# Tag Info

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Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae. The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the ...

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The $\sqrt{-1}$, complex analysis, and how real world problems could be solved "by these objects which don't exist."

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On this site one frequently finds under the linear-algebra tag questions of the kind: what is the determinant of a matrix $$\begin{pmatrix}a&b&b&b\\b&a&b&b\\b&b&a&b\\b&b&b&a\end{pmatrix}?$$ (I've just posted this question, which contains a list of such questions). It turns out finding an answer to this ...

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The realization that you can go on counting forever.

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You might be interested in the books Algebra and Trigonometry by Gelfand. Also, it's not dry or old, but the Precalculus textbook from artofproblemsolving.com won't spoonfeed, at least. From the book description: It includes nearly 1000 problems, ranging from routine exercises to extremely challenging problems drawn from major mathematics ...

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Yes I hear your point. Most books released these days look to spoon-feed. But that does not apply to all new books. I mean Spivak's books, Chapman Pugh's text on Analysis are examples. Now these are the books I perused during A Levels. I only got my hands on them because the government sells them for dirt cheap prices (I mean for less than 20 cents US). ...

1

I guess it depends on many things. First of all, what is the relevance of $T$ in its field? And secondly: what is the relevance of the new proof of $T$? In my field of research, nonlinear PDEs, it frequently happens that a researcher can solve a PDE by a "new" approach. However, this approach might be confined to that particular equation, and a referee may ...

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Mathematics courses geared towards engineers are (generally) more applied, focusing on applications rather than theory. For example, at UBC, the Linear Algebra course for engineers concerns mostly solving systems of equations, geometry, determinants, eigenvalues/eigenvectors, and applications of the aforementioned topics. Conversely, the course for ...

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it all depends. At the start of your undergrad, most math majors and engineers will take calculus 1 and 2 (the same one). However sometimes math majors may take the honours versions of those. By multivariable calc it may be a bit different with engineers focusing on vectors. and ofcourse engineers and math majors alike will take courses on ODE's and lots of ...

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One sometimes says "linear function" as verbal shorthand for "function whose value is linear in some particular variable(s)." In the case of $f(x) = ax + b$, the value $y = f(x)$ is linear with respect to the variable $x$; one might call it a linear function of $x$. Likewise, if we define $g(x) = ax^2 + bx + c$, one would likely say that $g(x)$ was a ...

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Is it misnomer to call a linear-equation a linear-function, or it is completely wrong to say that? Neither. The meaning of some mathematical terms (normal, regular, smooth, etc) is context-dependent. E.g., smooth function means $C^\infty$ in some papers/books and $C^1$ in others. In certain contexts, linear means "additive and commutes with scalar ...

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I thought about this once and came up with an "answer" of sorts. Notation: Given $z=a+ib$ and $w = c+id$ in $\mathbb{C}$, I will write $z \cdot w$ for the Euclidean dot product $ac + bd$. Ordinary multiplication of complex numbers will be denoted by juxtaposition. The basic observation is, given $z,w \in \mathbb{C}$, the quantity $\overline z w$ is ...

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For the purpose of applying set theory as a foundational subject, one must be able to express typical mathematical reasoning, such as Let $r$ be a real number... Let $n$ be an integer... Let $a$ be a sequence of real numbers... Let $f$ be a real-valued function of the reals... Let $S$ be a subset of the reals... And in general, if you have a kind of ...

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'Mathematical Foundation of Elasticity' written by Marsdan for any person who hopes to become an expert in strength of materials, explicitly states on its back cover that it will teach the functional analysis from the start. First edited in 1983 then reprinted by Dover publication at a low price. A 1.5-inch thick, yet not isoteric textbook.

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One cannot be realistic, honest to himself, in approaching Folland's manual on functional anlysis, with merely three or four light-weight calculus courses, two college-level linear algebra course and one ordinary differential equations course for biologist. At which profession do you focus? and at which university level? Folland's book is too deep (and too ...

