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12

Yes, a real analytic function on $\Bbb R$ extends locally to a complex analytic function, except that (in my opinion) "locally" doesn't/shouldn't mean what you say it does. If $f$ is real analytic on $\Bbb R$ then there exists an open set $\Omega\subset\Bbb C$ with $\Bbb R\subset\Omega$, such that $f$ extends to a function complex-analytic in $\Omega$. This ...


7

We indeed have a local extension, given by the Taylor expansion. That is, given $x_0\in\mathbb R$ there exists $\epsilon>0$ such that $$\sum_{n=0}^\infty\frac{f^{(n)}(x_0)}{n!}(z-x_0)^n$$ converges for $|z-x_0|<\epsilon$, and we call the limit $g(z)$. This defines an analytic function $g$ on a neighborhood of $\mathbb R$ which clearly extends $f$. (We ...


3

The traditional choice which I'm familiar with has been Awodey's Category Theory. The new edition has some solutions, and it's explicitly designed for students with less background than Maclane assumes. However, this might go deeper into the interests of computer scientists and logicians than you need. There's a newer book, Leinster's introduction to ...


3

Khan academy might be a good starting point. On youtube you can watch hundreds of their short videos on subjects like algebra, calculus, probability, linear algebra etc. When you feel like you master that material you can move on to MIT OpenCourseWare on youtube. Gilbert Strang's Linear algebra. David Jerison's Single variable calculus. Denis Auroux's ...


3

What you are asking for is the theory of hyperbolic 2-dimensional orbifolds. It's a very big theory. If you do not add additional hypotheses, it becomes somewhat unreasonable to expect a good description of the theory. Even if you add enough hypotheses to tame the question, it still requires a lot of mathematics to even describe the classification. Let me ...


2

Let me start by saying that I will try to explain by the end the level of the material and the order you can read and work through it. Algebra: Abstract Algebra, 3rd Edition by Dummit and Foote is a good option. It covers almost all the algebra you will need in your undergraduate and most of the Algebra in your graduate education. A link some question ...


2

Actually you don't need a new theorem. The usual trick is to add the equation $\mu'=0$, and so to consider the extended vector field $F(x,\mu)=(f_\mu(x),0)$. Notice that the new equation $$ \begin{cases} x'=f_\mu(x),\\ \mu'=0 \end{cases} $$ has an additional central direction and so its center manifold has one additional dimension (applying the center ...


2

usually graduate students study PDE (Partial Differential Equations) rather than ODE. You may want to try doing a project on Sturm–Liouville theory. https://en.wikipedia.org/wiki/Sturm%E2%80%93Liouville_theory It is a slightly advanced part of ODE that may be suitable for graduate students. If that is too hard, you can try Bernoulli differential equation: ...


2

Many possibilities.. Malthusian population model, Prey predator models, pursuit models, dynamic systems, or Black-Scholes model etc.


2

Take $R = \mathbb{F}_2[X]/(X^n)$. Then $R$ is finite and has exactly $n+1$ ideals. Indeed, ideals of $R$ are in canonical bijection with ideals of $\mathbb{F}_2[X]$ containing $X^n$, ie with polynomials dividing $X^n$ : these are the $X^k$ for $0\leqslant k\leqslant n$.


2

Yes, $d(n)$ can attain any positive integer for some $n$, consider $n=2^k$. Hence all ring numbers are of the form $d(n)$.


2

No reference, but here is a proof that $(\mathbb{C}^2, \ell^1)$ does not admit a linear isometric embedding into $(\mathbb{C}^k,\ell^\infty)$ for any $k$. Indeed, such an embedding can be described as a map $$(z,w) \mapsto (\alpha_j z+\beta_j w)_{1\le j\le k}$$ where $|z|+|w| = \max_j|\alpha_j z+\beta_j w|$. Plugging $(1,0)$ and $(0,1)$ for $(z,w)$ shows ...


2

Let's show that $\mathbb{C}^2$ with $\ell^1$ norm cannot be embedded in any space $\mathbb{C}^k$ with $\ell^\infty$ norm. Suppose otherwise, and let $T:(\mathbb{C}^2,\ell^1)\to(\mathbb{C}^k,\ell^\infty)$ be an embedding. We have $\Vert T(1,0)\Vert_\infty=\Vert(1,0)\Vert_1=1$, so by multiplying the coordinates of $T$ by suitable complex numbers of absolute ...


2

Take $$S_n=\sum_{k=1}^n\left|f_k\right|$$ and apply the monotone convergence theorem for $\left\{S_n\right\}_{n=1}^\infty$.


2

I would not limit yourself just to those universities if you want good analysis lectures. I would highly recommend looking into Francis Su's lectures from Harvey Mudd College. His is a one-semester course and he ends roughly when they get into single-variable differentiation. A first glance at MIT's Open Courseware doesn't show any video lectures in ...


