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The number of famous mathematicians in history who were also great mental calculators are few and far between. Euler, who became blind in his latter years, is said to have been able to perform calculations in his head. One story is that Euler passed away while he was mentally calculating the orbit of the moon. Emilie du Chatelet is also another famous ...


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please don't propose books that are impossible to find on internet, there are enough books and courses that are free to read on internet, and the scientifics that you are should understand it is a huge social progress that everyone can FREELY access to the scientific knowledge. look at any of the first links for : google / fourier transform pdf google ...


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It is a little known fact that much of Bernhardt Riemann's work had remained un-rigorous until Hilbert's arrival on the mathematical scene some fifty years after Riemann's death. Riemann was aware of his shortcomings, but many before him were not. Did you know that complex numbers were not properly accepted until the nineteenth century? The historian D.T. ...


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Another way to explain it based on a physical understanding: Consider our observable 3D space and double all lengths. a 1D object will have its mass multiplied by $2^1=2$. a 2D object will have its mass multiplied by $2^2=4$ a 3D object will have its mass multiplied by $2^3=8$ a object of dimension $h$ will have its mass multiplied by $2^h$ So ...


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This answer is something of a counterpoint to Samuel's enthusiastic answer. In short (and to directly answer your question), I would say that you are much better off just using Python (or other constituent programs) directly. Exactly which of these you might use depends very much on what you want to do. While Sage relies on Python, it is not a simple Python ...


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As you finish your undergraduate studies, you should have had at least some introduction or passing acquaintance to the following subjects: Algebra (the abstract kind) Discrete mathematics (maybe some number theory) Linear algebra Real/complex analysis (post-calculus) Topology Differential equations Statistics/probability Of these, which did you ...


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It is hard to get a good overview of all of mathematics. The best really is to take a good broad variety of classes. This you usually do (to some extent) the first couple of years in graduate school. Here you will learn the basic language of the main areas of mathematics. Here you will be get to meet various professors. For now, then, I wouldn't worry too ...


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If you are in the U.S., you could try attending AMS Sectional Meetings. These sorts of conferences happen often, and they feature talks in a huge number of different topics in current mathematics. Otherwise, I'm sure that in your locale, there are probably analogous conferences. Also, you could sign up for email notifications from arXiv mathematics. You ...


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I agree with programming/computation, probability/statistics, and linear algebra from Comments. Also, Optimization from Answer. Would add group theory. Find out what computer languages are currently in greatest use as the time graduation gets near, especially for managing and parsing large datasets; learn the basics of all. Unless the landscape changes ...


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An open problem I find surprising, the PAC (Perimeter to Area Conjecture) due to Keleti (1998): Conjecture: The perimeter to area ratio of the union of finitely many unit squares in the plane does not exceed 4. See for example Bounded - Yes, but 4? and references therein.


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Yes, there is a way to profitably read mathematical proofs, but it takes time. Here is an excerpt from the "note to the reader" in an excellent topology book: "It is a basic principle in the study of mathematics, and one too seldom emphasized, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it ...


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It is correct that in CS or when you work as a software engineer you rarely need to implement an optimisation algorithm yourself. However it is very important to recognise an optimisation problem when you see one in order to be able to apply the correct software packages. This is often not trivial and should be part of every course on optimisation.


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I think that part of this question answers itself when you get "into" a field. When I was writing my MSc. dissertation I really had no idea what I was interested in, but my supervisor pointed me in a few directions and I ended up working in the one I found most interesting. Then, as you read more and more papers and see the most recent research you find ...


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I have a specific application in mind, but Gunnar Carlsson gave a mini-course at this year's YTM, and in this he used many statistical methods along with topological methods to answer problems in computing. I think the broader title of this field would be somethign like Topological Data Analysis! Along this flavour, a paper by Gunnar Carlsson entitled ...


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Big Data is a very broad definition. If want to work in data-mining or machine learning my list would start with these Statistics/Measure theory Optimisation (in general and especially convex optimisation) Funcional Analysis


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One possibility is to look what gets published at arXiv.org. Skim the abstracts to get an overview and if something interests you, you can find more on that topic.


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Here are some advantages of Sage (also known as SageMath) relevant to your question. Sage supports all fields of mathematics, including those you mentioned (groups, finite fields, combinatorics, permutation groups, linear algebra over finite fields, linear programming, integer programming). Sage relies on Python. it is a widespread, well-designed ...


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Regarding everyday perception of number and the question of what "large" ought to mean, there is the field of Numerical cognition with many interesting experiments. E.g. when you flash a sheet with some dots for a second, people will be able to tell if there is only one dot, or two, or three. If there are three dots, people who see it for one second will not ...


