# Tag Info

2

There is a variant of ETCS with replacement, due to Colin McLarty, but I don't think this is really the right approach. For instance, for your specific problem of iterating the power object functor, the real issue is not replacement but rather that the induction principle you get out of the universal property of an NNO is too weak. Here's a non-parametrised ...

0

Dummit and Foote may not be the best book for Galois Theory (Chapters 13-14) but it contains numerous examples.

0

Try Galois Theory by Ian Stewart; one of the best book for beginners

3

Two-sided embedded surfaces which bound a smooth curve (also called a knot) $C$ are known as Seifert surfaces for the knot. I don't know a precise reference to van Kampen's result but the Wikipedia article seems to disagree he was the first to prove the existence of a Seifert surface for every knot. If you're looking for a proof, there's an algorithm for ...

0

I recommend 3Blue1Brown's Channel, just as an example, the last video: https://www.youtube.com/watch?v=cyW5z-M2yzw Also The Mathologer

1

I like the channel Tipping Point Math. Lots of interesting math made by a math professor.

4

I like [Hirschhorn, Model categories and their localisations]. First, a word of warning: the book is divided into two parts, but the first part depends logically on the second part. Thus, beginners should start at Chapter 7, not Chapter 1. The great thing about [Hirschhorn] is that all the definitions and results are stated carefully and clearly, so it ...

0

This is true. The problem becomes more transparent if we use the following characterization of $K$-Lipschitz functions: $f$ is $K$-Lipschitz if and only if both $Kx-f(x)$ and $Kx+f(x)$ are nondecreasing. Thus, it suffices to prove that any branch of a countable family of nondecreasing functions $\psi_n$ is also nondecreasing. Suppose that $f$ is a ...

1

Math and Music Kids of new generation should have different methods of training. They are capable to perceive information faster, with cross-modal processing, activating all senses at once : visual perception, audio analyzers, neuromotor functions. It is well known that the difficulties in the perception of any information, including musical ...

1

Somewhere between a joke and an essay, is Impure Math. http://www.snowman-jim.org/science/humor/impure-math.html

1

Personally, I would suggest "Vector Calculus" from Jerrold E. Marsden & Anthony J. Tromba. A very good book which contains all of what you are requiring (surface integrals, the theorems of vector calculus and vector analysis), the vector identities (for example for the usage of divergence and curl); changing order of integration or setting the bounds ...

0

Here are some thoughts on the subject. Using such a metric locally turns the Euclidean space into a Finsler manifold instead of a Riemannian manifold. If you are not familiar with differential geometry, you might want to ignore this. You should also check the link Noah Schweber gave in a comment: Wikipedia tells about $\ell^p$-norms in $\mathbb R^n$. One ...

