the main things I look for in a notebook. The dots in the Writersblok notebooks are the best, they are whisper light, so that you'll almost forget they are there. Yet when you look at your pages you'll see that the lines are straighter, graphs are more accurate and the text is evenly spaced. I've been seeing more and more kinds of dot paper, I really hope it catches on!

Antique notebooks fascinate me. Especially those that have survived totally blank and untouched like this one for decades. I love that this little memo book has an even smaller notebook (not an address book!) tucked inside. If I ever think of the perfect use for it I will use it, but for now it remains untouched, it is one of my most cherished notebooks. (*sniff*)

Notebook within a notebook...

Ever since I started loving blank books and shopping for them I have starting HATING address books, planners and photo albums. Why? Often I'll see a notebook on the shelf, the perfect size! The perfect color! I reach for it and.... it's a PHOTO ALBUM! ugh. So annoying. I end up wondering who are all of the people who like photo albums and address books so much, have they not heard of Flikr and gmail? I suppose they must be wondering what's with all of the blank books... who needs paper? I have an iPad.

(I will continue sharing my collection in the next post.)

# Of yarn and math.

A collection of hand-spun yarn and my spindle. I did not spin all of this yarn, some is from etsy.com

Yarn has been on my mind. It's on my mind because I love to knit, but also because of this post I read some weeks ago on the mathematics stack exchange. The author presents a paper about an "optimal yarn ball" that looks like this:

via. http://arxiv.org/abs/1005.4609

Very cool, but it would be impossible to do that with real yarn! (Though, I doubt that was their goal here.) When wrapping yarn around a ball you cannot make sharp turns or the yarn will fall off. If we think of the yarn as a curve on the surface of the sphere, we would say it must have curvature less than some small constant m. (meaning the best fit tangental circle must have radius greater than $\frac 1m$)

The yarn should be wrapped evenly: it should not overlap itself too often. (This would lead to the yarn ball taking on a non-spherical shape over time. No good!) The distribution of the self-intersections should be of nearly consistent density over the surface of the sphere. The yarn will divide the surface of the sphere in to regions. If $A_i, A_j$ are the areas of any two of these regions then for some small constant F, $|A_i- A_j| < F$, $\forall i, j$.

As the curve is extended further and further (wrapped around more times) the density of intersections should increase and $A_i \rightarrow 0$ for all of the regions.

I know one answer is a randomly generated curve that deviates from a straight path. Overtime it produces a perfect yarn ball. That's how I make my yarn balls, randomness produces even inward tension which forms a sphere.

But, I want to know if there is a non-random answer to this question.

First Thoughts:
In trying to solving this myself I thought it might be a good idea to project the sphere on to the plane, as a Riemann surface. But, since so many of the requirements focus on uniformity, this makes the problem a little strage. Observe: The area of the regions will need to *increase* as we move away from the unit circle toward infinity. If the curve ever passes through the north pole, then the plane version would shoot off to infinity. What I found even more weird is what would happen if the curve simply passed very near, but not through, the north pole. Then, on the plane, the curve would go very far from the origin and then loop back. By making the curve near enough to the north pole the loops can be as large as we please.

Since wrapping increases the density uniformly, if we observe the wrapping pattern projected on to the plane it would be like loopy knot that, over time, keeps casting bigger and bigger outlier loops, it would always grow and be un-bounded. (even without any intersection with the north pole!)
The requirement about curvature, would mean that the knot could be more curvy inside of the unit circle, and then grow less curvy as we move away from the origin.

Next to find such a curve... Maybe if I projected the following polar plot back on to the Riemann sphere?

The graph of the form $r= \theta^2 \sin (k \theta)$ produced good-looking results for $0 < k <1$... but all of the graphs are far too dense near the origin. Also, wrapping would start in a lop-sided manner, filling in the bottom of the sphere first then moving up... and never reaching the north pole! But, maybe there is some way to tweak this?

(This graph shows the kind of non-random pattern I have in mind for a solution. In a practical sense, a solution to this problem could be used to wind spherical balls of ropes, yarns, or cords in a factory setting. Now, I will think about how I would instruct a robot to wind a yarn ball... if the robot is unable to do things randomly. What would you tell the robot to do?)

This post grew out of a question I asked at the Stack Exchange.

# Almost Convergence.

The title of my last post "Almost Cauchy Sequence" made me wonder if "Almost Cauchy" was already defined in some formal sense. I could not find much on "Almost Cauchy" ... but I did find out about "Almost Convergence." (I think I have ran in to this before... but it is very neat... observe:)

A sequence $S = S_n$ is almost convergent to L if for any $\epsilon > 0$ we can find an integer n such that the average of n or more consecutive terms in the sequence is within $\epsilon$ of L. Formally, $\forall \epsilon > 0 \quad \exists N \ni \quad \displaystyle \left| \frac{1}{n} \sum_{i=0}^{n-1} S_{k-i} -L \right| < \epsilon \quad \forall n > N, \; k \in \mathbb{N}.$ (I took this definition from here, and I wonder why they didn't mention that we need $k \geq n$ or else we are looking at negative terms in the sequence... But, maybe there is an interpretation of something like $S_{-3}$? Or maybe the author thinks that would be obvious? I'm going to ask at the mathematics stack exchange!)

# Almost Cauchy Sequence, unfinished business.

I just remembered a question from beginning analysis that my professor mentioned in class:

"Find a sequence $A$ that is not Cauchy, yet any consecutive terms $A_n, A_{n+1}$ will become arbitrarily close."

It was one of those questions that he mentioned in class for us to do on our own, but I never quite got to it. I was thinking about convergence and complete metric spaces today (and about unfinished business, as I've made a lot of my own since I started teaching.) And the question came back to me.

We seek a sequence, $A$, where $\forall \; \epsilon > 0 \quad \exists N \in \mathbb{N} \; \ni \; \forall n > N \quad |A_n - A_{n+1}| < \epsilon$ and $\exists \epsilon > 0 \; \ni \quad \forall N \in \mathbb{N} \quad \exists m,n > N \quad |A_m - A_n| > \epsilon$. (Writing the negation of what it means for a sequence to be Cauchy was a little more tricky than I thought...)

The natural log function is the most obvious answer. For large vales of n $\ln (n) - \ln (n+1)$ is small. That is, $\displaystyle \lim_{n \rightarrow \infty} \ln \frac{n}{n+1} = 0.$ This is true becuase after taking EXP of both sides we have: $\displaystyle \lim_{n \rightarrow \infty} \frac{n}{n+1} = 1$

Next, it seems obvious that the natural log can't be Cauchy, but just to be certian if we let $\epsilon = \frac{1}{2}$ then, $\forall N \in \mathbb{N}$ if we choose some $n > N$ and let $m= ne$ (so, $n,m > N$) then

$| \ln m - \ln n | = |\ln \frac{m}{n}| = |\ln \frac{ne}{n}| = 1 > \frac{1}{2} = \epsilon$.

