Hot answers tagged reference-request
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When I first entered university, shortly before classes began, I met with an professor whose task was to advise me on which classes to take in my first semester. After hearing me describe my background, which included passing the college-credit calculus exam at age fifteen, he suggested that I take real analysis.
“But I took that already,” I protested. “I ...
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A couple of ‘entry level’ treatments that can be confidently recommended.
Herbert B. Enderton, The Elements of Set Theory (Academic Press, 1997) is particularly clear in marking off the informal development of the theory of sets, cardinals, ordinals etc. (guided by the conception of sets as constructed in a cumulative hierarchy) and the formal ...
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The key term is Bergman space. The space you introduced is usually denoted $A^1(\Omega)$ or $L^1_a(\Omega)$. When $\Omega$ is a disk, the dual of $A^1(\Omega)$ can be identified with the space of Bloch functions on the disk. This is proved in the book Bergman spaces by Duren and Schuster. (By the way, in the reflexive range $1<p<\infty$ we have ...
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I might be wrong, but I don't think it is completely right to call a controvariant functor "cofunctor" (at least if we want to stick to the convention of using the particle co- to mean and evoke duality) because the sentence "$T$ is a functor" is clearly self-dual, as Mac-Lane explicitely pointed out in his Category for the Working Mathematician. So, given a ...
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I think you are trying to ask for the distinction between defining the Riemann integral as a limit of supremum and infimum partitions vs just using a fixed family of partitions, in this case something akin to the dyadic intervals as Osgood does. The point is that both methods yield precisely the same result when the function $f$ is continuous on the whole ...
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One can define a function $f:[a,b]\to\Bbb R$ to be measured (not really Lebesgue, but seems a good translation of "reglada") if there exists a sequence $\{s_n\}$ of step functions such that $s_n\to f$ uniformly. Recall that if $s$ is a step function with constants $c_i$ and intervals of partition $[x_{i-1},x_i]$ for say $i=1,\dots, r$ we define its integral ...
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[Too long for a comment]. If you denote
$$ G(z)=\sum^\infty_{n=0} \frac{1}{n!} \frac{1}{n+1}(-z)^n {}_2F_2\left(m,n+1;\frac{1}{2},n+2; b z\right),$$
then certain combination of $G(z)$ and $G'(z)$ is a simpler single function:
$$\frac{d}{dz}\Bigl[zG(z)\Bigr]=e^{-z}{}_1F_1\left(m,\frac12;bz\right).$$
Therefore,
...
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The best rigorous treatment that I’ve seen at the senior undergraduate/first-year graduate level is Karel Hrbacek & Thomas Jech, Introduction to Set Theory. The first edition was already very good, and the third has been expanded to include some important topics that were not originally covered. I agree with the very favorable Amazon review by Michael ...
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Concerning references, I've seen the problem of first passage time distributions treated in two books, unfortunately both are in German:
Nollau, V.: Semi-Markovsche Prozesse, H. Deutsch (1981).
Störmer, H.: Semi-Markoff-Prozesse mit endlich vielen Zuständen, Springer Lecture Notes in Operations Research and Math. Systems, Vol. 34 (1970).
As the titles ...
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Mathscinet gives a favourable review of the book you mention, namely Partially Ordered Groups by A. M. W. Glass. The review says that it `will surely be an instant classic', and it has apparently been cited 78 times. It concludes saying that the book
will get the reader to the forefront of research in the field and would be suitable texts for students in ...
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It is always easier to work on an affine coordinate system, Barycentric coordinate system, when the integration is performed locally over a piecewise linear structure, than to work on the global Cartesian coordinates.
First let's replicate the formula you gave in a triangle (2-simplex). For the following triangle $T = \triangle ABC$:
denote the top ...
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Here is a suggestion, not a complete answer.
Since $n+q$ seems prevalent,
let $k = n+q$.
$\begin{align}
\sum^\infty_{n=0} \sum^\infty_{q=0}
\frac{1}{n!} \frac{1}{q!}
\frac{(m)_q}{(\frac{1}{2})_q}
\frac{(1)_{q+n}}{(2)_{q+n}}
(-z)^n (b z)^q
&=\sum^\infty_{n=0} \sum^\infty_{q=0}
\frac{1}{n!} \frac{1}{q!}
\frac{(m)_q}{(\frac{1}{2})_q}
...
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I will assume that the initial data is smooth. First we define the appropriate energy (this depends on oyur problem, but usually it is enough to take some high order Sobolev norm). For instance we take $E[u]=\|u\|_{H^1}$. Now we obtain an a priori bound for this energy. Multiplying the equation by u and integrating by parts we get
$$
...
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I guess here that you mean that the boundary of $D$ is a finite number of pairwise disjoint curves. In this case, this Cauchy formula follows directly from Mergelyan's Theorem about uniform rational approximation of holomorphic functions. Indeed, the formula holds for rational functions with poles outside $\overline{D}$, and Mergelyan's Theorem says that any ...
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Nathan Carter's Visual Group Theory.
Janet Chen's Group Theory and the Rubik's Cube.
(Available for free but a little more formal)
Wildberger's Informal Introduction to Abstract Algebra.
This question may help you to find out on the unsolvability of polynomials of degrees higher than 4.
This is the most informal and written in a moderately comprehensive ...
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I can highly recommend "A Book of Abstract Algebra", by Charles C. Pinter. You'll learn about groups, rings and fields. You will also learn enough Galois Theory to understand why polynomials of degree higher than $4$ are, in general, not solvable by radicals.
It is 'formal' in the sense that it is rigorous, but the author is also very good at explaining ...
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Great video lecture series by Benedict Gross at Harvard:
http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra
Outstanding short pieces on many facets of algebra by Keith Conrad at UConn. Very clear exposition and lots of intuitive discussion any plenty of warnings - teaching at its best(even one on Rubik"s cube):
...
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Arbib and Manes once made an attempt to apply the Automata Theory to Control Systems, see:
Manes, E.G.(ed.)
Category theory applied to computation and control. Proc. $1$st Internat. Symp., San Francisco, 1974. Lecture Notes in Comp. Sci., 25.
I am not an expert in the Control Systems and can not judge how successful this attempt is, but from the point ...
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It is a common misconception in regard to math, but this kind of misconception is common everywhere.
It's caused by the recursive properties of knowledge. The more you learn, the more you realize how little you know.; i.e:
Since it is known that less informed individuals see fields as more finite than informed individuals; While laymen may relate to the ...
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One way of proving it is to write
$$\int\frac{g^2}{\sqrt{f^2+g^2}+f}$$
and use CS to end up with
$$\int\frac{g^2}{\sqrt{f^2+g^2}+f}\int\sqrt{f^2+g^2}+f\ge\left(\int g\right)^2.$$
Now $\int\sqrt{f^2+g^2}+f\le \int f+g+f\le 2K+\int g.$ So our integral bounded below by $$\frac{\left(\int g\right)^2}{K+\int g}$$ which can be easily estimated via $\delta.$
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There are some significant typos in Delanoy's answer, but I'll accept it nonetheless as it pointed me in the right direction. (Sorry for the delay.)
Let $a^k+b^k+c^k = d^k+e^k+f^k,\;\; k =2,4\tag{1}$
Theorem 1 (Birck-Sinha): "Fact 1" should be instead, given (1) then,
\begin{aligned}&(a+b+c)^k + (-a+b+c)^k + (a-b+c)^k + (a+b-c)^k + (2d)^k + (2e)^k + ...
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