0

Are you asking for a reference for the recursion formula for the coefficients, or for the convergence on the unit disk? Note that the recursion imples that the odd coefficients are all positive (the even coefficients are $0$ by symmetry). By a result of Pringsheim, if such a series $g(z)$ has radius of convergence $r$ we must have $\lim_{z \to r-} g(z) = ... 1 Understanding numerical algorithms necessarily involves some arithmetic and algebra as well as just logic. I would strongly recommend you dip into Knuth's wonderful Art of Computer Programming. This is an encyclopaedic work but quite easy to use as a a very readable reference once you read the introductory sections. Volume 2 includes a discussion of the ... 5 These topological arguments involve the same basic idea: it's often easy to prove things for a subset of matrices which are dense in the space of all matrices. Any "continuous" fact (e.g. the assertion that two continuous functions are equal) can be proven for all matrices by proving it for this dense subset. For example, if$L$is diagonalizable with ... 2 Milnor "Morse Theory" contains an extremely well written introduction to the subject. 3 I'm a fan of Lee's Riemannian Manifolds: An Introduction to Curvature. It is definitely an introductory book; there are many deeper topics that it doesn't mention (compare to Peterson's Riemanninan Geometry). Here is an excerpt from the preface: "I have selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making ... 1 Have you tried Riemannian Geometry: A Beginners Guide, by Frank Morgan? 2 By far Gallot et al is a very good choice. 2 Edge contractions are not really so much algebraic graph theory as general graph theory. The subject is closely related to the subject of graph minors. Reinhard Diestel's book Graph Theory contains an entire chapter devoted to the famous graph minor theorem, including an outline of the proof. (The entire proof was published in a series of 20 papers.) The ... 2 In 2012 Feng and Wu showed that $$\limsup_{n} \delta_n \frac{\log \gamma_n}{2\pi}\geq 2.7327.$$ Note however that the quantity $$\mu=\liminf_n\delta_n \frac{\log \gamma_n}{2\pi}$$ is far more interesting. Unconditionally, the best bound is$\mu<0.525396$due to this recent paper of Preobrazhenskii, however even under the Riemann Hypothesis this cannot be ... 3 John Baez, Aaron Lauda, Higher-Dimensional Algebra V, arXiv Dan Mardsen, Category theory using string diagrams, arXiv Peter Selinger, A survey of graphical languages for monoidal categories, arXiv 2 You should be able to find a proof in most books covering homology theory. A specific reference is Theorem 4.59 of Hatcher's Algebraic Topology (p. 399-402). As Mike Miller commented, the idea of the proof is to show that any (co)homology theory can be computed by a "cellular chain complex" associated to that theory, and that these cellular chain complexes ... 2 It is a quotient space!$\mathbb{R}$here should be thought of as the one-dimensional subspace of$L^2(\Omega)$which consists of the (a.e.) constant functions. The norm given is the canonical norm on the quotient of a Banach space mod a closed subspace. This construction is discussed, for instance, in section III.4 of Conway's A Course in Functional ... 0 Okay, I think I have an idea: from Andres Caicedo's answer on this MSE post, we know that the composition of real analytic functions is real analytic. So, we can write $$\sqrt{f} = e^{\text{log}(\sqrt{f})} = e^{\frac{1}{2}\text{log }f}.$$ I think this solves the problem? 0 I found a good reference for the question. M.Taylor's "Introduction to Differential Equation", chapter 3 (Linear systems of differential equations) , section 6 (Second order systems) discusses precisely these questions that I was looking for. Thanks 3 Here are some examples: Ian Roulstone, John Norbury: Invisible in the Storm: The Role of Mathematics in Understanding Weather Vladimir Arnold: Catastrophe Theory Julian Havil: GAMMA David Harel: Computers Ltd George Szpiro: Kepler's Conjecture Malba Tahan: The Man Who Counted: A Collection of Mathematical Adventures. 2 I've only read the first couple chapters (so far), but I really like the Springer Undergrad Texts in Mathematics book Mathematics and its History by John Stillwell Also, I don't think you can do wrong with Newton's Philosophiae Naturalis Principia Mathematica - 'twas the book that first roused my interest in matters physick and mathematick. As well, I ... 0 Can't believe no one has (yet) mentioned George Polya's incomparable How to Solve It. It's very dissimilar to Feynman in that it covers very few (if any) specific subfields of mathematics - but it's very similar in that it attempts to give the student an understanding of how to approach the discipline, and to build their intuition so they can grapple with ... 2 My recommendations Taming the infinite by Ian Stewart. The great mathematical problems by Ian Stewart. Does God play dice by Mario Livio Golden Ratio by Mario Livio 1 Mathematics by David Bergamini is good. Some of it (especially the parts about computers) is dated, but much of it is just as valid today as it ever was. 4 I would recommend these books: Journey through Genius Dr. Euler's Fabulous Formula Prime Obsession The Music of the Primes Gödel's Proof (by Ernest Nagel) The Code Book 1 Cox concentrates on positive binary forms. I have given many, many answers on this site dealing with those or indefinite binary forms. Along with the first chapter of Conway, I like Duncan A. Buell, Binary Quadratic Forms. For me, the use has been that i have been able to write C++ programs to implement most of Buell. Oh, Stillwell's number theory book adds ... 1 Knuth's book: Surreal numbers builds Conways surreal number system up from the very beginning and steps through the very interesting proofs of it's basic properties. It's a bit more elementary and has a story element to it but there is certainly good mathematics in it too. Highly recommended. 1 John Conway's book: The sensual (quadratic) form is excellent and roughly fits the pattern you describe. Each chapter is about solving a new problem about quadratic forms, the machinery gets more and more sophisticated. Review: https://cms.math.ca/crux/v26/n3/page147-150.pdf 0 After searching several books on ODEs I think you could refer to a stronger result, the so called comparison lemma which appears in appendix E.4 of [1] W.J. Terrell, Stability and stabilization, Princeton University Press, 2009. Lemma E.4 [1] (Comparison Lemma) Suppose$w(t, u)$is continuous on a connected open set$\Omega$in$R^2$and the ... 1 One question is what you mean by "every Borel set can be approximated by compact sets from inside and by open sets from outside". The usual definition I take (adopted by Folland, who also provides essentially the example mentions below) is that$\mu(M) = \inf_{U \subset M}\mu(U)$(with$U$open) should hold for all Borel sets$M$, whereas$\mu(M)=\sup{K ...