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Euclid's "The Elements". Greek. Old. It doesn't get much more classic than that. As well as his other writings. http://en.wikipedia.org/wiki/Euclid#Other_works Also, the Principia Mathematica by Newton. As well as his other writings. http://en.wikipedia.org/wiki/Isaac_Newton#Mathematics Archimedes probably deserves a mention as well. You know, pi ...

2

Yes, using $\widehat p=\frac{Y}{n}$ instead of $p_0$ is valid (but your treacher's reason I think is incorrect). The reason that it is valid to use $\widehat p$ instead of $p_0$ is because the asymptotics are the same i.e. the hypothesis test is based on $$\sqrt{n}(\widehat p - p_0) \stackrel{d}{\rightarrow}N(0,p_0 (1-p_0))$$ By the continuous mapping ...

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I suggest you take a look at the Mathematics genealogy project and get lost following all fascinating links and finding famous mathematicians. The tree on the front page is already stunning.

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You probably already know this, but just in case: Mathematics Genealogy Project

1

When one is constructing a hypothesis test, you have to find a way to balance Type I error and Type II error the best. The problem is that Type I error and Type II error are usually inversely related (as in the case of this problem) which means that if you want decrease probability of Type I error in test you are going to increase Type II error. Thus the way ...

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Broad question! All I can offer is my own experiences regarding the above. How closely related are math and statistics? First of all, you need to make the distinction between applied math/stats and theoretical math/stats. I've found the applied/theoretical divide to be more substantial then the subject-level division per se. For example, take a look at a ...

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For hypothesis testing purposes, you always use the actual null distribution, which requires no estimation. Therefore, I am suspicious of your teacher's response. However, your teacher may have been referring to how confidence intervals are formed, in which case, you do use estimates. Your case appears to look like the beginnings of the basic "textbook" CI ...

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Not exactly what you were looking for (because these are "secondary sources"), but maybe interesting nevertheless: Mathematics and Its History by Stillwell walks you through the history of mathematics showing original problems in modern notation with many good exercises at an undergraduate level and with lots of pointers to the original sources. Euler - ...

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Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. Although not strictly speaking a purely mathematical book, surely I would put it among the classics.

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The best advice I can give you is practise, practise and practise! With time you develop the intuition and the skill. It may also help to join a learning group and to try to solve the exercises together. You can learn a lot from the way others tackle a problem (or from their instructive mistakes). It also is more fun and you gain confidence because you can ...

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If you want to learn algebraic geometry, a classical paper is Jean-Pierre Serre's FAC. See here for the French original, and here for the Englisch translation. See here for some advertisement by Georges Elencwajg.

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Three books: Euler's Introduction to the Analysis of the Infinite and Foundations of the Differential Calculus both translated by JD Blanton and published by Springer, also the very informative Analysis by its History by Hairer and Wanner. There are always the original papers by the biggies which are more often than not very interesting, illuminating and ...

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There are several Source Books that have made nice selections for you to pick from, e.g., Smith, Struik, Fauvel and Gray, Stedall. But for an extended read, you can do nothing wrong by immersing yourself in Gauss' Disquisitiones.

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For introductory Number Theory, you could go with Gauss, Disquisitiones Arithmeticae. Don't worry, you don't have to read Latin, it is available in English and other living languages.

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The first few chapters of GH Hardy's "A Course of Pure Mathematics" may be worth a read.

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As dry, old and rigorous as it gets "Advanced Mathematics Precalculus with discrete mathematics and data analysis." It's what I had in High School, although I had a modern textbook as a suppliment. There might be newer versions out, but I assume you want the older ones.

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If $C$ is invertible then, $Av = \lambda C v$ is equivalent to $C^{-1}\!Av=\lambda v$, so you're just looking and an eigenvector equation for a different matrix $C^{-1}\!A$ rather than for $A$.

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I think it is useful to start from applications of mathematics it self. In my view mathematics has the following uses. It has aesthetic value, i.e., it has beauty. Just as some people are attracted in the beauty of flowers, ornaments, young animals, music, painting,....mathematicians are attracted by the beauty of mathematics. its beauty lies in expressing ...