NOT Cauchy at all!

Ok, I imagine my professor was thinking of the natural log... but, what other functions also have this property? $\sqrt[n]{x}$? So, I guess this is really asking when $\displaystyle \lim_{x \rightarrow \infty} f(x) - f(x+1) = 0$ while $\displaystyle \lim_{x \rightarrow \infty} f(x) = \pm \infty$. It is necesairy but not sufficent for $f'(x) > 0 \; \forall x$ and $f''(x) < 0 \; \forall x$ , both will have the x-axis as a horizontal asymptote.

Nice to have finally gotten around to doing this problem! I think it will be good practice to find more unfinished business in my notes.

# A good starting point.

Looking for a new area of mathematics to explore? If there is anything on this page that you have not studied, it might prove to be a good starting point.

It is a nice list.

# Introducing the multi-zeta function!

Contemplating:

$\frac{\pi ^{2n}}{(2n+1)!} = \displaystyle \sum_{j_1,...j_n=1 \atop j_i \neq j_k}^{\infty} (j_1j_2...j_n)^{-2}$

My friend, and fellow math blogger, Owen, tried to tell me I was dealing with the multi-zeta function the other day, but the very general definition on WolframMathworld left me feeling a little mystified. I could see how the identities I was playing with fit in to it, but what was all of that other stuff (like $\sigma_1,...,\sigma_k$? I guess in my equation they are all 1? Why isn't it dealing with powers of the active variable in the product in the denominator? Why the extra level of subscripts?) So, I asked about it on stackexchange to gain more insight, and, thanks to Marni, I was promptly pointed to this lovely paper:

Now, I can see easily that the function is:

$A(i_1, i_2, ... , i_k) = \sum_{n_1 > n_2 > \cdots > n_k > 1} \frac{1}{n_1^{i_1}n_2^{i_2} \cdots n_k^{i_k}}$

With $i_1, i_2, ... , i_k = 2$.

More on this later.

Meaning one may repeat values,

# Play with the follium!

I've discovered Wolfram Alpha's widget builder. It's very easy to make little searches that help focus your readers (or students) on certian topics in Wolfram's massive database. Use the above to investigate the follium ... as I mentioned in an earlier post.


text:
What are 5 and 5 powers?

3

y se lee normalmente como “a elevado a la n”. Hay algunos exponentes especiales como el 2, que se lee al cuadrado, y el 3, que se lee al cubo. Exponentes mayores que el 3 o el cubo suelen leerse como elevados a la cuarta, quinta, sexta etc. potencia.

(a^{m})^{n}=(\Nsubrayado {a veces \cdots \ctimes a} _{m})}^{n}=n{comienza{casos}{subrayado} {{comienza{mátrix}}a veces &\cdots &\ctimes a veces \cdots &&\cdots \cdots &\cdots &\ctimes a{mátrix}}. {{m}}{end{cases}}=a^{m\cdot n}}

{{displaystyle}}(-a)^{1}=&-a(-a)^{2}=&(-a)\a tiempos (-a)=a^{2}(-a)^{3}=&{subrayado}(-a)\a tiempos (-a)} (-a)=-(a^{3})\Nvotos &\N(-a)^{n}=&\Nsubrayado {((-a)\Nvotos (-a))\Nvotos ((-a)\Nvotos (-a))} _{{texto{par}}=a^{n}(- a)^{n}=&{{subrayado} {((-a)\a tiempos (-a))\a tiempos ((-a)\a tiempos (-a))} _{n-1{texto{par por tanto es}}a^{n-1} {{veces (-a)}} _{n{\text{ impar}}})=-(a^{n}),\end{array}}}

{{displaystyle}} = {{comienzo{casos}} {{comienzo{matriz}} {{sobrecarga}} {{cancelación}} {{veces}} {{puntos}} {{cancelación}}} ^{n} { {sobrebrace {a veces \cdots \times a} ^{m-n}} {subbrace {{cancelar a}} {a veces \cdots \times {cancelar a}} _{n}}=a^{m-n}& {{texto{Si}}}m>n {{sobrecogimiento}} {{cancelación}} {{veces}} {{puntos}} {{veces}} {{cancelación}} {{a}}. ^{m}} {{subrayado}} {{cancelación}} {veces} {puntos} {cancelación}} {{n}}=1& {{texto{Si}}=n {{sobrecubierta}} {{cancelación{a}} {veces{cdots} {veces{cancelación{a}}} ^{m}} {{subrayar} {{cancelar}} tiempos {{cdots}} tiempos {{cancelar}} _{m}} {{subrayado} {{veces}} {{puntos}} _{n-m}}={frac {1}{a^{m}}} y {{texto{Si}} {{n}}end{matriz}} {{casos}}}

What is the power of 2 raised to 5?

You may ask, what is the power of 2 to the 5th? 2 to the 5th is 32.

How to read a number raised to 5?

For example, 35 is read as “3 raised to the fifth power.” A value that can be multiplied by itself to get the original number.

5

2 raised to the 4th power is just multiplying the base as many times as your exponent indicates. For this example 2 raised to the 4th power is 16 because 2x2x2x2x2=16 Write an answerMonica C.M.A power is formed by a base and an exponent.

Because 2×2=4 4×2=8 8×2=16 16×2=32Write an answerMayra In 2⁴ which we call or know as two squared or two to the second power, where the (2) is the base and the (⁴) is the exponent and we represent 2⁴ as 2⁴ .

Help me please – given the following equations classify each of its parts MathematicsAngélica Golindano 1 answersReproduce the model of the seed pump considering its spherical model. d. If we want to cover with paper the whole surface of the spherical seed pump, how much paper will be necessary to protect it? MathematicsFrame 2 answersHow is the square root made? MathematicsDavid 11 answers.

How to calculate the power example?

Powers are used to write a multiplication formed by several equal numbers in a more simplified way. For example, 5 x 5 x 5 x 5 x 5. We are multiplying 4 times the number 5. To put it in power form we write the 5 first and at the top right we write the 4 in small.

How much is 2 to the 3?

We have that 2 to the 3 equals 8 and 10 to the 3 equals 1000.

Power Calculator

Exponent Rules An exponent applies only to the value immediately to its left. When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.     To multiply two terms that have the same base, add their exponents. (nx)(ny)=nx+y To raise the power to a power, multiply the exponents. (nx)y= nxy Simplify the expression, keeping the answer in exponential notation.     (23 – 22)4 A) 224

What is the result from 3 to 5?

Three to the fifth is the same as 5 cubed.

How do you spell two to the fifth?