2

There already is at least one such list, compiled by Stephen Smale. See here

0

Relevant mathoverflow discussions: http://mathoverflow.net/questions/12709/are-there-any-books-that-take-a-theorems-as-problems-approach http://mathoverflow.net/questions/119621/learning-through-guided-discovery

2

\begin{align}\int_0^{2\pi} |f(re^{it})|^2dt &= \int_0^{2\pi}\overline{f(re^{it})}f(re^{it})dt\\ &=\int_0^{2\pi}\sum_m \overline{a_m}r^me^{-imt}\sum_n {a_n}r^ne^{int}\ dt\\ &=\sum_{m,n}\overline{a_m}a_nr^{m+n}\int_0^{2\pi}e^{it(m-n)}dt\\ &=\sum_{m,n}\overline{a_m}a_nr^{m+n}2\pi\delta_{mn}\\ &=2\pi\sum_n\overline{a_n}a_nr^{n+n}\\ ... 3 Harold Edwards agrees with you about reading original sources. His book on Galois Theory includes a translation of the seminal papers on Group Theory and an excellent account of the mathematics from both the original and the modern perspective. 0 I think I am now finally in some position to answer this question, but my answer will still be updated with time. Let me begin by stating that recognizing isomorphisms and homomorphisms as definitions in their own right is non-trivial, and historically took much time to develop. From Stillwell's Elements of Algebra (note that Stillwell is also the author of ... 0 I like the Schaum's book on Linear Algebra. For more information about the book, I refer you to the following link: http://www.amazon.com/Schaums-Outline-Linear-Algebra-Edition/dp/0071794565 Bob 0 This is a really underrated collection of mathematics and other video lectures http://nptel.ac.in/course.php For your purpose: http://nptel.ac.in/courses/111104024/ 0 We can apply a trinary Huffman code. If this is not clear enough, here is how to do it for the example given in the question: We have a sequence of N=4 "binary" questions (U_1,U_2,U_3,U_4). I put binary in quotes because we can answer yes, no, or I don't know. Take 0 as no, 1 as yes, and e as erased or I don't know. As in the question ... 0 I use Polish notation. The formation rules go: All lower case letters of the Latin alphabet qualify as significant expressions. If \alpha and \beta qualify as significant expressions, then so do N\alpha, C\alpha\beta$, A$\alpha$$\beta, and K\alpha$$\beta$. The significant expression CApqCCpqq can get proved in intuitionistic logic from the ... 0 Since Pre-calculus is an assembly of several basic mathematical concepts, you could always start studying from any of those books you recommended, there wont be too much of a difference. A book that I used was Precalculus by Ron Larson, which isn't a very good book for someone who is learning functions and their graphs for the first time, but since you are ... 0 As far as I've seen, pre-calculus is comprised in majority of past algebra. You could always begin practicing that (as it is essential). My past pre-calculus class had a chapter on matrix algebra and 2nd degree polynomials. I'm sorry to say I can't remember what the book was, as it was over a year ago. The book ended with an intro for calculus I, ... 5 This is not true. The scheme$X=\mathrm{Spec}(k\times k)$is projective and reduced over$k$but the global sections of the structure sheaf do not form a field. You need to assume$X$is connected too. In any case, by properness,$H^0(X,\mathscr{O}_X)$is a finite-dimensional$k\$-algebra (this is the hard part), hence a finite product of Artin local rings. ...

0

Writing proofs is essentially a sequence of statements and their justifications. We all learn some form of proof writing in geometry when we write two column proofs. With that being said, there are many techniques used in proof oriented problems. There are many good books which have already been mentioned. I advise you not to touch Munkres unless you ...

Top 50 recent answers are included