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Problem: Show there are finitely many primes. “Solution”: Suppose to the contrary there are infinitely many primes. Let $ P$ be the set of primes, and $ S$ the set of square-free natural numbers (numbers whose prime factorization has no repeated factors). To each square-free number $ n \in S$ there corresponds a subset of primes, specifically the primes which make up $ n$’s prime factorization. Similarly, any subset $ Q \subset P$ of primes..
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This is the first in a series of “false proofs.” Despite their falsity, they will be part of the Proof Gallery. The reason for putting them there is that often times a false proof gives insight into the nature of the problem domain. We will be careful to choose problems which do so. Problem: Show 1 = 2. “Solution”: Let $ a=b \neq 0$. Then $ a^2 = ab$, and $ a^2 – b^2 = ab – b^2$.
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This is the first in a series of “false proofs.” Despite their falsity, they will be part of the Proof Gallery. The reason for putting them there is that often times a false proof gives insight into the nature of the problem domain. We will be careful to choose problems which do so. Problem: Show 1 = 2. “Solution”: Let $ a=b \neq 0$. Then $ a^2 = ab$, and $ a^2 – b^2 = ab – b^2$.
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This is the first in a series of “false proofs.” Despite their falsity, they will be part of the Proof Gallery. The reason for putting them there is that often times a false proof gives insight into the nature of the problem domain. We will be careful to choose problems which do so. Problem: Show 1 = 2. “Solution”: Let $ a=b \neq 0$. Then $ a^2 = ab$, and $ a^2 – b^2 = ab – b^2$.
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Problem: $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 1$ Solution: Problem: $ \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots = \frac{1}{2}$ Solution: Problem: $ \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots = \frac{1}{3}$ Solution: Problem: $ 1 + r + r^2 + \dots = \frac{1}{1-r}$ if $ r < 1$. Solution: This last one follows from similarity of the subsequent trapezoids: the right edge of the teal(ish) trapezoid has length $ r$, a..
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Problem: $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 1$ Solution: Problem: $ \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots = \frac{1}{2}$ Solution: Problem: $ \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots = \frac{1}{3}$ Solution: Problem: $ 1 + r + r^2 + \dots = \frac{1}{1-r}$ if $ r < 1$. Solution: This last one follows from similarity of the subsequent trapezoids: the right edge of the teal(ish) trapezoid has length $ r$, a..
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Problem: $ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 1$ Solution: Problem: $ \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots = \frac{1}{2}$ Solution: Problem: $ \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots = \frac{1}{3}$ Solution: Problem: $ 1 + r + r^2 + \dots = \frac{1}{1-r}$ if $ r < 1$. Solution: This last one follows from similarity of the subsequent trapezoids: the right edge of the teal(ish) trapezoid has length $ r$, a..
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I just updated this blog to wordpress 3.2 which came out this week. Only a small glitch caused by me running an old theme which wasn’t 100% compatible. WordPress itself seems to be better. I had a quick look at the Twenty Eleven theme which comes packaged with wordpress and it looks nice even via … Continue reading July Update
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So after a fairly lengthy period of inattention to my Linkzie bookmark manager, I got frustrated on my current project and turned my focus to Linkzie today for some updates. There's a new menu bar shamelessly stolen from the new google bar. A brand new "beach house" color theme has been applied. The "organize" UI has been changed significantly. Lots of other minor UX tweaks have been made. For example, you can now click anywhere on the en....
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The SharePoint permission model uses user impersonation, where a typical asp.net application will run under the permissions of the…
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Jakob Nielsen recently published this post, where he (and I guess his team) analyze why the Wall Street Journal mobile app gets such bad customer reviews. It all stems from a horrible interface where the customers are led to believe that they need to pay separately for the monthly subscription to to the mobile app, when in fact it's free for existing subscribers.
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Google Web Fonts 2 i font gratuiti e ottimizzati per il Web
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nicolaiarocci.com
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14 years ago
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ita
Il progetto Google Web Fonts v2 offre a webmaster, designer e sviluppatori un accesso facile e gratuito a una ricca selezione di fonti tipografiche di qualità, ottimizzate per il web. Google sostiene che non ci dovrebbero essere barriere finanziare per la costruzione di siti web, e che il progetto Google Web Font contribuisce alla creazione di un web più bello, leggibile, accessibile e aperto. Secondo Google nei prossimi anni la gran part..
