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The Amazon reviews are correct about the relative strengths of the books by Rosen and Epp. Concrete Mathematics is a different kind of book altogether and doesn't really belong in the discussion. I recommend the book Mathematics: A Discrete Introduction by Edward Scheinerman: it's better written than the Rosen and has better coverage than the Epp.


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I suggest using a digitizing software to convert the graph into numbers. Once you have the graph as a data file, you can use whatever fitting algorithm you wish (usually a least square estimator provides good results). For the digitizing software, I use im2graph. im2graph is free and available for Linux and Windows. Converting graphs to data requires ...


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Since $$ {\rm erfi}(x)=\frac{2}{\sqrt{\pi}} e^{x^2} D(x), \qquad D(x)= e^{-x^2} \int_0^{x} e^{t^2} \, dt, $$ ${\rm erfi}(x)$ will be very much larger in magnitude than $D(x)$ for $x$ of large magnitude (in fact, $D(x)\sim 1/(2x)$ as $|x|\to\infty$, ${\rm erfi}(x)\sim e^{x^2}/(\sqrt{\pi} x)$ as $|x|\to\infty$.) So, for many large $x$, the value of $D(x)$ ...


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Persistently thinking about the problem, recording every little detail in the thought process, is the only way to get to a point where you can solve IMO 3s and 6s. The tendency is to ask people about how to become a master problem solver but the answer is to patiently climb the steep mountain of solving tough IMO problems even if it means that you have to ...


1

I'll start with the disclaimer that I've never been a math teacher and I'd probably be awful at it. First I'd want to know whether the beginner's interest in mathematics is more algebraic or more geometric. By algebraic, I'm talking about solving equations like $x^2 + 3y^2 = 5$ or $x^2 - 47y^2 = 37$. An example question would be whether these equations ...


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Manjul Bhargava Although older, they are still alive: the $4$ mathematicians who have won the Fields Medal, Abel Prize, and Wolf Prize: Pierre Deligne John G. Thompson John Milnor Jean-Pierre Serre


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Oftentimes in mathematics we find that by expanding our universe we can find out more about a subject that has its roots in something simpler. Look at polynomials over the real numbers, some of them don't have roots, but all of them have roots in $\Bbb C$, the complex numbers. Quadratic fields grew out of the early study of quadratic forms in two variables, ...


3

Off the top of my head, here is something you could try. (Everything you need should be in Fulton's book or Cox–Little–Schenck.) Explain how blowing up a subvariety of a toric variety corresponds to subdivision of the fan. Illustrate with simple examples, like blowing up a point in $\mathbf P^2$ and a line in $\mathbf P^3$. Now use this to ...


1

I think you could make the argument that not only is this possible, but it has been done, and 2300 years ago at that! I am referring to Euclidean geometry. We think of real numbers as measuring the lengths of line segments; or, to use more Euclidean language, telling us when to line segments are congruent. In the Elements, Euclid frequently refers ...


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As Asaf mentioned, you cannot obtain $\mathbb{R}$ as a completion of $\mathbb{Q}$ under these operations. However, there are still algebraic things to be said about $\mathbb{R}$. For example, every irreducible polynomial over $\mathbb{R}$ has degree less than or equal to two, which is trivial to check since $\mathbb{C}$ is algebraically closed, contains ...


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Is solving limits via algebraic manipulation still considered a rigorous proof? Yes. These are equivalent ways to show that a statement is true. There's no "most rigorous" way to prove a limit: fiddling with epsilons and deltas can feel very rigorous (hint: technical does not mean rigorous), but it results in a proof just as true as one that comes out ...


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It is still rigorous, if you state it accurately. Suppose we rewrite $f(x)$ defined for $x\neq a$ as $f(x)=g(x)$ for all $x\neq a$. Then if $g(a)=b$ AND $g(x)$ is continuous at $x=a$, we may conclude: $$ \lim_{x\to a} f(x)=g(a)=b $$ The reason is, that since $g(x)$ is continuous at $x=a$, we have given $\varepsilon>0$ that we can find $\delta>0$ so ...


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The manipulation rules are proven by epsilon-delta arguments, so everything always boils down to epsilon-delta in the end. Depending on your calculus teacher, they may or may not ever really go through that.


