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Algebraically, what's happening is that taking an ordinary generating function is a bijection between the vector space of sequences and the ring of formal power series. In the ring of formal power series you have additional algebraic structure coming from the multiplication operation (and the division operation, when applicable). Operations like ...

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Using your example, a matrix: $$A=\begin{bmatrix} a_{11}&a_{12}\\ a_{21}&a_{22}\\ a_{31}&a_{32}\\ \end{bmatrix}$$ is an element of a vector space that , in the standard basis, is represented as: $$A= a_{11} \begin{bmatrix} 1&0\\ 0&0\\ 0&0\\ \end{bmatrix}+ a_{12} \begin{bmatrix} 0&1\\ 0&0\\ 0&0\\ \end{bmatrix}+ a_{21} ... 5 you consider y as a function of x then \frac{dy}{dx}= y'(x) to compute your integral just consider the change of variable formula:$$\int n(y(x))y'(x)\, dx = \int n(y) dy $$4 First of, a functor F : \mathscr C → \mathrm{Set} is called representable (by C) if it's isomorphic (not necessarily equal) to \mathrm{Hom}(C, -) for an object C of \mathscr C. As for the term, an abstract functor F is represented by the very concrete action of \mathrm{Hom}(C, -). Take for example the functor L : \mathrm{Top} → \mathrm{Set} ... 4 Hint: Put -\alpha x^2=u \Rightarrow -2\alpha x \mathrm dx=\mathrm du,then x\mathrm dx=\frac {-1}{2\alpha}\mathrm du. you will have,$$\displaystyle\int xe^{-\alpha x^2}\,dx=\frac {-1}{2\alpha}\displaystyle\int e^u \mathrm du$$2 I don't know if it's interesting, but here is how you can interpret the facts you cited. The first thing, of course, is that the derivative vector b'(t) is in the direction of the tangent vector at parameter value t. But this is true of any parametric curve, not just Bezier curves. Take the n vectors \Delta b_n and relocate them so that their ... 2 You've essentially solved the problem by finding the elements of order 3 in \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/12\mathbb{Z}, but now you need to chase back through the various isomorphisms to find the corresponding automorphisms. You're right, to realize the isomorphism (\mathbb{Z}/7\mathbb{Z})^\times \times (\mathbb{Z}/13\mathbb{Z})^\times \cong ... 2 First, random and stochastic are synonyms. A stochastic process is a collection of random variables X=(X_i)_{i\in I}. You have two important cases: the discrete case: I=\mathbb{N} or \mathbb{Z} (or a subset of those). Then the process is a sequence of random variables. the continuous case : I=\mathbb{R} or [0,+\infty ) for example. Then the ... 1 Maybe it will be useful to consider an example of two norms F and G of a vector space X not being equivalent to each other. What it means is that at least one of the quantities \sup\limits_{x \in X}\frac{F(x)}{G(x)} or \sup\limits_{x \in X}\frac{G(x)}{F(x)} is unbounded, i.e. there is a sequence (x_n)_{n \geq 0} of vectors in the space such that ... 1 Suppose you have a sequence which converges in G . The lower bound implies it converges in F  to the same limit. Suppose you have a sequence which ddoes not converge in G . The upper bound implies it does not converge in F . That's all you need, since metric spaces are sequential spaces. 1 You can start with "the norms induce the same topology". Then use the fact that a linear transformation is continuous if and only if it is bounded. And this is one of your inequalities. For the other direction, use the inverse of that linear transformation. 1 I can't think of a perfectly suitable description of what AA^{T} and A^{T}A can represent (certianly out of my comfort zone with regards fluid dynamics)other that to say that for some special types of groups the transpose matrix is actually the inverse, the rotation matrices have this property.So geometrically the rotation matrices preserve length. A ... 1 Note SpamIAm's method yields: (2,0) \mapsto \phi_{79} and (0,4) \mapsto \phi_{29}. Explicitly calculating \langle \phi_{79},\phi_{29}\rangle, we find the subgroup of \text{Aut}(\Bbb Z_{91}) isomorphic to \Bbb Z_3 \times \Bbb Z_3 is: \{\phi_1,\phi_{79},\phi_{53},\phi_{29},\phi_{22},\phi_{16},\phi_9,\phi_{81},\phi_{74}\} So your \phi_9 and ... 1 Because the base b representation of a number is "invertible". Let a number be$$n=\sum_{i=0}^dd_ib^i,$$where b is the base and the d_i are the digits, such that 0\le d_i<b. For example,$$443_5=4\cdot5^2+4\cdot 5+3=123.$$By the remainder theorem, there is a unique quotient q and a unique remainder r such that$$n=qb+r\text{, and }0\le ...

