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First of all, I would recommend taking a course. You can find one at many universities in Canada where I am. It is unlikely they will cover all of these topics though. Here are some references. population dynamics: Otto and Day http://press.princeton.edu/titles/8458.html Linda Allan ...


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The statement is known as Lie's third Theorem: Every finite-dimensional real Lie algebra $L$ is integrable, that is, there exists a Lie group $G$ with $Lie(G)\cong L$. For a proof see, for example, the note by J. Ebert Van Est's exposition of Cartan's proof of Lie's third theorem, and the references.


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Answer to question 1: In a general setting, Radon Nikodyn derivatives always refer to a couple of measures. So the question rephrases to: what are $P_{Y|X}$ and $Q_{Y|X}$? If these are measures in some sense, you can define the generalized Radon Nikodyn derivative. What you find is that $P_{Y|X}$ and $Q_{Y|X}$ are Regular Conditional Probabilities ...


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Koblitz, Algebraic Aspects of Cryptography, by the co-inventor of Elliptic Curve Cryptography is a good choice.


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Let me try to outline the proof, and you can fill in the details. We take an $n$-dimensional smooth quadric hypersurface $Q$ sitting in $\mathbf P^{n+1}$. By the Lefschetz hyperplane theorem, for $k \neq n$, the restriction maps $$H^k(\mathbf P^n, \mathbf Z) \rightarrow H^k(Q,\mathbf Z)$$ are isomorphisms, and so the cohomology groups of $Q$ in these ...


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I've seen in the literature the notation $C$ with some additional specifications for the contraction maps of all sorts, but the amount of decorations on the symbol $C$ varied depending on the context. See, e.g., A.Gray, Tubes, p.56, where these maps are used in the case of somewhat special tensors, and therefore the notation is simpler. In general, there is ...


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The analogous theorem does hold if $A$ is a connected subset of a Banach space. The following is the statement of result $(8.6.3)$ in my edition of Dieudonné's "Foundations of Modern Analysis": Let $A$ be an open connected subset in a Banach space $E$, $(f_n)$ a sequence of differentiable mappings of $A$ into a Banach space $F$. Suppose that: (1) there ...


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Omar Pol is the original author of the diagram, put online circa 2007. It appears that other similar works were derived from Pol's original diagram. A larger diagram (n = 1..16) is online at: http://www.polprimos.com/imagenespub/poldiv01.jpg (Source: Personal correspondence with Omar Pol.)


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Long comment: The commutative monoid $(\mathbb{N}\setminus\{0\},\times,1)$ is freely generated (as a commutative monoid) by the prime numbers. In other words, for any commutative monoid $X$ and any function $f:\mathbb{P} \rightarrow X$, there's a unique homomorphism $\hat{f}:(\mathbb{N}\setminus\{0\},\times,1) \rightarrow X$ such that the restriction of ...


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The function $\ell$ on $\Bbb N$ is the sum of prime factors (function), or a little less literally, the integer logarithm. P.A. MacMahon calls $ell(n)$ the potency of $n$ (see P.A. MacMahon, Properties of Prime Numbers Deduced from the Calculus of Symmetric Functions, Proc. London Math. Soc. (1925) s2-23 (1): 290-316.) The values $\ell(1), \ell(2), \ldots$ ...


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Question 1: Is the logarithm on base 2 or base 10? As entropy is in bits, it should be in base 2. However, the Lyapunov Exponent (LE) for Tent Map = 0.69 approx (please correct me if wrong). If this relationship applies to all maps, then log2(2) and log10(2) would give entirely different result. The Lyapunov exponent is commonly defined using the ...


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You might wanna check out Mathologer.


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More usually with one of these intractable problems, it is not the theorem itself which has some important consequence, rather the method of proof. The intractability of the problem represents some shortcoming in our mathematical methods, or some failure to recognise two fields or structures within mathematics to be equivalent to each other. By the act of ...


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Consider the case of $n$ players, which includes $3$ players. When pairs of players play bimatrix games (two-player strategic form games), and must play the same strategy across these games, and each player's payoff is the sum of payoffs across the all two-player games, then this is called a polymatrix game. For more info see, e.g., this tutorial: ...