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$$\int_{0}^{1} \frac{\ln(x^2+1)}{x^2+1}$$ I really like this. It seems to be posted again and again. Solutions can be found here: Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$


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Specialization is the name of the game in mathematics today, just as is the case in medicine. If you ever have the misfortune of breaking both a shoulder and an ankle, you'll have to talk to twenty different specialists. There is a joke that maybe you've heard before and you'll certainly hear again. Your local university once invited the world's most ...


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Typically, a nice feature of Ito SDE's is that they are not forward looking. As a result, mathematical finance relies on Ito calculus, not Stratonovich calculus.


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Not necessarily true. For example, $u'=u$ and $u'-xu = 1-x$ both are saisfied by the same family $u = Ae^x$.


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In theory, there could be an infinite number of differential equations which have a particular function as a solution. Abel's formula would seem to indicate this. Off the top of my head, for $f(x)=x^2$, I can come up with $y''-2=0$ and $y(y'')-y'=0$.


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Say we have $\frac{5}{2}$, we can either think of this in terms of "How many lots of 2 fit into 5?", as you said, but you could also think of it in terms of "If I break 5 into 2 groups, how many will be in each group?". The 2nd definition makes your question much more clear. If I travel $100km$ in $2hrs$, rather than thinking "How many lots of $2 hrs$ can I ...


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I have found the identity below from which can be deduced infinitely many others of increasing degree by the elementary theory of elliptic curves. The 6-tuples indicate the coefficients of, respectively $n^5$, $n^4$, $n^3$, $n^2$, n and 1 A = (1, 10, -8, 16, 64, -32), B = (1, -10, -8, -16, 64, 32), C = (-1, 8, 8, -16, 80, 32), D = (-1, -8, 8, 16, 80, ...


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Yes, it is exactly division. You divide $x$, whose units are kilometers, by $y$, whose units are hours, to get $\frac{x}{y}$, whose units are kilometers per hour or km/h. When you are in a car, and the spedometer says 100 kph or km/h, that means your rate of travel is 100. If you maintain that rate for one hour, you will travel 100 kilometers. That is: ...


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A humble mathematician might admit he really does not understand any mathematics (I'm being very deliberate with my choice of pronoun here). What are some mathematical concepts that you think you understand today which you did not understand ten years ago? Review these concepts and ask yourself how complicated and advanced these would have appeared to you ...


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In data analysis, the eigenvectors of a covariance (or correlation matrix) are usually calculated. Eigenvectors are the set of basis functions that are the most efficient set to describe data variability. They are also the coordinate system that the covariance matrix becomes diagonal allowing the new variables referenced to this coordinate system to be ...


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If you are new to linear algebra ,then you should use "Introduction to Linear Algebra" by Gilbert Strang.In case you posses some knowledge of LA then you can use " Matrix Theory and Linear Algebra" by I.N. Herstein .There are many books on pure linear algebra and computational linear algebra,you can choose as per your requirement and interest. The two ...


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Microsoft OneNote should be ideal for this purpose. You can use it to write notes, add pictures, and everything is stored in the cloud (and will be uploaded/updated while you are taking notes). Best of all, it is available for free. For your additional question, I would recommend to use the free flash card programs like Anki. You can create your own flash ...


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Would you like an app that facilitates annotating electronic books, or an app that functions as "rough-work sheets"? For the former one, GoodReader and iAnnotate are the two most powerful readers, with highlighting, annotating, and freehand drawing enabled. For the latter, paper-based note taking apps, like bamboo paper, paper 53 are all good apps, but ...


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For me, it feels very simple and logical, and I am always slightly surprised others think it is hard. It gives me some degree of inner balance, slight feeling of fullfilness, satisfaction. I believe your feelings would gradually change as your knowledge grows, however this is good, and is one of the most attractive aspects of learning. You would go from ...


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Recall the species of set partitions with the constituents marked which is $$\mathfrak{P}(\mathcal{U}\mathfrak{P}_{\ge 1}(\mathcal{Z}))$$ which gives the generating function $$G(z, u) = \exp(u(\exp(z)-1)).$$ It follows that the exponential generating function of Bell numbers is given by $$G(z) = \exp(\exp(z)-1).$$ Suppose we are trying to compute ...


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The parity of the order of a finite group plays an enormous role in finite group theory! The Feit-Thompson Theorem settled Burnside's conjecture that every non-Abelian simple group has even order. There were numerous stepping-stones along the way, and a big part of the classification of finite simple groups was devoted to studying how involutions ...