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The best description of this process I have seen (in terms of clarity) comes from Howard Eves' Introduction to the History of Mathematics: If we have a number expressed in the ordinary scale, we may express it to base $b$ as follows. Letting $N$ be the number, we have to determine the integers $a_n,a_{n-1},\ldots,a_0$ in the expression $$... 1 Choose any integer b such that b\geq 2. Choose some integers d_k,d_{k-1},\ldots,d_1,d_0, each of which satisfies 0\leq d_i<b, and define n to be$$n=d_k\cdot b^k\;+\;d_{k-1}\cdot b^{k-1}\;+\;\cdots\;+\;d_1\cdot b\;+\;d_0$$(In other words, define n to be the base-b number (d_k\ldots,d_1d_0)_b\;.) Recall what the division algorithm says: ... 1 Are you familiar with integrals of the form \int f'(x).g(f(x)) \text{d}x, where f and g are ''well-behaved'' functions? If yes, try to see how you can apply this idea when g(x)=e^x and f(x)=\alpha x^2, where \alpha is a constant of course. If not, find out how to use this method because it is an essential tool. Actually, knowing about ... 1 If \alpha=0 the integral diverges (it is \infty). Now if \alpha \not =0, the derivative of \frac{-1}{2\alpha}e^{-\alpha x^{2}} is xe^{-\alpha x^{2}}. Now \int\limits_{0}^{\infty} xe^{-\alpha x^{2}}dx=\frac{1}{2\alpha}. The answer should be a real number, and not a function of x 1 Find the derivative of x\mapsto e^{-\alpha x^2}. You'll see that it's easy to find an antiderivative of x\mapsto xe^{-\alpha x^2} and thus to solve this integral, which by the way is not a function of x. 1 Hint: I don't think integration by parts is the right strategy (and I don't understand the way you used it). Try performing the substitution u=x^2, du = 2xdx. 1 The differential of a function f at x_0 is simply the linear function which produces the best linear approximation of f(x) in a neighbourhood of x_0. Specifically, among the linear functions l that take the value f(x_0) at x_0, there exists at most one such that, in a neighbourhood of x_0, we have:$$f(x_0+h)=f(x_0)+l(h)+o(h^2) It is the ...

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The theory that explains details of $\mathrm{d}x$ rigorously and formally is quite abstract and complex and not one that I would recommend trying to grasp whilst still going through calculus or real-analysis. However, a quick read through will provide some nice clarification. To that end, I suggest reading this excellent blog post on math.blogoverflow.com: ...

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Here's my earlier answer to a question that is not exactly the same, but it may shed light: What is $dx$ in integration?

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The tangent line is the first order Taylor polynomial. The n:th derivative can be "visualized" in the same sense as the n:th order Taylor polynomial. This will only give a localized significance to the derivative. There are many other important properties of derivatives. For instance when solving differential equations the exponential functions are very ...

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From a point of view of a general topologist, these spaces are nice. For topologists which work with cardinal invariants, many of them collapse for these spaces. By Tichonoff-Urysohn Theorem, a second countable space is metrizable iff it is regular (in this case it is homeomorphic to a subspace of the Hilbert cube $[0,1]^\omega$). First countable spaces are ...

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