# Tag Info

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What I can come up with is two books on linear algebra: "Krypa, gå" (Crawl, go) by Peter Hackman "Boken med kossan på" (The book with the cow on the front) by Peter Hackman they have both been used at introductory courses at the university.

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Geometry is a pretty new app on Mac OS X for making geometric constructions and check angles etc. Contrarily to Latex or others, you can move points and lines etc interactively and see how the drawing evolves based on the construction constraints. I like that this app is lightweight, with easy keyboard shortcuts, and that it is associated with a website ...

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"Do you have an example of a mathematical theory (i.e. not an isolated theorem, but a coherent set of mathematical concepts and theorems) that you believe will be of no use, ever, to let's say engineers or physicists or non-mathematician scientists in general ?" Take Laplace Transform (LT) and Fourier Transform (FT) for examples. Both are used widely in ...

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It's fairly common to define $\sin(x)$ and $\cos(x)$ as the solutions of differential equation, $$(1) \quad {{dy^2} \over {dx^2}}=-y$$ The initial conditions require that $\sin(x) \$ satisfies $[y'(0),y(0)]=[1,0]$ and $\cos(x) \$ satisfies $[y'(0),y(0)]=[0,1]$. Using Picard's Method, we can get series expansions for the solution of $(1)$. We have, ...

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Start with $$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)\\ \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$$ Differentiate with respect to $y$: $$\sin'(x+y)=\sin(x)\cos'(y)+\cos(x)\sin'(y)\\ \cos'(x+y)=\cos(x)\cos'(y)-\sin(x)\sin'(y)$$ Let $y=0$ and note that $\cos'(0)=0$ because $\cos$ has a maximum at $0$: $$... 1 Instead of using usual definitions, I tried to start from the trigonometric identities, and 'redefine' them as the special functions which satisfy those identities. Claim. Suppose that there exist differentiable functions f, g:\mathbb{R}\to\mathbb{R} such that \forall x,y\in \mathbb{R}; \; f(x)g(y)+g(x)f(y)=f(x+y) \forall x,y\in \mathbb{R}; \; ... 0 I sometimes find the following very useful:$$\int \Re f(x) dx = \Re \int f(x) dx \int \Im f(x) dx = \Im \int f(x) dx \int \sum_{i=0}^{k}f(i,x) dx = \sum_{i=0}^{k} \int f(i,x) dx $$The third one works even when k=\infty, which is the main reason it is useful. Combine it with a good knowledge of series. 2 To the first order, the tangent is a good approximation of the boundary of a shape. So can you approximate (\cos(\theta+d\theta),\sin(\theta+d\theta)) in terms of (\cos\theta,\sin\theta)? In other words, what's: (\cos(\theta+d\theta),\sin(\theta+d\theta))-(\cos\theta,\sin\theta) Well, it's tangent to the circle, and hence points in the perpendicular ... 2 I'm suprised that the following two haven't shown up: What is the smallest Riesel number? What is the smallest Sierpiński number? In both cases we know they exist because they are smaller than or equal to 509,203 and 78,557 respectively. 17 Consider the image from this "proof without words", Asymptotically, the angle between the black radius and the red vertical line is complementary to both angles marked as \theta. Thus, asymptotically, those angles are equal, and the two red triangles are similar. Therefore, by similar triangles,$$ ...

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The constant in the Berry-Esseen theorem: If we have a bunch of i.i.d. random variables $(X_j)_{j\geq 1}$ with a finite third moment, that is $E[|X_j|^3]<\infty$ (and thus they also have some mean $\mu$ and variance $\sigma^2$), then we can prove without too much trouble that their scaled average, $A_n := \frac{(\sum_{j=1}^n X_j)-n\mu}{\sigma \sqrt{n}}$ ...