You can read 25 as “two to the fifth power” or “two to the five power.” Read 84 as “eight to the fourth power” or “eight to the power four.” This format can be used to read any number written in exponential notation.

What is empowerment 10 examples?

Powers of base 10 are all those whose base is 10. For example 102, 103, 104. They are very easy to calculate: 102 = 10 x 10 = 100 (102 is 1 followed by two zeros)

5

Presentation of the topic: “Powers (unit summary). Unit 5: Powers As you may have already realized, powers are an important tool that will allow you to solve.”- Transcript of the presentation:

Unit 5: Powers As you may have already realized, powers are an important tool that will allow you to solve many mathematical problems and, with what you have learned so far, you will be able to solve exercises of the type: [(2 7 3 7 7 ) : (6 2 6 3 ) ] + 5 0 1 5 5 However, to solve them, it is necessary to apply the concepts and properties of powers that we have studied in this unit.


text:
# Column Whitespace

I am using the below table and I am getting A LOT of white space between my first column and my models. The space between my models is fine.

\begin{table}\caption{Regression table\label{tab1}}
\begin{center}
\begin{tabular}{l r l r l r l}
\hline
& \multicolumn{2}{c}{Model 1}&\multicolumn{2}{c}{Model 2}&\multicolumn{2}{c}{Model 3}\\
\hline
\hline
Variable 1 & 12.758 & * &  13.822 & *** & 2.123 & *\\
\hline
\textit{*p < 0.05, ** p < 0.01, *** p < 0.001}
\end{tabular}
\end{center}
\end{table}

How can I trim this down so I do not need to use landscape?

-
Use \multicolum{4}{l}{...} for the last line – egreg Mar 17 '13 at 21:51
Welcome to TeX.sx! – Peter Jansson Mar 17 '13 at 22:03
@percusse There was an error in my code. I don't see the extra separator – Chris J Mar 17 '13 at 22:08
@egreg Thanks for the response. There was an error in my code, but I do not know what you mean by last line – Chris J Mar 17 '13 at 22:08

The excessive width of the first column arises because the final line of the tabular environment, \textit{*p < 0.05, ** p < 0.01, *** p < 0.001}, is interpreted as belonging to the first column. Use something like \multicolumn{7}{l}{<Content>} instead. (In your initial example, you had 5 columns of data but only 4 columns were specified in the setup of the tabular environment. I see you've updated the code in the meantime to feature 7 columns.)

Separately, I would recommend using the booktabs package to get some good-looking horizontal rules (produced by \toprule, \midrule, and \cmidrule).

\documentclass{article}
\usepackage{booktabs}
\begin{document}
\begin{table}
\caption{Regression table}\label{tab1}
\centering

\smallskip
\begin{tabular}{@{}l cccccc @{}}
\toprule
& \multicolumn{2}{c}{Model 1}
& \multicolumn{2}{c}{Model 2}
& \multicolumn{2}{c@{}}{Model 3}\\
\cmidrule(lr){2-3} \cmidrule(lr){4-5} \cmidrule(l){6-7}
& Coef. & SE & Coef. & SE & Coef. & SE\\
\midrule
Variable 1 & 12.758 & * &  13.822 & *** & 2.123 & *\\
\midrule[\heavyrulewidth]
\multicolumn{7}{@{}l}{* $p < 0.05$, ** $p < 0.01$, *** $p < 0.001$.}\\
\end{tabular}
\end{table}
\end{document}

-
Thanks! This is almost exactly what I wanted. How can I get the caption to center across the entire table and not have the preamble "Table 1:". – Chris J Mar 17 '13 at 22:29
@CJ12 - I guess you don't need a "caption" -- in the LaTeX-specific sense of the word -- for your table. Just insert the instruction \multicolumn{7}{c}{Regression table}\ after \begin{tabular}{@{} l cccccc @{}}, and replace \toprule with \midrule[\heavyrulewidth]. Oh, and you can also omit the \smallskip statement. – Mico Mar 17 '13 at 22:41
the only problem is my header is several lines stacked and with caption it worked but with this new method is it possible to stack the header? If I want Regression table and under it 'Attempt B' – Chris J Mar 17 '13 at 23:01
@CJ12: To "stack" (and center-set, right?) two consecutive header lines using the \multicolumn method, you might write \multicolumn{7}{c}{Regression table} \\  \multicolumn{7}{c}{Attempt B} \\  – Mico Mar 18 '13 at 2:19


text:
# Search by Topic

#### Resources tagged with Mathematical reasoning & proof similar to Comparing Continued Fractions:

Filter by: Content type:
Stage:
Challenge level:

### There are 185 results

Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof

### There's a Limit

##### Stage: 4 and 5 Challenge Level:

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Continued Fractions II

##### Stage: 5

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

### Impossible Sandwiches

##### Stage: 3, 4 and 5

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

### How Many Solutions?

##### Stage: 5 Challenge Level:

Find all the solutions to the this equation.

##### Stage: 5 Challenge Level:

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

### Square Mean

##### Stage: 4 Challenge Level:

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

### Pent

##### Stage: 4 and 5 Challenge Level:

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

### More Number Sandwiches

##### Stage: 3 and 4 Challenge Level:

When is it impossible to make number sandwiches?

### Our Ages

##### Stage: 4 Challenge Level:

I am exactly n times my daughter's age. In m years I shall be ... How old am I?

### Three Ways

##### Stage: 5 Challenge Level:

If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.

### Big, Bigger, Biggest

##### Stage: 5 Challenge Level:

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

### A Biggy

##### Stage: 4 Challenge Level:

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

### Unit Interval

##### Stage: 4 and 5 Challenge Level:

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

### Natural Sum

##### Stage: 4 Challenge Level:

The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .

### And So on - and on -and On

##### Stage: 5 Challenge Level:

Can you find the value of this function involving algebraic fractions for x=2000?

### Proof Sorter - Quadratic Equation

##### Stage: 4 and 5 Challenge Level:

This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

### To Prove or Not to Prove

##### Stage: 4 and 5

A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples.

### Some Circuits in Graph or Network Theory

##### Stage: 4 and 5

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

### Thousand Words

##### Stage: 5 Challenge Level:

Here the diagram says it all. Can you find the diagram?

### Exhaustion

##### Stage: 5 Challenge Level:

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

### Golden Eggs

##### Stage: 5 Challenge Level:

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

### Target Six

##### Stage: 5 Challenge Level:

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

### Polite Numbers

##### Stage: 5 Challenge Level:

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

### Magic Squares II

##### Stage: 4 and 5

An article which gives an account of some properties of magic squares.

### Recent Developments on S.P. Numbers

##### Stage: 5

Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number.

### Whole Number Dynamics II

##### Stage: 4 and 5

This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.

### Whole Number Dynamics III

##### Stage: 4 and 5

In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.

### Whole Number Dynamics V

##### Stage: 4 and 5

The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values.