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The symbols $\gg$ and $\ll$ don't have a formal definition. Usually they are used to compare two extremely big numbers, for example $\mathrm{Graham's \; number} \ll \mathrm{TREE}(3)$ or something like that. They are only used because someone wants to make clear that one of them is so much greater. The numbers are indeed just small compared to everyday ...


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There is a series of math children books in Russian by Владимир Артурович Лёвшин. To list some: Магистр рассеянных наук (translates roughly as Master (as in M.Sci) of the absent-minded sciences, though google translates it as Master scattered Sciences), Новые рассказы Рассеянного Магистра (New stories of the absent-minded Master), Путешествие по Карликании и ...


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I think it's important to know first how deeply you want to study differential geometry/differentiable manifolds. I agree completely with Mike Miller's comment above, but would like to add a few thoughts. There is a big distinction between just studying differentiable manifolds and differential geometry. Geometry relies entirely on the definition of ...


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Ramsey's theorem For any given number of colors c, and any given integers n1, . . . ,nc, there is a number, R(n1, ..., nc), such that if the edges of a complete graph of order R(n1, . . . , nc) are colored with c different colors, then for some i between 1 and c, it must contain a complete subgraph of order ni whose edges are all color i. The two ...


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Strictly speaking, equation $x^2=y$ has two solutions, in both real and complex domain. Exceptional $y$ is $0$ when the equation has only one solution. The difference between the complex and real case arises when we want to choose a solution $x(y)$ to depend CONTINUOUSLY on $y$. In the real domain, if we restrict $y$ to some interval that does not contain ...


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You might have a good intusion for science, but to understand modern science you also need the language of mathematics. Learning (other) languages can be boring but it's worth it being able to use it. Perhaps there are more urgent disciplines than discrete math to start with? Calculus and algebra are more important at the begining in science and is more ...


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I majored in math in college over 50 years ago-- this is from my distant experience of that time. A previous writer (littleO)had an important comment about finding the right book. Some authors seem to be trying to demonstrate their cleverness rather than trying to really convey to the reader the motivation and intuition for the math. So if one text doesn't ...


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To help hone your intuition, it might help to think about the following simple example, which shows that the Blumenthal law is using much more than just the continuity of Brownian motion. Let $Z$ be a random variable with $P(Z=1) = P(Z=-1) = 1/2$, i.e. a coin flip. Set $X_t = tZ$, so that the process $X_t$ just moves with constant speed $\pm 1$ depending ...


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There are many books for graph theory, and you should select the topics that suit the education goals of the course. For example, in computer science the Graph theory algorithms is highly considered as main topic in each graph theory course. Personally, I prefer the following two books: Introduction to Graph Theory Graph Theory (Graduate Texts in ...


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If you consider the study of separable metric spaces a valid field, that provides an example (although by the wrong reason): They have cardinality bounded by that of $\mathbb{R}$.


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A metric space is an example of a category enriched over the monoidal category of non-negative real numbers, with morphisms given by $\leq$ and the monoidal product given by addition. Instead of "hom-sets" between objects, we assign a non-negative real $d(x,y)$ to each pair of objects. Then the composition law says that, given three objects $a,b,c$, there ...


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Metric spaces are an example of a topological space where the topology is determined by the endowed metric.


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Sorry, this isn't a full answer, rather some algebraic hints at how you might relate the cross product to your intuition about the cup product (wrt intersections) and to linear algebra. It should really be a comment but it's too long. First, a quick review! The cross product, $\times$, is actually part of the cup product: $$x \smile y := \Delta^*(x \times ...


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Introduction to Topology. Pure and Applied by C Adams & R Franzosa


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How to overcome the temptation to read many books covering the same topics[?] Simply by working. I can't tell you how many things I am able to push out of my mind when thinking about the mathematics itself. I have no idea what you've been doing with respect to analysis, but over the past few months, you have asked several questions about analysis ...


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Yep. If you read one book over another you'll likely miss problems that are covered in only one of those books but not the other. However, if you read one whole book and have a solid understanding of the general theory, it won't be too hard to learn other topics that you missed if needed. How should I overcome such temptations and anxiety? Don't worry ...


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Ask a professor or more advanced math student (whose opinion you trust) what book is best on the subject. Find the best book, understand it cover to cover (make sure you understand all pre-requisites first), then advance to a higher level book on that subject (if so desired).


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It seems that what you're looking for is an interpolation space : https://en.wikipedia.org/wiki/Interpolation_space I don't remember if these interpolation spaces between $BV$ and $L^1$ are "usual" spaces or not


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The work done by Donaldson on 4-manifolds used techniques borrowed from gauge theory. Witten also used techniques from Physics to solve mathematical problem, including if I remember correctly, a simpler proof of the index theorem using supersymmetry.