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MySQL turns out to be a good Swiss Army Knife for persistence, if used wisely. Understanding disk access patterns driven by your storage…
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Jakob Nielsen recently published this post, where he (and I guess his team) analyze why the Wall Street Journal mobile app gets such bad customer reviews. It all stems from a horrible interface where the customers are led to believe that they need to pay separately for the monthly subscription to to the mobile app, when in fact it's free for existing subscribers.
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MySQL turns out to be a good Swiss Army Knife for persistence, if used wisely. Understanding disk access patterns driven by your storage…
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We assume the reader is familiar with the concepts of determinism and finite automata, or has read the corresponding primer on this blog. The Mother of All Computers Last time we saw some models for computation, and saw in turn how limited they were. Now, we open Pandrora’s hard drive: Definition: A Turing machine is a tuple $ (S, \Gamma, \Sigma, s_0, F, \tau)$, where $ S$ is a set of states, $ \Gamma$ is a set of tape symbols, including a ..
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We assume the reader is familiar with the concepts of determinism and finite automata, or has read the corresponding primer on this blog. The Mother of All Computers Last time we saw some models for computation, and saw in turn how limited they were. Now, we open Pandrora’s hard drive: Definition: A Turing machine is a tuple $ (S, \Gamma, \Sigma, s_0, F, \tau)$, where $ S$ is a set of states, $ \Gamma$ is a set of tape symbols, including a ..
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We assume the reader is familiar with the concepts of determinism and finite automata, or has read the corresponding primer on this blog. The Mother of All Computers Last time we saw some models for computation, and saw in turn how limited they were. Now, we open Pandrora’s hard drive: Definition: A Turing machine is a tuple $ (S, \Gamma, \Sigma, s_0, F, \tau)$, where $ S$ is a set of states, $ \Gamma$ is a set of tape symbols, including a ..
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I have noticed that many people don’t know what a blog is, so I would like to talk about that briefly. A blog is a type of website or part of a website. Blogs are usually maintained by an individual with regular entries. (…) Entries are commonly displayed in reverse-chronological order. Blog can also be used as a verb, meaning to maintain or add content to a blog. Source: Wikipedia A blog is a website. A blog has many entries or..
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I have noticed that many people don’t know what a blog is, so I would like to talk about that briefly. A blog is a type of website or part of a website. Blogs are usually maintained by an individual with regular entries. (…) Entries are commonly displayed in reverse-chronological order. Blog can also be used as a verb, meaning to maintain or add content to a blog. Source: Wikipedia A blog is a website. A blog has many entries or..
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I saw this article on Hacker News today. It's very similar to a blog post I was planning to write, but never did.
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I saw this article on Hacker News today. It's very similar to a blog post I was planning to write, but never did.
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The first step in studying the sorts of possible computations (and more interestingly, those things which cannot be computed) is to define exactly what we mean by a “computation.” At a high level, this is easy: a computation is simply a function. Given some input, produce the appropriate output. Unfortunately this is much too general. For instance, we could define almost anything we want in terms of functions. Let $ f$ be the function which..
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The first step in studying the sorts of possible computations (and more interestingly, those things which cannot be computed) is to define exactly what we mean by a “computation.” At a high level, this is easy: a computation is simply a function. Given some input, produce the appropriate output. Unfortunately this is much too general. For instance, we could define almost anything we want in terms of functions. Let $ f$ be the function which..
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The first step in studying the sorts of possible computations (and more interestingly, those things which cannot be computed) is to define exactly what we mean by a “computation.” At a high level, this is easy: a computation is simply a function. Given some input, produce the appropriate output. Unfortunately this is much too general. For instance, we could define almost anything we want in terms of functions. Let $ f$ be the function which..
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Check out my new little automation utility belt app for the keyboard-centric power users. It is hosted on a github project here .
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Problem: Prove that for all $ n,k \in \mathbb{N}, k > 1$, we have $$\sum \limits_{i=0}^{n} k^i = \frac{k^{n+1}-1}{k-1}$$ Solution: Representing the numbers in base $ k$, we have that each term of the sum is all 0’s except for a 1 in the $ i$th place. Hence, the sum of all terms is the $ n$-digit number comprised of all 1’s. Multiplying by $ k-1$ gives us the $ n$-digit number where every digit is $ k-1$.
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Problem: Prove that for all $ n,k \in \mathbb{N}, k > 1$, we have $$\sum \limits_{i=0}^{n} k^i = \frac{k^{n+1}-1}{k-1}$$ Solution: Representing the numbers in base $ k$, we have that each term of the sum is all 0’s except for a 1 in the $ i$th place. Hence, the sum of all terms is the $ n$-digit number comprised of all 1’s. Multiplying by $ k-1$ gives us the $ n$-digit number where every digit is $ k-1$.