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Two possible techniques. First technique: use other people's work. The distribution in question appears to be, from @BGM's comment, a multivariate hypergeometric distribution, which is a very enjoyable name to pronounce very quickly 5 times in a row. More precisely, $(X,Y,k)\sim \operatorname{MultivHypergeom}_3(\underbrace{(\gamma N,\gamma N,(1-2\gamma) ...


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I'm using the second edition. I think that theorem $9$ should be Definition. An $m\times m$ matrix is said to be an elementary matrix if it can be obtained from the $m\times m$ identity matrix by means of a single elementary row operation. instead of Definition. An $\color{red}{m\times n}$ matrix is said to be an elementary matrix if it can be ...


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I've got a strange result for expectation, so feel free to downvote it: At first, lets split $k$ balls taken into two parts - one with red and blue balls, and other - with green balls. So we get the random variable $U$: $$P(U=u) = Hypergeometric(u, k,2\gamma N,N) $$ which means that drawing $k$ balls from the box with $N$ balls that contains $2\gamma N$ ...


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I recommend you to take a look at Symbolic Dynamics & Coding by Lind & Marcus. Other nice options are Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts by B. Kitchen and Substitutions in Dynamics, Arithmetics and Combinatorics by P. Fogg.


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If you meant to include numbers with abundancy index $$I(x) = \dfrac{\sigma(x)}{x}$$ equal to an integer which is at least $3$, then you can check out The Multiply Perfect Numbers Page, last updated by Achim Flammenkamp on 2014-01-25 19:35 UTC+1.


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Some remarks on infinite-dimensional manifolds. There are two approaches which to me still feel very manifold-ish (there are others yet: Frolicher spaces and diffeological spaces, which feel a bit less so); the approach of Banach manifolds and the much more general approach of Frechet manifolds. The former is almost precisely the same theory as the theory ...


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I've found the source of this interview. It's in Paul Erdos' biography by Paul Hoffman, " The Man Who Love Only Numbers"(1998). Starting from page 78, the book describes Erdos left Hungary for Cambridge in 1934 due to the raging Hungarian Fascism. It was at Cambridge, the second day of his arrival, that he met G. H. Hardy, and inquired him about Ramanujan. ...


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This is functional in general. Let $X$ be a set an let $\mathcal V\subseteq\wp(X)$. Then the collection of finite intersections of sets in $\mathcal V$ automatically has the basic properties of a base of a topology. This in the understanding that the empty intersection equals $X$. Denoting the collection by $\mathcal{V}^{\stackrel{\cap}{f}}$ we have ...


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We have: $$\left(\bigcup_{i \in I} A_i\right) \cap \left(\bigcup_{j \in J} B_j\right) = \bigcup_{(i,j) \in I \times J} (A_i \cap B_j)$$ so that the intersection of two unioned families is again a unioned family, over a bigger index set, and if the $A_i,B_j$ come from a family which is closed under finite intersections, the latter is also a union from that. ...


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With the question by the OP asking for a reference I will try to do just that, providing links to the OEIS. Surprisingly enough even the OEIS does not offer the usual variety of references here, suggesting that this problem is open. We compute generating functions $T_{\le h}(z)$ for the height being at most $h$ and the desired count is then given ...


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I do not have a reference, but isn't this easy to prove? Any morphism from $\mathbf A^1$ to an abelian variety $X$ is constant. Indeed, such a morphism extends to a morphism $f$ from $\mathbf P^1$ to $X$. For all global differential forms $\omega$on $X$, the pull-back $f^\star\omega$ is a global differential form on $\mathbf P^1$. Therefore, ...


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Some obvious things are less obvious than others.... Let $X$ be a rank two graph consisting of a disjoint pair of circles plus an arc with one endpoint on each circle. Then no matter what $v \in X$ you pick, $\pi_1(X,v)$ is not generated by simple closed curves through $v$.


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By the universal UHF algebra people usually mean the UHF algebra with the associated supernatural number: $$\prod_{p\in \mathbb{P}}p^\infty.$$ In other words, UHF algebra is the the unique UHF algebra $A$ with $K_0(A)=\mathbb{Q}$. A good reference is Rørdam's Classification of Nuclear C${}^\ast$-Algebras.


