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For linear algebra I recommend Linear Algebra and its Applications by Peter Lax.


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I have heard that Isaac Newton, after developing calculus, did not want to use the methods in publications, so proved all about the theory of gravity using only Euclidean geometry. However, I don't have a source for this. (I was hoping someone else would point this example out with a better source, but since no-one has, I'll post this.)


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The best general advanced linear algebra book I know is Module Theory An Approach to Linear Algebra by T.S.Blyth. It's beautifully written, very careful and modern.It has probably the most detailed treatment of multilinear algebra you'll find outside of a graduate algebra text. It may be a bit too difficult for your level, though. Check it out and judge for ...


1

A beautiful example is the question, when an antiderivative is an elementary function. It was proved by Liouville in XIXth century. Methods were purely analytic. (J. Liouville. Mémoire sur l’intégration d’une classe de fonctions transcendantes, J. Reine Angew. Math. Bd. 13, p. 93-118. (1835)) About hundred years later computer symbolic integration begun. ...


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The prime number theorem. First proved using complex analysis by Hadamard, and later Erdos and Selberg found an elementary proof.


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A bit different look, to build different intuitions: Paul R. Halmos, A Hilbert Space Problem Book. It cannot be the main textbook for you, but seems to be a good source of exercises on the border of linear algebra and geometry, from near elementary to very advanced ones.


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Plenty of exercises is Topology Without Tears, for which I add that is a topology without fears.


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The idea is to consider the two sides $\text{Quot}_{X_T/T}(\mathcal{H}_T, P)$ and $ T\times_S\text{Quot}_{X/S}(\mathcal{H}, P)$ as $T$-schemes, to show that their functors of points are isomorphic and to apply Yoneda. In fact, one does not even need to know that the functors are actually representable, and the argument works even in situations when they are ...


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We have an exact sequence $0\to I\cap J\to I\oplus J\to I+J\to 0$. If $I+J=R$, then we are done. Let's prove that there is an ideal $I'$ such that $I\simeq I'$ and $I'+J=R$. First chose an ideal $K$ such that $IK=aR$ and $K+J=R$. This can be done as follows: write $I=\prod p^{i_p}$, and for every $p$ appearing in $I$ and $J$ pick an element $x_p\in ...


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For the "foundational" debate of the '30s, see : Marcus Giaquinto, The Search for Certainty : A Philosophical Account of Foundations of Mathematics (2002). For some impact on mathematics of one of the above philosophical "schools" (e.g. Intuitionism and related : Intuitionistic Logic, The Development of Intuitionistic Logic, Luitzen Egbertus Jan Brouwer, ...


1

In this paper you can find a numerical treatment of the Bethe equations for the isotropic spin-$\frac{1}{2}$ Heisenberg quantum spin chain with periodic boundary conditions. A conjecture on the number of solutions with pairwise distinct roots of these Bethe equations in terms of numbers of so-called singular (or exceptional) solutions is formulated. More ...


3

Maps into $\mathbb{A}^1$ are just regular functions, and it's likely that you've shown that the only functions defined on all of $\mathbb{P}^1$, or more generally on a connected projective variety, are constant. In the setting of complex manifolds this is just the maximum principle. Algebraically, you're looking for $f(x) \in k[x]$ and $g(y) \in k[y]$ such ...


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$\Bbb{P}^1(k)$ is connected, and regular map from connected projective variety to affine line must be constant. let $(a:b)\in \Bbb{P}^1(k)$, then regular function at $(a:b)$ is $$ f(x:y)=\frac{p(x,y)}{q(x,y)}$$ where $ p,q\in k[x,y]$ is homogeneous, the same degree and $q(a,b)\ne 0$. let $f=p'/q'$ be another representation of $f$. then $$ p/q=p'/q'$$ and ...


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Russell and Whitehead's Principia Mathematica has no formulas, but it's written in morse code. On a serious note, it's almost impossible to talk rigorously about anything but very simple mathematics without using mathematical notation at some point. Writing a math book without using mathematical notation is like writing a novel without using the letter ...


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You can also see easily without an explicit construction that the intersection of all equivalence relations containing $R$ is an equivalence relation.