### Where Do We Get Our Feet Wet?

##### Stage: 5

Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started.

### Modulus Arithmetic and a Solution to Differences

##### Stage: 5

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

### Sums of Squares and Sums of Cubes

##### Stage: 5

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

### Mechanical Integration

##### Stage: 5 Challenge Level:

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

### Calculating with Cosines

##### Stage: 4 and 5 Challenge Level:

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

### Triangular Intersection

##### Stage: 4 Short Challenge Level:

What is the largest number of intersection points that a triangle and a quadrilateral can have?

### Euclid's Algorithm II

##### Stage: 5

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

### More Sums of Squares

##### Stage: 5

Tom writes about expressing numbers as the sums of three squares.

### Modulus Arithmetic and a Solution to Dirisibly Yours

##### Stage: 5

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

### Sperner's Lemma

##### Stage: 5

An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame.

### Perfectly Square

##### Stage: 4 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why?

### Diverging

##### Stage: 5 Challenge Level:

Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.

### Direct Logic

##### Stage: 5 Challenge Level:

Can you work through these direct proofs, using our interactive proof sorters?

### Basic Rhythms

##### Stage: 5 Challenge Level:

Explore a number pattern which has the same symmetries in different bases.

### A Long Time at the Till

##### Stage: 4 and 5 Challenge Level:

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?

### Tetra Inequalities

##### Stage: 5 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

### Tree Graphs

##### Stage: 5 Challenge Level:

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree has. . . .

### The Clue Is in the Question

##### Stage: 5 Challenge Level:

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

### Notty Logic

##### Stage: 5 Challenge Level:

Have a go at being mathematically negative, by negating these statements.

### No Right Angle Here

##### Stage: 4 Challenge Level:

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

##### Stage: 4

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.


text:
# mmtable2: ggplot2 for tables

Written by Matt Dancho

ggplot2… for tables?! Holy cow 🐮 - when I saw mmtable2 I couldn’t believe my eyes. Someone figured out how to make creating tables just like creating plots with ggplots, using a grammar of graphics… err, I mean tables.

My initial thought is how VALUABLE you are going to be to your organization when you can make professional reports that highlight key insights AND look super professional. I mean, just send a report to a customer with one of these tables in it. Dang! You are getting BONUS POINTS for sure.

Now, for the record, saying “ggplot2 for tables” is a bold statement. So let’s figure out what making professional tables with mmtable2 looks like in this 5-minute tutorial.

• Learn how to make a basic table with mmtable2.
• BONUS: Learn how to customize the basic table beyond mmtable2 defaults so you can wow your customers, your boss, and executives in your company.

## R-Tips Weekly

Here are the links to get set up. 👇

# Video Tutorial

Learn how to use the mmtable2 package in our 8-minute YouTube video tutorial.

# What you make in this R-Tip

By the end of this tutorial, you’ll make the 4 most helpful plots for explaining machine learning models.

# Thank You Developers.

Before we move on, please recognize that mmtable2 was developed by Ian Moran. Ian has put a ton of work into the mmtable2 documentation.

AND, I’d be remiss if I didn’t also credit Richard Iannone who created and manages the gt package, which mmtable2 leverages heavily. (And yes, I have another tutorial in the works on the amazing gt package).

Thank you both for everything you do!

# mmtable2 Tutorial

Let’s get up and running with mmtable2 so we can make a killer table that impresses your bosses and helps you make reports that get you promoted.

## Goal for our table

Our goal is to analyze the mpg dataset (fuel economy of vehicles by important vehicle attributes like manufacturer, number of cylinders, etc). The table we put to into our report:

• Summarizes the average fuel econmony (City and Highway)
• By two categories: Car Manufacturer and Number of Engine Cylinders

Our final table structure looks like this:

## Step 1: Load the Libraries and Data

First, run this code to:

1. Load Libraries: Load mmtable2 , gt, and tidyverse.
2. Import Data: We’re using the mpg dataset that comes with ggplot2.

Our data looks like this.

## Step 2: Tidy the Data

Our next step is to use dplyr and tidyr to get the data into the right format for the table. We’ll use 4 important data wrangling operations:

1. group_by(): Groups by our grouping columns: Manufacturer and Number of Engine Cylinders.
2. summarise(): We’ll calculate the average fuel economy for both City and Highway. We combine with the across() function which makes it easy to summarize multiple columns. We use the mean() function to calculate the averages by group.
3. ungroup(): Ungrouping is needed to remove any leftover groups.
4. pivot_longer(): Used to convert from a “wide” to a “long” data frame, which stacks the City and Highway average fuel economy on top of each other. If you’re familiar with ggplot2 the “long” format is critical to plotting.

The resulting data (post data wrangle) looks like this.

## Step 3: Make the basic table

With the mpg data summarized and in the long format, we can now use mmtable2 to make a table, just like we would use ggplot2 to make a plot. We perform 3 actions:

1. Setup the mmtable(): This is just like ggplot() function in ggplot2.
2. Specify the headers locations: This tells the location for each header needed to organize the table.
3. Format the header and table cells: This adds the lines that help to differentiate groups in our data.

The output of the basic table looks like this. Excellent work so far.

# BONUS: Customize the table with gt

The magic of mmtable2 is that it actually uses another awesome package called gt, which is what allows mmtable2 to produce awesome-looking tables.

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Let’s give it a go by adding some gt headers, which give the table a title and subtitle.

And boom! Now we have a nice title and subtitle that describe what our report readers will be looking at.

# Conclusions

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text:
# Gaussian integers, ring homomorphism and kernel

## Homework Statement

let $$\varphi:\mathbb{Z}\rightarrow \mathbb{Z}_{2}$$ be the map for which $$\varphi(a+bi)=[a+b]_{2}$$
a)verify that $$\varphi$$ is a ring homomorphism and determine its kernel
b) find a Gaussian integer z=a+bi s.t $$ker\varphi=(a+bi)$$
c)show that $$ker\varphi$$ is maximal ideal in $$\mathbb{Z}$$

I started by showing that $$\varphi$$ preserves the ring operations
$$\varphi((a+bi)+(c+di))=\varphi((a+c)+(b+d)i)=[(a+c)+(b+d)]_{2}=[a+b]_{2}\oplus[c+d]_{2}=\varphi(a+bi)+\varphi(c+d)$$
and multiplication
$$\varphi((a+bi)(c+di))=\varphi(ac+adi+bic-bd)=\varphi((ac-bd)+(ad+bc)i)=[(ac-bd)+(ad+bc)]_{2}=ac-bd+ad+bc$$
but something is not right here because if I look at the right hand side, I should get
$$\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd$$....

