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This is the "reflection property" of the hyperbola. In the context of "academic references", it is good practice to cite the original source, which is • Apollonius of Perga, 200 BCE, Treatise on Conic Sections, book III, proposition 48; English translation by T.L. Heath, page 116 (Cambridge University Press, 1896). Alternatively, you may ...


4

E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie [Hypercomplex Quantities and the Theory of Representations], Math. Zeit. 30 (1929), 641–692; informal translation of section 23. Please feel free to edit as you see fit original Determinant of a hypercomplex system. Let $\mathbb{o}=a_1P+\ldots+a_h P$ be a hypercomplex system. I adjoin to $P$ a ...


3

This is not an answer and I cannot comment due to lack of reputation, but it seems to first pop up in his Lemma 4 on p. 50. This occurs after a long, very technical discussion about double coset operators. The goal is to show that two specific representations of the Hecke algebra are equivalent (one whose image lies in $End_{\mathbb{C}}(S_{k+1/2}(N, \chi))$ ...


3

The lectures for Steven Strogatz's (ongoing) course on "Asymptotics and Perturbation Methods" are posted to his YouTube channel: https://www.youtube.com/user/stevenstrogatz1/videos. Strogatz is an outstanding lecturer.


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This may not be a viable route, but if your friend is familiar with Python, a hands-on course might be an effective way to explore Markov chains. The course designed by Sargent and Stachurski guides you step-by-step through the basics and a variety of applications from economics. No prior knowledge is needed beyond elementary probability theory and linear ...


3

I haven't yet got an idea of what sort of answer would be satisfactory; it probably depends on how you are thinking of the automorphisms. Here's one attempt, just to have something written; it is entirely elementary, so probably unsatisfactory, but you can let me know how it falls short of the goal, and we can see if it can be fixed. You probably think of a ...


2

Here’s a course on C*-algebras and compact quantum groups by in my opinion a genius lecturer https://youtube.com/playlist?list=PLq3E5oubNNoAZi6W7968tNkSnIGTt4hEn He also has a course on functional analysis (see the HSE youtube channel where the above lectures are uploaded). It is different from the most because he takes more algebraic approach.


2

The rate of decrease of the tails of a probability distribution has to do with the degree of smoothness of the corresponding characteristic function (c.f.). The rate of decrease of the tails has nothing to do with the positivity of the c.f. E.g., the uniform distribution over the interval $[-1,1]$ has zero tails, but its c.f. $f$, given by $f(t)=\dfrac{\sin ...


2

As Igor Belegradek commented, the correct statement is as follows: Theorem (classification of closed simply connected manifold with nonnegative curvature operator): A closed simply connected manifold with nonnegative curvature operator is isometric to a Riemannian product of standard spheres with metrics of nonnegative curvature operator closed Kahler ...


2

What you would like to say is that $G \cong A \times T$, which is obviously not quite true. But it is true smooth-locally, and that's enough to conclude. Indeed, note that the quotient map $G \to A$ is faithfully flat with smooth fibres, hence smooth [Tag 01V8]. In fact, $G \times_A G \cong T \times G$ via $(g,h) \mapsto (g-h,g)$, and likewise $G \times_A G \...


1

For example, if we take any basis $e_1,e_2,e_3,e_4$ of a 4-dimensional vector space, and if $F$ is the span of $e_1,e_2$, so $F\in Gr_2$ and $F'$ is the span of $e_3,e_4$, so $F'\in Gr_2$, then $F\cap F'$ is the set of vectors which are both of the form $ae_1+be_2$ and of the form $ce_3+de_4$. Every vector admits a unique representation as a linear ...


1

We have $$\prod_{k\geq 2} (1+x^k) = \frac{\prod_{k\geq 1} (1+x^k)}{1+x} \equiv \sum_{j\geq 0} \frac{x^{j(3j+1)/2} + x^{(j+1)(3j+2)/2}}{1+x}$$ $$\equiv \sum_{j\geq 0} x^{j(3j+1)/2}\frac{1 - x^{2j+1}}{1-x}\equiv \sum_{j\geq 0} x^{j(3j+1)/2}(1+x+\dots+x^{2j}) \pmod{2}.$$


1

Let ${\frak g}$ be a simple Lie algebra over $\Bbb C$, and let $\theta$ be an inner involution of ${\frak g}$, that is, an inner automorphism of ${\frak g}$ of order dividing 2. Such automorphisms are classified by Kac labelings of the extended Dynkin diagram ${\widetilde D}={\widetilde D}({\frak g})$. We fix a Cartan subalgebra ${\frak t}\subset{\frak g}$ ...


1

This is covered in detail in Endomorphisms of Algebraic Groups by Robert Steinberg, AMS Memoirs #80, 1968. What you are getting is a quotient of the root system (obtained by restricting roots to $H^\theta$, or equivalently looking at $\theta$-orbits of roots). The book by Onischik and Vinberg (Lie Groups and Algebraic Groups, 1990) has a nice treatment (see ...


1

The following paper by de Boor suggests that this is the case, although he develops the proof only up to degree 6 splines. de Boor, C., On the convergence of odd-degree spline interpolation, J. Approximation Theory 1, 452-463 (1968). ZBL0174.09902. The paper below by Swartz, and the references in it, also claim that this can be proved, at least under some ...


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Ben Webster gave an introductory symplectic geometry course this past term which I think was very good. The presentation was pretty elementary. The course was offered through the Fields Institute, and their YouTube channel also has links to some other courses by some professors around Ontario which were offered in the same way.


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This course on Real Algebraic Geometry from Konstanz Universität's professor Markus Schweighofer is a really good introduction to the topic.


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Factoring as Optimization could be a useful link in this direction as well.


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Daniel Halpern-Leistner is teaching a foundational course on moduli theory based on stacks at Cornell. Very nice addition to the literature in my opinion.


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