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Imagine a particle moving around a circle with a position vector $\textbf r(t) = \textbf{i}\cos t + \textbf{j}\sin t$. The velocity vector $\textbf r'(t)$ must point in the direction tangent to the circle. A tangent to a point on a circle is always perpendicular to the radial line, so $\textbf r(t) \perp \textbf r'(t)$. Now you need to determine $|\textbf ... 7 My two favorite ways: Alternative A. Assuming that you are familiar with the Taylor series of both sine and cosine (some purists think that you should not rely on pictures when defining these functions, and power series are the way to go). Consider the matrix $$A=\left(\begin{array}{cc}0&-1\\1&0\end{array}\right).$$ We can use this as an ... 7 This answer uses the following definitions: $$e^{ix} = \cos(x) + i \sin(x)$$ $$\sin(-x) = -\sin(x)$$ $$\cos(-x) = \cos(x)$$ Rearranging gives the complex exponential definition of cosine: $$\cos(x) = (e^{ix} + e^{-ix})/2$$ $$\Rightarrow \cos'(x) = i(e^{ix} -e^{-ix})/2$$ The complex exponential definition of sine: $$\sin(x) = (e^{ix} ... 0 Elements of Discrete Mathematics by C. L. Liu is Helpful for the Introduction ofNumber Theory,Combintorics,Logic. 2 How would a basis for \mathbb R as a field over \mathbb Q look like? By the axiom of choice we know one must exist, but I have no idea how it would look. It would be quite odd, as it must be large enough so that every real number can be expressed as a finite sum of its elements, but it must be small enough to be linearly independent. For a simpler ... 3 There are many examples of objects whose existence can be proven using Probability Theory. One example is the existence of matrices with the restricted isometry property: https://en.wikipedia.org/wiki/Restricted_isometry_property. If one wants matrices of certain dimensions, the only known way to construct them is to create matrices at random using a ... 0 I have been quite surprising when I discovered that the elliptic curves of the shape X^3+Y^3=AZ^3 , A cube-free integer, does not have nonzero rational point if and only if all its quadratic points (there are always an infinity) are equivalent to (I called it) a conjugate point which have the form (X,Y,Z)= (a+b\sqrt m, a-b\sqrt m, c) where a,b,c,m ... 2 It can be proved, for instance,that x^2+x+1 is irreducible in \mathbb F_p[x] for p=29 which gives two "irrational" elements (the two roots of the polynomial) in a quadratic extension of \mathbb F_{29}. What kind of object is each of these two roots? Absolutely non idea. And for all finite field there are in general infinitely many of these ... -2 If we raise \sqrt{2} to the \sqrt{2} power, and raise the result to the \sqrt{2} power, then we have raised an irrational number to an irrational power and gotten a rational, but we don't know in which step we did it... 4 Let p=2n+1 be an odd prime, and consider the 2^n expressions you get by all possible choices of signs in$$\pm1\pm2\pm3\pm\cdots\pm n$$They can't be perfectly uniformly distributed among the p residue classes modulo p, since p doesn't divide 2^n, but the are distributed as uniformly as they can be, in that each nonzero residue class comes up the ... 4 I was about 10 when I called my dad to the blackboard to see my theorem that if the square root of an integer was not also an integer then the decimal expansion did not terminate. It was a proof by cases based on the least significant digit of the square root, so didn't prove the square root was irrational. 1 This is really obvious but I realized something I had not been tought, from this : Let P(x) be a polynomial of degree n, than: P(x+1)-P(x) is degree n-1. This was not surprising, in fact this just came from the binomial theorem. What was surprising to me is the fact that we can model quantitative data that is taken in constant intervals just ... 1 I was teaching myself about tensor algebra and playing around with Einstein summation notation, and I surprised myself by deriving a closed formula for the principal invariants of a second-order tensor, using the generalized Kroenecker deltas.$$\det (A-\lambda ... 6 Many problems that are just computationally hard. Only about 40 or so Mersenne primes are known (not a very good example, because there is no proof they are infinite, so we might know all of them. ). Take the sequence of primes p where the gap between p and the next larger prime is larger than any earlier gap between consecutive prime numbers. These two ... 0 I stumbled on Gödel's incompleteness theorem falling down for second order logic without understanding what second order logic is. Due to the way I bootstraped into it, I needed somebody else to tell me it was second order logic. 7 What is the least infinitely repeating prime gap? See the recent work by Zhang, et. al. 1 ZFC proves that there exists a well-ordering of the real numbers. (Many such, in fact.) Nobody has a clue what one is like. 5 A function$\mathcal{G}(n)$that, for each positive integer$n$, gives the length of the Goodstein sequence for$n$. We know such a function is well-defined and finite for all$n \in \mathbb Z^+$, but the function values get so large for small$n$that it is difficult to compute. Moser's worm problem for a convex set. Blaschke's selection theorem ... 7 The functions given by the Riemann mapping theorem. A simply connected region can be mapped bijectively and holomorphically onto the open unit disk. Take some slightly weird shape, say, a square with 4 half-circles attached to the sides, and it is likely there is no nice description on how this map looks like. Similarly, there are a lot of such examples ... 0 Well, by disproving conjectures. But joking aside, by finding construction proofs of (ie, finding ways to calculate) something for which only existence proofs are known. Btw, this answered the original question (which asked what contributions could be made besides proving theorems) rather than the question as it is currently worded. 