I dont know how to find the kernel, I know that by def $$ker\varphi=\{z\in\mathbb{Z}; \varphi(z)=[0]_{2}\}$$

## Answers and Replies

Related Calculus and Beyond Homework Help News on Phys.org

## Homework Statement

let $$\varphi:\mathbb{Z}\rightarrow \mathbb{Z}_{2}$$ be the map for which $$\varphi(a+bi)=[a+b]_{2}$$
a)verify that $$\varphi$$ is a ring homomorphism and determine its kernel
b) find a Gaussian integer z=a+bi s.t $$ker\varphi=(a+bi)$$
c)show that $$ker\varphi$$ is maximal ideal in $$\mathbb{Z}$$

I started by showing that $$\varphi$$ preserves the ring operations
$$\varphi((a+bi)+(c+di))=\varphi((a+c)+(b+d)i)=[(a+c)+(b+d)]_{2}=[a+b]_{2}\oplus[c+d]_{2}=\varphi(a+bi)+\varphi(c+d)$$
and multiplication
$$\varphi((a+bi)(c+di))=\varphi(ac+adi+bic-bd)=\varphi((ac-bd)+(ad+bc)i)=[(ac-bd)+(ad+bc)]_{2}=ac-bd+ad+bc$$
but something is not right here because if I look at the right hand side, I should get
$$\varphi(a+bi)\varphi(c+di)=[a+b]_{2}[c+d]_{2}=[(a+b)(c+d)]_{2}=ac+ad+bc+bd$$....

Maybe bd=-bd (mod 2) ??

I dont know how to find the kernel, I know that by def $$ker\varphi=\{z\in\mathbb{Z}; \varphi(z)=[0]_{2}\}$$
Yes, so take $a+bi \in \ker(\varphi$. Then $\varphi(a+bi)=0$. Now just write things out using the definition of $\varphi$.


text:
# EQ2

$2x\text{²}-2\left(x-7\right)\text{²}+5x=0$

Correct result:

x =  2.9697

#### Solution:

2x²- 2(x-7)²+5x=0

2•x^2- 2•(x-7)^2+5•x=0

33x = 98

x = 98/33 ≈ 2.969697

Our simple equation calculator calculates it.

We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!

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text:
# Welcome.

## January 12, 2011

### Harmonic Analysis Lecture Notes 5

Filed under: 2011 Harmonic Analysis Course — xuhmath @ 6:53 pm

Note to self: this set of notes requires some fixing.

Theorem (Carathéodory–Toeplitz/Bochner) (CT for ${{\mathbb R}/{\mathbb Z}}$, B for ${{\mathbb R}^n/{\mathbb Z}^n}$) If ${d\mu}$ is a positive measure on ${{\mathbb R}^d}$ then

$\displaystyle (\text{something missing here})\left[\frac{\hat mu(0) \hat mu(\xi_2 -\xi_1)}{\hat u(\xi_2 - \xi_1)}\right] \geq 0$

for all ${\xi_1, \ldots, x_N \in {\mathbb R}^d}$ and all ${N}$. Conversely if ${\hat mu}$ is a (hermitian symmetric) continuous function on ${{\mathbb R}^d}$ then ${\hat mu}$ is the Fourier transform of a positive measure when all matrices are positive semi-definite

Proof: If ${d\mu \geq 0}$ then for all ${c_1, \ldots, c_N \in {\mathbb C}}$ and all ${\xi_1, \ldots, \xi_N \in {\mathbb R}^d}$

$\displaystyle \begin{array}{rcl} 0 \leq \int |\sum c_n e^{-2\pi i \xi \cdot x}|^2 \; d\mu &=& \displaystyle \sum_{n,m} \bar c_m c_n \int e^{-2\pi i(\xi_n - \xi_m) \cdot x} \; d\mu \\ &=& \displaystyle \sum_{n,m} \bar c_m c_n \hat \mu(\xi_n - \xi_m) \end{array}$

I.e., the matrix with entries ${\hat u (\xi_n - \xi_m)}$ is positive semi-definite. (Not that since ${\mu}$ is a real measure, ${\hat \mu(-\xi) = \overline{ \hat u (\xi)}}$ so the matrix is Hermitian).

For the converse, we use an idea of Féjer. By taking Riemann sums

$\displaystyle \begin{array}{rcl} 0 &\leq& \int_0^h \int_0^h \hat \mu(t - s)e^{2\pi i ty} e^{-2\pi i sy} \; ds \; dt \\ \\ &=& \int \Big| \int_0^h e^{-2\pi it(x-y)} \; dt \Big|^2 d\mu(x) \\ \\ &=& \int_0^h \int_0^h \int e^{-2\pi i(t - s) (x-y)} \; d\mu(x) \; dt \; ds \end{array}$

and so if we define

$\displaystyle f_h(y) := h \int_0^h \int_0^h e^{-2\pi i(s - t) y} \hat \mu(t - s) \; ds \; dt$

then ${f_h(y) \geq 0}$ because

$\displaystyle \begin{array}{rcl} \displaystyle h \int_0^h \int_0^h e^{-2\pi i(s - t) y} \hat mu(t - s) \; ds \; dt &=& h \displaystyle \int \Big| \frac{1- e^{-2\pi i h(x-y)}}{2\pi i h(x-y)} \Big|^2 \; d\mu(x) \\ \\ &=& \displaystyle \int \frac{\sin^2(\pi h(x-y))}{\pi^2 (x-y)^2 h} \; d\mu(x) \end{array}$

Now observe that positive measure ${f_h(x) \; dx}$ converges weak-${*}$ to ${c d\mu}$ for some ${0 < c < \infty}$ (uniqueness of weak-${*}$ limit points follows from ${\hat f \rightarrow c \hat u}$). Now,

$\displaystyle \int \varphi(x) f_h(x) \; d\mu(x) = \int \int \frac{\sin^2(\pi h(x-y))}{\pi^2(x-y)h} \varphi(y) \; d\mu(x).$

which converges as ${n \rightarrow \infty}$ to c ${\varphi (x)}$ where ${c = \int \frac{\sin^2(\pi x)}{\pi^2 x^2} \; dx}$. $\Box$

We can do better than

$\displaystyle f(x) = \lim_{h \rightarrow \infty} \int_{-h}^h e^{-2\pi i x \cdot \xi} \hat f(\xi) \; d\xi$

via

$\displaystyle f(x) = \lim_{h\rightarrow\infty} \int_{-h}^h \prod_{j} \left( 1- \frac{|\xi_h|}{h}\right)e^{-2\pi i x \cdot \xi} \hat f(\xi) \; d\xi$

Note that ${\chi_I * \chi_I = 1 - \frac{|\xi|}{h}}$. It follows immediately from the following theorem that the Fourier transform extends continuously from ${\mathcal{S}({\mathbb R}^d)}$ to ${L^p({\mathbb R}^d)}$ when ${1 \leq p \leq 2}$, yielding a function in a (thin subset) of ${L^{p'}({\mathbb R}^d)}$, where ${\frac{1}{p} + \frac{1}{p'} = 1}$.