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Problem: Prove that for all $ n,k \in \mathbb{N}, k > 1$, we have $$\sum \limits_{i=0}^{n} k^i = \frac{k^{n+1}-1}{k-1}$$ Solution: Representing the numbers in base $ k$, we have that each term of the sum is all 0’s except for a 1 in the $ i$th place. Hence, the sum of all terms is the $ n$-digit number comprised of all 1’s. Multiplying by $ k-1$ gives us the $ n$-digit number where every digit is $ k-1$.
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So, Google Plus launched, the first truly viable Facebook competitor. The timing is quite interesting, given Google's recent failures with the Buzz microblogging platform, and the impending Facebook IPO . After a bit of time with Plus, here are some thoughts: Google already knows everything I do, so sharing stuff there feels less risky The UI is pretty and a lot less bloated than Facebook's Messages and comments can be edited, saving fr....
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Another cache buffers chains latch contention troubleshooting example using LatchProf
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tanelpoder.com
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14 years ago
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eng
One of my blog readers recently dropped me an email noting that he had noticed some cache buffers chains latch contention recently and successfully troubleshooted it with LatchProf . I asked if he’d like to blog about it and here’s the article: http://web.archive.org/web/20111113062613/http://orapsdba.wordpress.com/2011/06/21/another-latchcache-buffer-chains-troubleshooting Cache buffer chains latch contention typically shows up wh..
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Another cache buffers chains latch contention troubleshooting example using LatchProf
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tanelpoder.com
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14 years ago
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eng
One of my blog readers recently dropped me an email noting that he had noticed some cache buffers chains latch contention recently and successfully troubleshooted it with LatchProf . I asked if he’d like to blog about it and here’s the article: http://web.archive.org/web/20111113062613/http://orapsdba.wordpress.com/2011/06/21/another-latchcache-buffer-chains-troubleshooting Cache buffer chains latch contention typically shows up wh..
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Additional Patterns Last time we left the reader with the assertion that Conway’s game of life does not always stabilize. Specifically, there exist patterns which result in unbounded cell population growth. Although John Conway’s original conjecture was that all patterns eventually stabilize (and offered $50 to anyone who could provide a proof or counterexample), he was proven wrong. Here we have the appropriately named glider gun, whose ma..
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Additional Patterns Last time we left the reader with the assertion that Conway’s game of life does not always stabilize. Specifically, there exist patterns which result in unbounded cell population growth. Although John Conway’s original conjecture was that all patterns eventually stabilize (and offered $50 to anyone who could provide a proof or counterexample), he was proven wrong. Here we have the appropriately named glider gun, whose ma..
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Additional Patterns Last time we left the reader with the assertion that Conway’s game of life does not always stabilize. Specifically, there exist patterns which result in unbounded cell population growth. Although John Conway’s original conjecture was that all patterns eventually stabilize (and offered $50 to anyone who could provide a proof or counterexample), he was proven wrong. Here we have the appropriately named glider gun, whose ma..
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Cellular Automata There is a long history of mathematical models for computation. One very important one is the Turing Machine, which is the foundation of our implementations of actual computers today. On the other end of the spectrum, one of the simpler models of computation (often simply called a system) is a cellular automaton. Surprisingly enough, there are deep connections between the two. But before we get ahead of ourselves, let’s se..
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Cellular Automata There is a long history of mathematical models for computation. One very important one is the Turing Machine, which is the foundation of our implementations of actual computers today. On the other end of the spectrum, one of the simpler models of computation (often simply called a system) is a cellular automaton. Surprisingly enough, there are deep connections between the two. But before we get ahead of ourselves, let’s se..
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Cellular Automata There is a long history of mathematical models for computation. One very important one is the Turing Machine, which is the foundation of our implementations of actual computers today. On the other end of the spectrum, one of the simpler models of computation (often simply called a system) is a cellular automaton. Surprisingly enough, there are deep connections between the two. But before we get ahead of ourselves, let’s se..
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I just noticed that Jonathan Lewis has announced that he’s writing a new Oracle (fundamental) internals book , due to be out in November. So, I’m happy to add to Jonathan’s announcement, that I’m the tech reviewer of that book! After all the hard work on the Exadata book , I didn’t want to hear about working on any book again (even if it’s just tech reviewing work), but as this is Jonathan’s book, about exactly these topics I love and..
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