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Well, someone asked for so there is a compact solution (I can provide more details of course) : First, since $f$ is one to one we deduce that $f'(z) \neq 0$ for all $z \in \bar D$. Then we set $D_r = \{ z, \; |z| \leqslant r\}$. Then if $A_r$ denotes the area of $f(D_r)$ we have : $$A_r= \iint_{f(D_r)} \mathrm{d}x \, \mathrm{d}y =\iint_{D_r} |f'(x+iy)| ...


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An example of a frame operator depends of course on the frame. For a frame $(f_n)_{n \in \mathbb{N}}$ in a Hilbert spaces $\mathcal{H}$, the frame operator $S : \mathcal{H} \to \mathcal{H}$ is, as you mentioned, just defined as $$ S f = \sum_{n \in \mathbb{N}} \langle f, f_n \rangle f_n. \quad \quad (*)$$ To my knowledge, this is the only form in which ...


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Here is a way to understand complex Fourier series representation. Consider $L^2 [-\pi,\pi]$, the set of square integrable complex valued functions on the interval $[-\pi,\pi]$. Define the inner product on this space: $$\langle u(x),v(x)\rangle=\int_{-\pi}^{\pi} u(x)\overline{v(x)}dx.$$ In addition, we can define the norm: $$\|u(x)\|=\sqrt{\langle u(x),u(x) ...


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Professor A. E. Hamza and me published the article for power quantum difference operator which name is "Leibniz’s Rule and Fubini’s Theorem Associated with Power Quantum Difference Operators". This operator generalizes $q$-difference operator and yields $q$-difference operator when $n=1$.


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Let $f$ be a twice differentiable function. Observe $$ \text{det} \left[ \begin{array}{ccc} y & y' & y'' \\ f(x) & f'(x) & f''(x) \\ e^x & e^x & e^x \end{array} \right] =0$$ is a 2nd order linear ODE with horrible coefficients which takes $y=f(x)$ as a solution (and $y=e^x$ as a second solution). But, $f(x)$ is nearly arbitrary, so ...


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By the Schreier conjecture, now generally accepted as a corollary of the classification of finite simple groups, the outer automorphism group of a finite simple group is a solvable group. Thus a finite almost simple group is an extension of a solvable group by a simple group. This is how far one can get, since it is very general.


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Some scholars have argued that Cauchy's conception of the number line incorporated infinitesimals and his notion of convergence therefore imposes additional conditions (for example, convergence at infinitesimal points). Thus, historian and mathematician Detlef Laugwitz argued that Cauchy's theorem was correct as stated. Actually Cauchy himself published ...


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No, in any characteristic. Let $f:\mathbb{P}^1\times\mathbb{P}^1=X\to Y=\mathbb{P}^2$ be such a morphism. Then, the map can not be finite, since it is of degree one and $Y$ is smooth would imply that $f$ is an isomorphism. This is not true by Picard group considerations or many other considerations. So, one must have an irreducible curve $E\subset X$ such ...


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From $$\alpha(t)=\int_0^t e^{is^2}\>ds$$ it follows that $\dot \alpha(t)=e^{it^2}$ has absolute value $1$, hence $\alpha$ is a locally isometric immersion. From ${\rm arg}\bigl(\dot\alpha(t)\bigr)=t^2$ it then follows that the curvature $$\kappa(t)={d\over dt}{\rm arg}\bigl(\dot\alpha(t)\bigr)=2t$$ is strictly monotonically increasing for ...


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I found the assignments from the dead link previously posted. Here is the list in case the link dies again. HOMEWORK Assignment 1: Due in class, Thursday August 25, 2011 required: Chapter 1: 1, 2, 5, 6, 10, 11, 13, 14, 15, 20, 25 extra: Chapter 1: 16, 17, 22 Assignment 2: Due 4 pm, Friday September 2, 2011 required: Chapter 2: 1,2, 3 parts a ...


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We have $$\vartheta_{4}(q) = \prod_{n = 1}^{\infty}(1 - q^{2n})(1 - q^{2n - 1})^{2}\tag{1}$$ It is easily seen that the above product can be written as $$\prod_{n = 1}^{\infty}(1 - q^{2n})\cdot\frac{(1 - q^{n})^{2}}{(1 - q^{2n})^{2}} = \prod_{n = 1}^{\infty}\frac{(1 - q^{n})^{2}}{1 - q^{2n}} = \prod_{n = 1}^{\infty}\frac{1 - q^{n}}{1 + q^{n}}\tag{2}$$ ...