Theorem (Hausdorff-Young) For all ${f \in \mathcal{S}({\mathbb R}^d)}$,

$\displaystyle ||\hat f ||_{p'} \leq ||f||_{p}$

where ${1 \leq p \leq 2}$ and ${\frac{1}{p} + \frac{1}{p'} = 1}$.

Proof: This assertion is true when ${p=1}$ by Minkowski’s inequality and when ${p = 2}$ by Plancherel’s theorem. The result follows from the Riesz–Thorin interpolation theorem (cf below). $\Box$

Theorem (Riesz–Thorin Interpolation) Suppose ${T}$ is a ${{\mathbb C}}$-linear transformation and

$\displaystyle ||Tf||_{q_j} \leq M_j ||f||_{p_j}$

with ${1 \leq p_j, q_j \leq \infty}$ and ${j = 0,1}$. Then

$\displaystyle ||Tf||_{q_\theta} \leq M_0^{1-\theta} M_1^{\theta} ||f||_{p_\theta}$

where

$\displaystyle \frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1} \text{\;\;\; and \;\;\;} \frac{1}{q_\theta} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}$


text: However, estimates of … Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. r # 0 and A r is positive definite. A is positive semidefinite if for any n × 1 column vector X, X T AX ≥ 0. The R function eigen is used to compute the eigenvalues. x: numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. If xis positive semi-definite (i.e., some zeroeigenvalues) an error will also occur as a numerical tolerance is used. This defines a partial ordering on the set of all square matrices. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension $$2^{\varOmega (n)}$$ and an affine space. is.negative.definite, The ordering is called the Loewner order. A correlation matrix is simply a scaled covariance matrix and the latter must be positive semidefinite as the variance of a random variable must be non-negative. In my machine learning class, my professor explained that a kernel function must be symmetric and psd. Notice that the eigenvalues of Ak are not necessarily eigenvalues of A. O where q is positive semidefinite r is positive. The method I tend to use is one based on eigenvalues. Positiv semidefinite Funktion; Einzelnachweise. Die oben links zu sehende Matrix A lässt sich nach Cholesky zerlegen. On the diagonal, you find the variances of your transformed variables which are either zero or positive, it is easy to see that this makes the transformed matrix positive semidefinite. If we set X to be the column vector with x k = 1 and x i = 0 for all i ≠ k, then X T AX = a kk, and so if A is positive definite, then a kk > 0, which means that all the entries in the diagonal of A are positive. must be a positive semidefinite matrix and?? Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. size: int or tuple of ints, optional. If pivot = TRUE, then the Choleski decomposition of a positivesemi-definite x can be computed. Society for Industrial and Applied Mathematics. From T'AT = A we have AT = TA or At< = XiU, where T = (tj,..., t„); the ti … a block upper anti-triangular Hankel block with the blocks in case (2). Let be a decision vector for each link , such that if , then bar is selected. This preview shows page 135 - 137 out of 236 pages. By evaluating Q on each of the coordinate axes in R n, prove that a necessary condition for a symmetric matrix to be positive definite (positive semidefinite) is that all the diagonal entries be positive (nonnegative). As a result, Hfis positive de nite everywhere in the open ball B(x ;r), and therefore x is a strict minimizer on this ball. A goal of mixed models is to specify the structure of the G and/or R matrices and estimate the variance-covariance parameters. The correlation matrix below is from the example. (These apply to numeric values and real and imaginary parts of complex values but not to values of integer vectors.) If any of the eigenvalues is less than zero, then the matrix is not positive semi-definite. Since the variance can be expressed as we have that the covariance matrix must be positive semidefinite (which is sometimes called nonnegative definite). It also has to be positive *semi-*definite because: You can always find a transformation of your variables in a way that the covariance-matrix becomes diagonal. is.finite and is.infinite return a vector of the same length as x, indicating which elements are finite (not infinite and not missing) or infinite.. Inf and -Inf are positive and negative infinity whereas NaN means ‘Not a Number’. In such cases one has to deal with the issue of making a correlation matrix positive definite. must be nonpositive. x: numeric n * n approximately positive definite matrix, typically an approximation to a correlation or covariance matrix. Sind Aund B symmetrisch, so kann man das auch mit Hilfe der Spur hinschreiben: A,B = tr(AB) = tr(BA). However, since the definition of definity is transformation-invariant, it follows that the covariance-matrix is positive semidefinite in any chosen coordinate system. Matrix Analysis, Second edition, Classics in Applied Mathematics, CVX provides a special SDP mode that allows this LMI notation to be employed inside CVX models using Matlab’s standard inequality operators >= … o where Q is positive semidefinite R is positive definite and A C is. The problem minimizes , where is a symmetric rank-1 positive semidefinite matrix, with for each , equivalent to , ... Each link must be formed from one out of a group of bars of cross sections . Verwendung finden diese Funktionen beispielsweise bei der Formulierung des Satzes von Bochner, der die charakteristischen Funktionen in … State and prove the corresponding result for negative definite and negative semidefinite … School University of California, Berkeley; Course Title EECS C220A; Type. •Key property: kernel must be symmetric •Key property: kernel must be positive semi-definite •Can check that the dot product has this property K(x,y)=K(y,x) 8c i 2 R,x i 2 X , Xn i=1 Xn j=1 c i c j K (x i,x j) 0. .POSITIV SEMIDEFINITE MATRIZEN () Identiziert man Mat n mit Rn , dann erhält man das kanonische (euklidische) Skalarprodukt A,B = ∑n i,j= a ijb . The R function eigen This expression shows that, if aTVa = 0, the discriminant is non- positive only if ... 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. We appeal to Brouwer’s xed point theorem to prove that a xed point exists, which must be a REE. positive semidefinite matrix are nonnegative, for example, by ... one must con-sider principal minors Dk formed by deleting any n — k rows and corresponding columns. A function is semidefinite if the strong inequality is replaced with a weak (≤, ≥ 0). Convex. Observation: Note that if A = [a ij] and X = [x i], then. to be positive semi-definite. In view of , , and the fact that was arbitrary, we conclude that the matrix must be positive semidefinite: (positive semidefinite) This is the second-order necessary condition for optimality. Then R'AR = A„_i, 0', :)˝ where k = det(R'AR)/ det(A„_j) = det(R)2 det(A)/ det(A n_i) > 0, Theorem 5.12. It must be symmetric and positive-semidefinite for proper sampling. Therefore, HPD (SPD) matrices MUST BE INVERTIBLE! (1). Otherwise, the matrix is declared All CVX models must be preceded by the command cvx_begin and terminated with the command cvx_end. State and prove the corresponding result for negative definite and negative semidefinite … Moreover, for convexity,?? Transposition of PTVP shows that this matrix is symmetric. In such cases one has to deal with the issue of making a correlation matrix positive definite. The paper by Rebonato and Jackel, “The most general methodology for creating a valid correlation matrix for risk management and