11 A partition of the 3D ball into 5 distinct pieces such that, through only translations and rotations, the pieces can be moved and reassembled into two balls, each of equal size to the original. This is the Banach-Tarski paradox. The existence of such a partition depends on the axiom of choice, so in particuar there is no way to say exactly what the ... 12 The leading (decimal) digit of the ludicrously huge number$TREE(3)$. (See https://cp4space.wordpress.com/2012/12/19/fast-growing-2/ and http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html.) OK, one might object "But that's an absolutely uninteresting object!" That may be; I'd argue, though, that it is meta-interesting in the following sense. The ... 7 (1) From the set-theory axiom system called ZFC we can prove that the set of reals has a cardinal number$\mathfrak{c}$. But, assuming that ZFC is consistent, we don't know what$\mathfrak{c}$is: it cannot be proven or disproven from ZFC that$\mathfrak{c}$is the least uncountable cardinal, or the second or even the$\mathfrak{c}$-th. (2) Perhaps someone ... 12 Collisions in cryptographic hashes must exist due to the pigeonhole principle. Although some collisions have been found for some hash functions, we "have no idea what they are" in the sense that they aren't readily calculated. 38 Not sure if this satisfies the requirement that we "have no idea what they are", but the extremely strange Mill's constant seems worth mentioning here: There is supposed to be some real number$r>0$with the property that the integer part of $$r^{3^n}$$ is prime for every natural$n$. It is not known if$r$is rational and as far as I know not even a ... 24 There are a number of games, like Hex and Chomp, for which it is easy to prove a first player win by strategy stealing but we do not generally know the winning strategy. 19 Take objects which existence proof uses the axiom of choice, e.g: Each vector space has a basis (the standard existence proof uses Zorn's lemma). How does a concrete basis of$C[a,b]$look like? What about$\mathbb R$as a$\mathbb Q$-vector space? Ultrafilter, which are used in the construction of the hyperreals: Does the sequence$(0,1,0,1,0,1,\ldots)$... 8 This is a copy of my answer to How to prove that$\lim\limits_{x\to0}\frac{\sin x}x=1$?: From the above picture,$\arcsin y$is twice the area of the orange bit. The area of the red bit is${1 \over 2}y\sqrt{1-y^2}$. The area of the red bit plus the orange bit is$\int_{0}^y \sqrt{1-Y^2} dY$. So $$\arcsin y = 2\int_{0}^y \sqrt{1-Y^2} dY - y\sqrt{1-y^2}.$$ ... 4 This answer uses following definition: $$\cos(x)=\Re(e^{ix}) \quad \mathrm{and} \quad \sin(x)=\Im(e^{ix})$$ It are known facts that $$\Re f'(x) = [\Re f(x)]'$$ $$\Im f'(x) = [\Im f(x)]'$$ Therefore $$\cos'(x)=[\Re(e^{ix})]' = \Re[e^{ix}]' = \Re (ie^{ix}) = \Re (i\cos(x)+i^2\sin(x)) \\ = \Re (i\cos(x)-\sin(x)) = -\sin(x)$$ ... 4 This answer uses the geometric definition. Using the geometric definition one can prove that $$\sin(\alpha+h)= \sin(\alpha)\cos(h)+\cos(\alpha)\sin(h)$$ $$\cos(\alpha+h)= \cos(\alpha)\cos(h)-\sin(\alpha)\sin(h)$$ $$\sin^2(x)+\cos^2(x)=1$$ Now we want to prove the following limits: $$\lim_{x\to0} \frac{\sin(x)}{x} = 1 \quad \mathrm{and} \quad ... 0 My long-time favorite, which I have mentioned here a number of times, is "Mathematics for the Million" by Lancelot Hogben. It is available for 15 from Amazon: http://www.amazon.com/Mathematics-Million-Master-Magic-Numbers/dp/039331071X/ref=sr_1_1?ie=UTF8&qid=1442883932&sr=8-1&keywords=hogben+lancelot Once you have that, I like "What Is ... 0 The balls with radii \frac{1}{n} and center at a rational number form a basis for the euclidean topology. This family is countable as it is a countable union of countable sets. We have found a countable basis, so we are done. 2 This somewhat relates to point 8 in the question, but the presentation is very different. For the geometric series below we have:$$1+5+5^2+\cdots+5^{n-1}=\frac{5^n-1}{5-1}=\frac{5^n-1}{4}$$The LHS is a sum of integers, so the RHS must be an integer. Then 5^n-1 must be a multiple of 4. 2 While in high school I discovered a formula for addition of cosines of ascending multiples of an angle:$$\sum_{k=0}^{n}{\cos k\theta}=\frac{1-\cos\theta+\cos n\theta-\cos[ (n+1)\theta]}{2(1-\cos\theta)} \tag{A}$$by: using de Moivre's formula, treating the complex exponential sum as a geometric series, making the denominator real, then equating real ... 0 I once found out that my one-sided sandwich did not only not fall with jam side down but also got the cheese slices perfectly on top when I subsequently dropped the cheese. Don't have any evidence so I don't expect anyone here to believe me though. Mathematically wise I must say exploring non-linear things with (sparse) metrics in linear algebra is ... 0 I was very surprised the first time I discovered this result,the fact that some q-continued fraction I had discovered was related to gauss's continued fraction for pi,blew my mind. Using the continued fraction$$\psi^2(q^2) = \cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q^2(1-q^2)^2}{1-q^5+\cfrac{q^3(1-q^3)^2}{1-q^7+\ddots}}}}$$And then multiplying both ... 1 It's hard to find anything as useful as the one you mention, but this might be of interest. http://mathoverflow.net/questions/115192/power-log-distance-between-matrices/115227#115227 (ps. again I would have made this a comment rather than an answer, but I still can't do that) 1 In some sense, this is a nice generalization of some of your facts: If$T$is some complete theory, and$M \models T$,$|M|< \omega$, then if$N \models T$, we have that$M \cong N$. Example: Let$L= \{<\}$, and let$M$be a finite total order. Let$N \models Th_L(M)$. We show that$M \cong N$. Recall that we can say that$M$has exactly$m\$ ...

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All simple transcendental extensions of a given field are isomorphic.

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