2

It seems to be The Art of Combinatorics, Volume IV: Arrangements and Methods by Douglas B. West; see here: http://www.math.illinois.edu/~dwest/. “Four advanced graduate textbooks and research references on classical and modern combinatorics. Preliminary versions available by special arrangement for use in specialized graduate courses; not available for ...


2

The paper is: Bronisław Knaster (1893-1980) and Kazimierz [Casimir] Kuratowski (1896-1980), Sur quelques propriétés topologiques des fonctions dérivées [On some topological properties of derivative functions], Rendiconti del Circolo Matematico di Palermo (1) 49 (1925), 382-386. JFM 51.0208.01 I mentioned it in my answer to Examples of dense sets in the ...


2

This is a general fact about operads. Let $\mathtt{P}$ be any operad (e.g. the pre-Lie operad) in a symmetric monoidal category $\mathsf{C}$. If I understand your question correctly, you have two definitions of "free algebra" you want to compare: The forgetful functor $U : \mathtt{P}\text{-alg} \to \mathsf{C}$ has a left adjoint $F_\mathtt{P}$, and a "free ...


1

$$\frac{d^2y}{dx^2} +y=\tan(x)$$ The first task is to solve the related homogeneous equation : $$\frac{d^2Y}{dx^2} +Y=0$$ I suppose that you can solve it. The second task is to find one particular solution of $\frac{d^2y_p}{dx^2} +y_p=\tan(x)$. The index $p$ means that we look for only one function (any one) among the infinity of functions $y$ which are ...


1

Following @JJacquelin, by variation of the constant, the equation becomes $$\sin(x)f''(x)+2\cos(x)f'(x)-\sin(x)f(x)+\sin(x)f(x)=\tan(x).$$ Then with $g(x)=f'(x)$, $$\sin(x)g'(x)+2\cos(x)g(x)=\tan(x).$$ We multiply by $\sin(x)$ to get an exact differential on the left $$\sin(x)^2g'(x)+2\sin(x)\cos(x)g(x)=(\sin^2(x)g(x))'=\sin(x)\tan(x).$$ Now we ...


1

Note that $D^2+I=(D+i)(D-i)$ and, using integrating factors, we get $$ (D+i)f(x)=e^{-ix}D\left(e^{ix}f(x)\right) $$ and $$ (D-i)f(x)=e^{ix}D\left(e^{-ix}f(x)\right) $$ To invert the $D+i$ operator, first multiply by $e^{ix}$ and integrate: $$ \begin{align} \int e^{ix}\tan(x)\,\mathrm{d}x ...


1

You might be interested in Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values, by Jianqiang Zhao.


1

The online notes on differential geometry by Balazs Csikos are really quite good for self-study: http://www.cs.elte.hu/geometry/csikos/ under Lecture Notes, then under BSM Lecture Notes. Assumed prerequisites are multivariate calculus, linear algebra, and just a little topology.


1

For the model defined in the following way: Fix some graph $G$. For any $p \in [0,1]$, each vertex $v$ in $G$ is in the random induced subgraph with probability $p$ independent of other vertices. This model is an instance of site percolation. Two popular introductory textbooks to Percolation Theory are Percolation by Bela Bollobás and Oliver Riordan and ...


1

As David C. Ullrich and Jason pointed out, the answer is not always affirmative. Let me quote a Theorem that gives a partial affirmative answer to your question, with somewhat different assumptions. Theorem: Assume that $f$ satisfies the Fourier inversion formula, i.e. $f(x) = \int_{\mathbb{R}}\hat{f}(\xi)e^{2\pi x\xi}\,d\xi$ and that $$|\hat{f}(\xi)| ...


1

I think one reason Complex Analysis is so nice is because being holomorphic/analytic is an extremely strong condition. As opposed to real analysis, differentiability is a rather weak condition, so we have functions that are differentiable once but not twice etc. Real analysis is full of nasty counterexamples like the Weierstrass function which is continuous ...


1

The answer to the first question is no. Let $H$ be any group of Fitting length $n > 1$ and $G = H \times H$. Then, taking $N = H \times \{1\}$, $f(N)=f(G)=n$, but $G/N \cong N$ has Fitting length $n$, so it is not nilpotent.


1

The topics that you want to study use mostly the very essential ideas from Linear Algebra. Yes, over $\mathbb{R}$ and $\mathbb{C}$ is all you need. Since you seem to be a theoretically minded person with interest in geometry related subjects, I would recommend Gelfand's "Lectures on Linear Algebra". Strang's textbook is excellent, but probably not the style ...


1

For each pair of disjoint open intervals $U,V\subset\mathbb{R}$ with rational endpoints, $f^{-1}(U)$ and $V$ are both $F_\sigma$, so $f^{-1}(U)\times V$ is an $F_\sigma$ subset of $\mathbb{R}^2$. Furthermore, since $U$ and $V$ are disjoint, $f^{-1}(U)\times V$ is disjoint from the graph of $f$. I claim that $\bigcup_{(U,V)}f^{-1}(U)\times V$ is equal to ...


1

Well, any triangle-free regular graph has constant link. Or take the line graph of such a graph.



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