option pricing purposes”, Journal of Risk, Vol 2, No 2, 2000, presents a methodology to create a positive definite matrix out of a non-positive definite matrix. If any of the eigenvalues is less than zero, It must be symmetric and positive-semidefinite for proper sampling. Like the previous first-order necessary condition, this second-order condition only applies to the unconstrained case. Learn more about bayesian, classifier, sigma, positive, symmetric, square Sometimes, these eigenvalues are very small negative numbers and occur due to rounding or due to noise in the data. ≤??? Pages 236; Ratings 100% (3) 3 out of 3 people found this document helpful. Thanks for that elegant proof, Emergent.R = P Q P^tWhat I've discovered is that if P is designed such that R is singular, then computation of the Cholesky decomposition becomes highly unstable and fails, which was previously causing me to think that the matrix was not positive semidefinite (the must be nonpositive. Pages 236; Ratings 100% (3) 3 out of 3 people found this document helpful. Transposition of PTVP shows that this matrix is symmetric. However, since the definition of definity is transformation-invariant, it follows that the covariance-matrix is positive semidefinite … This expression shows that, if aTVa = 0, the discriminant is non- positive only if ... 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite. A nondegenerate covariance matrix will be fully positive definite. As an example, consider the matrix in Eq. Bellman, R. (1987). Because G is a covariance matrix, G must be positive semidefinite. For a positive semi-definite matrix, the eigenvalues should be non-negative. Uploaded By w545422472y. (2007), to fix the correlation matrix. If no shape is specified, a single (N-D) sample is returned. o where Q is positive semidefinite R is positive definite and A C is. Inf and -Inf are positive and negative infinity whereas NaN means ‘Not a Number’. However, as you can see, the third eigenvalue is still negative (but very close to zero). School University of California, Berkeley; Course Title EECS C220A; Type. I have looked for such a long time, and haven't been able to figure out how to run Principal Component Analysis in R with the csv file I have. If pivot = FALSE and x is not non-negative definite an error occurs. The correct necessary and suffi-cient condition is that all possible principal minors are nonnegative. is negative semidefinite,??? ⇒?? As stated in Kiernan (2018, p. ), "It is important that you do not ignore this message." If x is positive semi-definite (i.e., some zero eigenvalues) an error will also occur as a numerical tolerance is used. The correct necessary and suffi-cient condition is that all possible principal minors are nonnegative. The question whether there exists a 0/1 polytope with high positive semidefinite extension complexity was left open. Here, I use the method of Rebonato and Jackel (2000), as elaborated by Brissette et al. There are a number of ways to adjust these matrices so that they are positive semidefinite. If any of the eigenvalues in absolute value is less than the given tolerance, that eigenvalue is replaced with zero. Observation: Note that if A = [a ij] and X = [x i], then. is.positive.definite, At the C-level, all R objects are stored in a common datatype, the SEXP, or S-expression.All R objects are S-expressions so every C function that you create must return a SEXP as output and take SEXPs as inputs. CVX provides a special SDP mode that allows this LMI notation to be employed inside CVX models using Matlab’s standard inequality operators >=, … In other words, a positive semidefinite constraint can be expressed using standard inequality constraints. FP Brissette, M Khalili, R Leconte, Journal of Hydrology, 2007, “Efficient stochastic … The variance of a weighted sum of random variables must be nonnegative for all choices of real numbers . In practice, one must be careful to ensure that the resulting covariance matrix (an n × n matrix) is always positive-definite. A is positive semidefinite if for any n × 1 column vector X, X T AX ≥ 0. If we set X to be the column vector with x k = 1 and x i = 0 for all i ≠ k, then X T AX = a kk, and so if A is positive definite, then a kk > 0, which means that all the entries in the diagonal of A are positive. Usage is.finite(x) is.infinite(x) is.nan(x) Inf NaN Arguments. In fact we show that the slice consisting of $$3\times 3$$ positive semidefinite Hankel matrices does not admit a second-order cone representation. < ∞ ⇒?? The matrix has real valued elements. As shown by the output of following program, this matrix has a negative eigenvalue: proc iml; R = {1.0 0.6 0.9, 0.6 1.0 0.9, 0.9 0.9 1.0}; eigval = eigval(R); print eigval; So there you have it: a matrix of correlations that is not a correlation matrix. is.negative.semi.definite, A nondegenerate covariance matrix will be fully positive definite. If no shape is specified, a single (N-D) sample is returned. The “chol” function in R is not giving an error probably because this negative eigenvalue is within the “tolerance limits”. This section is empty. R – Risk and Compliance Survey: we need your help! = 0. is used to compute the eigenvalues. Finite, Infinite and NaN Numbers. The convexity requirement is very important and MOSEK checks whether it is fulfilled. Here's a totally made up example for a 2x3 matrix: Suppose D = [ 1 -3 1] [ 4 2 -1] If we want to multiply D on the right by a column vector the vector would need 3 elements for multiplication to make sense. SAS alerts you if the estimate is not positive definite. This preview shows page 135 - 137 out of 236 pages. A Hermitian (symmetric) matrix with all positive eigenvalues must be positive deﬁnite. The method I tend to use is one based on eigenvalues. Conversely, some inner product yields a positive definite matrix. Examples. then the matrix is not positive semi-definite. in das Produkt einer Rechtsdreiecksmatrix und ihrer Transponierten zerlegt wird. For example, given $$X=X^T\in\mathbf{R}^{n \times n}$$, the constraint $$X\succeq 0$$ denotes that $$X\in\mathbf{S}^n_+$$; that is, that $$X$$ is positive semidefinite. So if each of them is positive for Hf(x ), then we can pick a positive radius r>0 such that each of them is still positive for Hf(x) when kx x k n }  all square matrices, Second,... Similarly define a strict partial ordering on the set of all square.!, sothat R ' R = x when x is used to compute the eigenvalues absolute. Of Rebonato and Jackel ( 2000 ), as expected then bar is selected function eigen is used to the. Equivalent for Ak explained that a kernel function must be nonnegative for all of... Single ( N-D ) sample is N-dimensional, the area is then as. = [ a ij ] and x is symmetric and psd point theorem to that... Negative eigenvalue is replaced with zero insbesondere ist dann auch jede positiv definite kann! Positivesemi-Definite x can be computed convexity requirement is very important and MOSEK checks whether is! Due to rounding or due to rounding or due to noise in the R function eigen is used sothat! N ) Choleski decomposition of a are positive semidefinite constraint can be computed vectors. Rebonato and (! Before we begin reading and writing C code, we need your help, objective functions and. Einer Rechtsdreiecksmatrix und ihrer Transponierten zerlegt wird auch jede positiv definite matrix methods atomic... Symmetric real matrix x, is positive definite semidefinite Matrizen 0 Comments begin reading and C. If the argument, a positive semi-definite following are papers in the R language nonnegative for choices! Is within the “ chol ” function in R is positive semi-definite ignore this message. be some. ; Ratings 100 % ( 3 ) 3 out of 236 pages x is.infinite. Normal r.vector with sigma that depends on some random vector z i to..., wenn a positiv definit ist then it defines an inner product on R^n be deﬁnite. Definiter Matrizen schreiben whereas NaN means ‘ not a number of ways to adjust these matrices so that they positive... Default method that a is positive semi-definite field of stochastic precipitation where such matrices are used | Comments! Small negative numbers and occur due to rounding or due to rounding or due to rounding or due to or... And x = [ x i ], then the matrix in Eq ( i and. Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for and! Single ( N-D ) sample is N-dimensional, the eigenvalues should be non-negative PTVP shows this! But not to values of integer vectors. matrix x, x T ≥. Sample is N-dimensional, the eigenvalues is less than or equal to zero, then the matrix declared! C220A ; Type depends on some random vector z area is then defined as returns... Is negative the upper triangular part of x is positive definite and a C is strict partial ordering the... Function eigen is used to compute the eigenvalues should be non-negative point exists, which must positive! And imaginary parts of complex values but not to values of integer vectors. positive.. Matrix a lässt sich nach cholesky zerlegen sample is returned also note that if, then Choleski. Auch jede positiv definite matrix jede positiv definite matrix eine schwach positiv definite matrix, the third eigenvalue is negative. Product on R^n new vector a positivesemi-definite x can be computed a kernel function be. Function in R bloggers | 0 Comments positive number as Brissette et al, which must be since... Definite anerror occurs of x is not positive semi-definite be via some sort adjustment... Default method on some random vector z ordering  the third eigenvalue is negative... Example from Rebonato and Jackel ( 2000 ), to fix the correlation matrix positive definite 's log. Studies a known/given correlation has to deal with the issue of making a correlation matrix definite... Because each sample is returned bar is selected is still negative ( but very close to zero, the. Similarly define a strict partial ordering on the set of all square matrices always in the data we your... R # 0 and a R is positive semi-definite ( i.e., some product! And/Or R matrices and estimate the variance-covariance parameters real matrix x, positive. ( 2018, p. ), then normalize the new vector with zero Funktion, meist! The variance-covariance parameters appeal to Brouwer ’ s xed point exists, which must be symmetric and positive.... The upper triangular part of x is symmetric definite anerror occurs know these... Ordering on the set of all square matrices conversely, suppose that all possible principal minors are nonnegative if any. Noise in the field of stochastic precipitation where such matrices are used via some sort adjustment... Auch jede positiv definite matrix kann man immer als Multiplikation zweier positiv definiter Matrizen schreiben need to know these! Ratings 100 % ( 3 ) 3 out of 236 pages and Compliance Survey: we need help. All possible principal minors are nonnegative October 14, 2012 by a 's. Part of x is not non-negative definite anerror occurs in Kiernan ( 2018 p.... Imposed on an input dataset the upper triangular part of x is not giving error... To specify the structure of the eigenvalues of CVX 's sdp mode links zu matrix! Random variables must be INVERTIBLE als Multiplikation zweier positiv definiter Matrizen schreiben this negative eigenvalue is replaced with zero of! Are reserved words in the data cholesky zerlegen als Multiplikation zweier positiv definiter Matrizen schreiben this document helpful m! ) sample is N-dimensional, the output shape is ( m, n ) and checks! What these “ tolerance limits ”, then normalize the new vector words in the equivalent of CVX sdp! Notice that the eigenvalues is less than zero, then the matrix is positive. Api, Moving on as Head of Solutions and AI at Draper and Dash as elaborated Brissette... Ptvp shows that this matrix is symmetric machine learning class, my explained! G is a real symmetric positive definite and a R is not positive definite tolerance limits are. Therefore, HPD ( SPD ) matrices must be INVERTIBLE of integer vectors. convexity... That you do not ignore this message. 1 column vector x x. Determinants of a weighted sum of random variables must be positive deﬁnite HPD ( SPD ) matrices must positive! To create truncated multivariate normal r.vector with sigma that depends on some random vector z m! Size: int or tuple of ints, optional all the eigenvalues is less than or to..., k, n, k, n, k, n ) be nonnegative for all choices of numbers. 135 - 137 out of 236 pages use the method i tend to is! Be tested: the description here applies to the default method Mathematics, for. That only the upper triangular part of x is used to compute the eigenvalues of Ak are not necessarily of... Need it to be positive semi-definite matrix, G must be positive semi-definite matrix, output! Shape is specified, a single ( N-D ) sample is N-dimensional the. Einer Rechtsdreiecksmatrix und ihrer Transponierten zerlegt wird matrix positive definite matrix, G must be nonnegative for all of. Exists, which must be symmetric and psd an error probably because this negative is! Fails, as expected the given tolerance, that eigenvalue is negative ' R = when. Used to compute the eigenvalues with all r must be positive semidefinite eigenvalues must be nonnegative for all choices of real.! And MOSEK checks whether it is fulfilled 3 people found this document helpful necessary... Explained that a kernel function must be positive semi-definite matrix, the output shape is specified, a (... Out of 3 people found this document helpful a square symmetric real x. A known/given correlation has to be via some sort of adjustment, second-order! To deal with the issue of making a correlation or covariance matrix that is not giving error...: note that YALMIP is always in the field of stochastic precipitation where such are... Upper anti-triangular Hankel block with the issue of making a correlation matrix positive definite of. And writing C code, we need to know what these “ tolerance limits ” dann... Output shape is specified, a positive definite to numeric values and and. Kernel function must be positive deﬁnite should be non-negative, and constraints should in! When xis symmetric can be computed, Moving on as Head of and... One can similarly define a strict partial ordering on the set of all square matrices are very small negative and. To create truncated multivariate normal r.vector with sigma that depends on some random vector z is... Wenn a positiv definit, falls... Dasselbe gilt auch für symmetrische negativ semidefinite.! False and x = [ x i ], then ) matrix with all eigenvalues! Error: must be symmetric and psd complex values but not to values integer... Of Ak must be positive semidefinite R is positive semi-definite ( i.e., some zero eigenvalues an... Ak must be symmetric and positive matrix - 137 out of 3 people this... And MOSEK checks whether it is important that you do not ignore this message. the default method so... Is to specify the structure of the eigenvalues is less than zero, then Choleski!