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It gives the same decomposition, because the equation $Av=\lambda Cv$ only gives an eigenvalue equation $Av=\lambda ' v$ if $Cv=\mu v$. Otherwise $w:=Cv$ and $v$ are not collinear, and solving $Av=\lambda w$ is not related to eigenvalues of $A$. So we may suppose that $Cv=\mu v$. Now $$Av=\lambda Cv=\lambda \mu v=\lambda' v,$$ which is again an eigenvalue ...

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If you have solved an open problem and can monetize it then this question is easy. For example, consider factorization. If you really developed an algorithm for efficiently factoring large numbers, you could write a paper in latex describing your algorithm and proving its correctness. You could send that paper to me. It is likely I will not understand ...

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Writing a chess engine that plays only moderately well, is a considerable programming challenge. Imo, very few people have the discipline, ability and patience to actually organize and execute something like that from scratch, programming-wise. There are some open sources which you can see and read, like that of GNU chess, Borland's Turbo chess and several ...

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Start here.

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The Knights tour is a famous chess/math problem.

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A very satisfying visualization of the area of a circle.

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It looks a bit like a deltoid, which belongs to the family of Kakeya sets. It was thought to be the solution to the problem of finding the minimal area you need to rotate a needle in the plane.

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The logo dates back to Adobe Reader v1.0. (link) I always thought it was chosen simply because it looks like the letter A for Acrobat. A parametric function that approximates the logo is: $$x(t)=5cos(t)\times \left( sin(t+.5)+cos(3t+.5) \right) \\ y(t)=5sin(t)\times \left( sin(t+.5)-cos(3t+.5) \right)$$

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It bears a strong resemblance to (in cartesian coordinates) $x = \cos(t) + 3 \cos(2 t)/4, y = \sin(t) - 3 \sin(2 t)/4$ but not quite as symmetric.

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"Decision mathematics" was spurred by the first war regarding resource allocation, linear programming for example. My A-level teacher introduced us with this topic. Cryptography also progressed in leaps and bounds as well as computing because of the need for cryptography. The Curta mechanical calculator was invented by a Jew (Curta) in a concentration ...

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There is a very entertaining book by T.W. Korner called The Pleasures of Counting in which he discusses among other things how the Allies nearly lost WW2. It's been a while since I read the book but the naval effort was rescued by mathematicians in an ingenious and essentially simple way. Cryptography is another field in which WWII may have spurred ...

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Operations research was explored during WW2. See, e.g., http://en.wikipedia.org/wiki/Operations_research#Second_World_War

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Game theory's development accelerated at a record pace during World War II/Cold War. If one nation changed the balance of power (by building a missile-defense shield, for instance), would it lead to a strategic blunder that resulted in nuclear war? Governments consulted game theorists to prevent such imbalances.

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Asaf was referring to André Weil, see here: http://en.wikipedia.org/wiki/Andr%C3%A9_Weil

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Consider the following integral $\displaystyle\int_{0}^{1}\dfrac{x^7-1}{\ln x}\,dx$. All of our attempts at finding an anti-derivative fail because the antiderivative isn't expressable in terms of elementary functions. Now consider the more general integral $f(y) = \displaystyle\int_{0}^{1}\dfrac{x^y-1}{\ln x}\,dx$. We can differentiate with respect to ...

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I recall something like this coming up when evaluating certain summations. For example, consider: $$\sum_{n=0}^{\infty} {n \over 2^n}$$ We can generalize this by letting $f(x) = \sum_{n=0}^{\infty} nx^n$, so:  \begin{align} {f(x) \over x} &= \sum_{n=0}^{\infty} nx^{n-1} \\ &= {d \over dx} \sum_{n=0}^{\infty} x^n \\ &= {d \over dx} {1 \over ...

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The solution to the Monty Hall problem Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, follows the fixed protocol of opening another door, say No. 3, which has a goat. He then says to you, ...

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