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Hardy-Littlewood's second conjecture does appear in the paper you cite.$^*$ The article is 70 pages long and the idea is briefly noted at pp. 52-54. The article is cited for Hardy-Littlewood's second conjecture in dozens of places and fortunately one gave the pages. At page 52 the authors introduce a difference. "The general case raises very ...


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You will love this book: http://ksu.edu.sa/sites/py/ar/mpy/departments/math/learnResources/ResourceCenter/Documents/euclidegeometry.pdf a rigorous introduction. knowledge of high school math is more than enough.


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1. We first talk a bit about the underlying method of the proof. We want look for $(I + \epsilon T)^{-1}$ as an infinite series, similar to the way we would expand $1/(1+x)$ into a power series for $|x| < 1$ over the real or complex field. We then show that our infinite series of operators converges. Along the way, we need to use the fact that $\|TS\| ...


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Let us assume that $V$ is complete (otherwise, this is probably false). Note that finite dimensional normed vector spaces are always complete. Also, let us assume that $T$ is not an arbitrary linear transformation, but a bounded linear transformation, which means that $$ \Vert T \Vert := \sup_{\Vert x \Vert \leq 1} \Vert Tx \Vert < \infty. \qquad ...


1

A matrix is invertible iff the determinant is not equal to zero. Determinant is a continuous function of the entries of a matrix.


-3

I remember when I was in your situation trying to find the right source for good studying and intuitive thinking. I recommend this MIT course with full video lectures, notes, problem sets, practice tests, and challenge problems (the best!). I personally like this course as a whole because it develops you intuition over you reasoning, which is what a ...


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See also Advanced Linear Algebra, Steven Roman, chapters 9 and 14, just as a supplement.


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N.Jacobson. Lectures in Abstract Algebra, vol 2. Linear Algebra. M.M. Postnikov. Lectures in Geometry: Semester II. Linear Algebra and Differential Geometry. Linear Algebra and Geometry Shafarevich, Igor R., Remizov, Alexey Translated by Kramer, D.P., Nekludova, L. 2013, http://www.springer.com/mathematics/algebra/book/978-3-642-30993-9


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There is no one answer to your question. It depends where you are in your mathematical journey. Take linear algebra for example. Perhaps one should begin with: (this may be too silly for some of us, not me) The Manga Guide to Linear Algebra Easy reading, fun, maybe good for highschool. A bit later, say after you've had some calculus at the university, you ...


0

We can prove the well-ordering property if we assume the principle of induction (and similarly we can prove the principle of induction if we assume the well-ordering property). For a proof, let $A\subset\mathbb{N}$, and suppose it has no least element. Let $B$ be the set of all natural numbers $n$ such that $1,\dots,n\not\in A$. Then clearly $1\in B$ ...


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For a list of great books, see The Mathematics Autodidact’s Aid.


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I found one called College Mathematics that appears promising.


1

Of course the well-ordering principle can be proved for the natural numbers in standard set theory, and you can find a proof with google. If so, what would a proof look like? Usually one uses set theory to prove induction, and then induction to prove well-ordering of $\Bbb N$. I guess it depends on what the "properties of arithmetic" are that he ...


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To echo others, the question leaves a lot open to interpretation; however, my vote would have to go to Gosset, the inventor of the "t Distribution." William Sealy Gosset (13 June 1876 – 16 October 1937) was an English statistician. He published under the pen name Student, and developed the Student's t-distribution. (wiki) Given that Gosset published ...


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I've recently discovered Lara Alcock's 'How to think about analysis'. It isn't really a textbook, it's more of a study guide on how to go about learning analysis, but I believe it also covers the key ideas.


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What they are saying is somewhat true: A connected graph can be regarded as a metric space (when equipped with the graph metric) and as such can be used for approximation of geometry of Riemann surfaces (say, hyperbolic ones) as well as higher dimensional Riemannian manifolds. When done properly, one can derive some conclusions about, say, spectral ...


1

It is possible that the following paper would be useful Kauffman


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Harmonic analysis means different things to different people, but to me it's describing the decomposition into irreducibles of the regular representation of a group $G$ on the space $L^2(G)$ of square-integrable $f:G\rightarrow\Bbb{C}$ via some appropriate "Plancherel theorem''. This may be something you're familiar with, but I'll say it anyway: the idea is ...