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Let's see two simple examples of classifications of coverings. As usual, let $p:E\rightarrow B$ be a covering map with $E$ path connected and $B$ locally path connected, path connected and semi-locally simply connected. Let $b_0 \in B$ and $x_0 \in p^{-1}(b_0)$. Example 1 Take the trivial subgroup of $\pi_1(B,b_0)$, let's call it $0$. What is the class of ...


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Regarding your third question, set $x(t) = \int_0^t \cos(s^2) \, ds$. The function $x$ is continuous so to show that $x$ is bounded it is enough to show that $\lim_{x \to \pm \infty} x(t) = \pm \int_0^{\infty} \cos(s^2) \, ds$ exists. We have $$ \int_0^{\infty} \cos(s^2) \, ds = \int_0^1 \cos(s^2) \, ds + \int_1^{\infty} \cos(s^2) \, ds = \int_0^1 \cos(s^2) ...


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There is no other convenient word. A numeral, in English, means a representation of an integer. A decimal, in English, means a number with some digits after the decimal point. It is far easier to abstract away the ten-ness of "decimal" when necessary than to invent a hitherto unused specialist term and expect readers to understand it easily. You might ...


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Let $\kappa$ be a Mahlo cardinal in $V$, i.e. let $\kappa$ be inaccessible such that $S:= \{ \alpha \in \kappa \mid \alpha \text{ is regular} \}$ is stationary in $\kappa$. First, note that $L \models \kappa \text{ is inaccessible}$. Indeed, if $L \models \kappa \text{ is not a cardinal}$, then there is some $\mu < \kappa$ and some $f \in L$ such that $L ...


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This is Corollary 5.2.17 in the book "An introduction to Banach space theory" by Megginson, where the term "uniformly rotund" is used instead of "uniformly convex". For completeness, I sketch the proof. We may assume $x=0$, by translation. Let $d = \|y_n\|\to\inf \{\|y\| : y\in C\}$. We may replace $C$ with $\{y\in C: \|y\|\le 2d\}$ without changing ...


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These all have english subtitles and are are highly enjoyable: Discrete Mathematics. Arsdigita University. Instructor: Shai Simonson The Fourier Transforms and its Applications. Standford. Professor Brad Osgood Probability. Harvard Probability Primer. Mathematicalmonk's channel General topology from the very basics, including set theory, techniques for ...


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I would recommend "An Introduction to Linear Analysis" which can be downloaded for free here: https://archive.org/details/AnIntroductionToLinearAnalysis. This was actually the book we used when I took a class on PDE's. It introduces all of the linear algebra needed for the classical theory on PDE's and in my opinion explains everything quite clearly. You can ...


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The definition of $\pi$ is based on the route you take to define the circular functions $\sin x, \cos x$. The traditional approach based on the circle (that's why the name circular functions) is rigorous/intuitive/fruitful/easy. Many believe that the geometric definition based on circle is not rigorous, but I have shown in this answer that it is a fully ...


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Your list seems to be ok. Although I would rather take the series definition $e^x = \sum\limits_{n=0}^\infty \frac {x^n}{n!} $. This gives you a definition for exponentials of complex numbers right from the beginning. Your limit definition for Euler's number follows easily. But of course your way works fine too. As for $\pi$, a "standard" definition is ...


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When it comes to topological fixed point theories, such as Brouwer fixed point theorem, Borsuk-Ulam fixed point theorem and Lefschetz fixed point theory, a great start is Guillemin & Pollack's Differential Topology. It's a wonderful book, and you can read it if you have some background in multivariable calculus or a bit of differential geometry.


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There are syntactic interpretations of PA into ZF-I and of ZF-I into PA. For the purposes of equiconsistency, it does not matter whether these are inverses of each other, an issue that is discussed by Kaye and Wong in this paper. They are worried about something stronger than mere equiconsistency. I will assume we are talking about interpretations in their ...


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Richard R. Goldberg, Methods of Real Analysis, 2nd Ed., Wiley and Sons, 1976



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