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I don't know if it's interesting, but here is how you can interpret the facts you cited. The first thing, of course, is that the derivative vector $b'(t)$ is in the direction of the tangent vector at parameter value $t$. But this is true of any parametric curve, not just Bezier curves. Take the $n$ vectors $\Delta b_n$ and relocate them so that their ...


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I would say it is the standard. When a mathematician says $C[0,1]$ (read "the continuous functions on $[0,1]$) this is usually the set of functions $f:[0,1]\to\mathbb C$ that are continuous.


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I find that symbols are only really helpful to the extent that they don't actually change the sentence. Writing "Let epsilon be..." and "Let $\varepsilon$ be..." are identical, because the collection of strokes translates to the same sentence in my head, which is also how I would naturally describe the idea to myself and others. The second form may be ...


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Count the number of ways of placing them in line, in a circle; how many ways to form a line of, say, 5 of the students. How many ways to choose, say, 5 of them gives binomial coefficients. How many ways are there to make up a 5-person group out of 2 boys and 3 girls, this generalizes to Vandermonde's identity. Fool around with a 3-way split of the class, and ...


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All my math related university textbooks made large use of logic symbols. The worst ones were actually almost only logic symbols, which is not bad per se, but as someone noted symbols only state things, a little writing in a less formal language (note: implying math symbols are a language) can help understand. Conclusion: best is to state with symbols and ...


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First of, a functor $F : \mathscr C → \mathrm{Set}$ is called representable (by $C$) if it's isomorphic (not necessarily equal) to $\mathrm{Hom}(C, -)$ for an object $C$ of $\mathscr C$. As for the term, an abstract functor $F$ is represented by the very concrete action of $\mathrm{Hom}(C, -)$. Take for example the functor $L : \mathrm{Top} → \mathrm{Set}$ ...


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Because to obtain our chain complex for a space $X$ we look at arbitrary maps $\Delta^n \to X$ which are not necessarily nice, i.e. singular. In contrast we look at nice embeddings when dealing with simplicial homology.


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This is not quite what was asked for, but I thought it worth mentioning: Below is (the non-trivial) Proposition 11.1.9 in Kalton and Albiac's Topics in Banach Space Theory: $\ \ \ $(i) For $1\le p\le2$, $L_q[0,1]$ embeds in $L_p[0,1]$ if and only if $p\le q\le2$. $\ \ \ $(ii) For $2< p<\infty$, $L_q[0,1]$ embeds in $L_p[0,1]$ if and only if $q=2$ ...


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Here are some interesting facts about the relations between different $L^p$-spaces over the same measure space $(X,\Sigma,\mu)$ (based on Section 6.1 of Folland, 1999): If $0<p<q<r\leq\infty$ and if $f\in L^q$, then there exist $g\in L^p$ and $h\in L^r$ such that $f=g+h$. If $0<p<q<r\leq\infty$, then $L^p\cap L^r\subseteq L^q$. If ...


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Here are some examples of relations between the spaces: If $p,q\in[1,\infty)$, there is a bijection $L^p\to L^q$, namely $L^p\ni f\mapsto |f|^{p/q-1}f\in L^q$. If $A$ has finite measure, then $p\geq q$ implies $L^p(A)\subset L^q$. If $f\in L^p$ and $g\in L^q$ so that $1/p+1/q=1/s$ (assuming $p,q,s\in[1,\infty]$), then the pointwise product $fg$ is in ...


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Corollary 3 of Chapter 7 of Royden: If $E$ is measurable with finite measure and $1\leq p_1< p_2\leq \infty$, then $L^{p_2}(E)\subseteq L^{p_1}(E)$. Furthermore, $||f||_{p_1}\leq c||f||_{p_2}$ for all $f\in L^{p_2}(E)$ where $c=[m(E)]^{\frac{p_2-p_1}{p_1p_2}}$ if $p_2<\infty$ and $c=[m(E)] ^{\frac{1}{p_1}}$ if $p_2=\infty$. Royden remarks in an ...


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As claimed here any child can be turned into a genius. The secret is early education. So, you should start to learn math at a young age. The younger you are, the less hard wired your brain is and that allows you to hard wire "mathematical aptitude" in a way that is impossible to do at a later age. It is very unfortunate that the school curriculum for math ...


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I empathize with your username @iLearnSlow. The main reason I'm decent at trigonometry, and I daresay this goes for many people here, is because I've done many, many, many problems. Like many people, I really struggled at first, and I definitely didn't grok the material until the second or third time through. I first did problems at home with my father. ...



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