5

In my opinion, having a basic knowlegde of algebra (Axler is very good, for sure), I would bet on learning different small topics from different books, because it is rather difficult to find everything in one book. If you want to have deeper insight in linear algebra with applications to geometry, I would suggest you to study the following (advanced) topics: ...


4

For more on the geometry, take a look at Berger's wonderful two-volume text Geometry. It's a very sophisticated treatment of classical geometries (not differential geometry), full of linear algebra. You might also look at Pedoe's beautiful book, Geometry, A Comprehensive Course, which uses multilinear algebra as well as linear algebra. There's also the ...


2

Only in the twentieth century did a substantial amount of mathematics come into being that was not connected with physics. mathematical logic / proof theory; (some) abstract algebra; computability; type theory; ... Thus it is now possible to have a mathematical career untouched by physics. There is at least one substantial body of such people. We call ...


3

As darij grinberg comments above, there's Linear Algebra and Geometry by Suetin, Kostrikin, and Manin; it's fairly difficult, but it should be accessible, given the time between now and when you originally asked this question. I didn't read all of it, but quite liked what I did. It has plenty of good exercises. On a different level entirely (and not helpful ...


0

Here is an online link for the paper. Due to copyright reasons the link shall be disabled in a few days. Enjoy!


4

If $A$ is a subgroup of ${\rm Aut(G)}$ with $C_G(A)=1$, then one says that $G$ acts fixed-point-freely upon $G$ by automorphisms. The fixed-point-free automorphism conjecture asserts that if a finite group $G$ admits a fixed-point-free automorphism group $A$ (and, if $A$ is noncyclic, further suppose that $gcd(|G|, |A|) = 1$), then G is soluble. Several ...


1

The following is (an adaptation of) theorem 10 of section 6.5 in Ahlfors "Complex Analysis". Suppose $\Omega \subset \Bbb C$ is an $n$-connected domain for which none of the connected components of $S^2 \setminus \Omega$ are a single point. Then for some real numbers $\lambda_1, ..., \lambda_{n-1}$, there is a one-to-one conformal mapping of $\Omega$ ...


2

A good introductory reference explaining the relationship between the four models mentioned by Zhen Lin is the survey paper Julia Bergner, A survey of (infinity,1)-categories, in J. Baez and J. P. May, Towards Higher Categories, IMA Volumes in Mathematics and Its Applications, Springer, 2010, 69-83, pdf. A good idea would be first to study enriched ...


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In effect, the phrase "$(\infty, 1)$-category" is a cover term for a family of related concepts which are very closely related. Quasicategories are certain special simplicial sets. The theory has been extensively developed by Joyal and Lurie, and [Higher topos theory] covers a lot. As you note, Lurie simply calls these "$\infty$-categories". Simplicially ...


1

Here's three possible strategies. Prove that $A\ge 0$ and $A\le 0$. Prove that $A^2=0$. Prove that $-A=A$.


1

I can recommend the following introduction: Moritz Groth, A short course on infinity-categories, http://arxiv.org/abs/1007.2925


1

See in : http://mathworld.wolfram.com/BesselDifferentialEquation.html Eq.(3) and reference to Browman.


0

You will probably want to start off with a Discrete Math textbook, which will introduce you to relevant mathematical preliminaries as a CS major. Certain combinatorial arguments rely on your understanding of functions and basic set theory, plus it helps formalize intuition. So this is really a good starting point. Epp's Discrete Math text is pretty friendly ...


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As mentioned on Kiran Kedlaya's webpage: Those [official solutions] appear in the official exam summary, along with results and statistics, in the American Mathematical Monthly sometime in the year following the competition, usually in October. (One typically also finds solutions in Mathematics Magazine in early spring.) A set of unofficial solutions, ...


3

You do not need to know any calculus to do geometry. I think you are all set to read it. By the way, I am also reading Hartshorne!, but his other geometry book, also no calculus required there either.


1

Petri nets model networks and distributed systems, but also chemical reactions. They can be seen as symmetric monoidal categories. The Azimuth Project, Petri net V. Sassone has several publications about this connection between Petri nets and category theory.


0

One book that fits the bill really well is Calculus-1 by Apostol.It does treat you as an intelligent reader and moreover,it gives you an idea of what you are really doing when you integrate/differentiate.I also suggest you to look at these two Coursera materials: https://www.coursera.org/course/precalculus https://www.coursera.org/course/sequence The ...



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