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The case of even perfect numbers is settled since long ($P=M(M+1)/2$ for $M$ a Mersenne prime). That of odd perfect numbers is still open.


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In even-dimensional Euclidean spaces, spherical waves have trailing edges; in odd-dimensional spaces they don't.


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I am not sure one can give a justification from scratch. However, an important algebraic motivation of adjunction is to formalize the "free construction" on a set in a given algebraic theory. Leaning on the basic free constructions (groups, abelian groups, modules, rings, etc.) that your student should know, you have a panel of adjunctions $F \dashv U$ to ...


2

Notice that \begin{align} S & = \sum_{n = 0}^{\infty} \frac{n^{2}}{n!} \\ & = \frac{0^{2}}{0!} + \frac{1^{2}}{1!} + \frac{2^{2}}{2!} + \frac{3^{2}}{3!} + \cdots \\ & = \frac{1}{0!} + \frac{2}{1!} + \frac{3}{2!} + \frac{4}{3!} + \cdots. \end{align} Then \begin{align} S - e & = \left( \frac{1}{0!} + \frac{2}{1!} + \frac{3}{2!} + ...


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Here's a nice "generator" of the result I commented in the comments.


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An example: $$ 1=\sqrt{2-1} \longrightarrow1=\sqrt{2-\sqrt{2-1}} $$ Repeat infinitely many times: $$ 1=\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2...}}}} $$ You can check this is true: $$ x=\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2...}}}} \longrightarrow x=\sqrt{2-x} $$ $$ x^2=2-x \longrightarrow x_1=1 $$ $x_2=-2$ can be discarded as we're taking a square root. It's worth to ...


3

Contour integration is line integration with a complex variable. You might first try to understand line integrals (which are pretty easy to understand). A line integral looks like $$ \int_C f ds = \int_a^b f(\gamma(t))\gamma'(t)dt $$ where $C$ is the curve being integrated along and $\gamma(t)$ is a parametrization of the curve. This can be interpreted like ...


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I've known about contour/line integrals for a long time. When I saw this gif it blew my mind, it made the concept so much more intuitive. (taken from wikipedia article on line integrals)


0

One of the books that i would say would be very good is 'Discrete mathematics by Rosen', it covers a lot of basics like graph theory, set theory, combinatorics, propositional logic,number theory, it covers most of the topics in a very self contained way, regarding the time needed to complete could vary a lot considering some of the more demanding parts are ...


0

The day, the tradition of writing the phrase "it is easy to see that..." in all graduate-level textbooks in math, becomes a thing of past, it would be a great victory day for the mathematics lovers in all countries.


2

I try to explain this concept several different ways, using some of the more mathematical approaches above. Sometimes I run into students who still just don't get it, so I offer them this analogy: Pretend your car represents a mathematical function. The inputs are the gas you put in the tank, and the output is how far you can drive. Let's say that you ...


1

One humble suggestion: Read chapter 1.8.2 of the famous "Feynman Lectures on Physics." In order to introduce the concept of instantaneous speed, Feynman quotes this joke there: The cop stops the lady and says: "Lady, you were going 60 miles per hour!" She says, "That is impossible, sir, I was travelling only for seven minutes..." The conversation between ...


1

Imagine that you are in a room with Albert (A) and Bernard (B). The three of you are perfectly rational. The first thing to realise is that if B has number 18 or 19, then he automatically knows the whole date. Albert: "I know that B doesn't know." The only way A can possibly know this is if A does not have May or June (Since, if A has May or June, then (as ...


2

Case 1: The reason is because Bernard told him. If that was the case, the dialog would start with: Bernard: I don't know when her birthday is. In mathematics we never make any extra assumptions, is always understood that no information is withhold. Bernard telling something to Albert, information which is needed to solve the problem, would automatically ...


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I want to briefly sketch how this kind of problems can be solved in a systematic way. One does not have to rely on intuition, scribbling and hand-waving. The wording of the problem is indeed somewhat imprecise. It is not clear how Albert comes to his first statement. The problem statement should be changed to the following (notice point 2): 1) Cheryl gives ...


1

It seems that you are looking for mathematics that is not too technical. In this regard, I do recommend the textbook 'Comprehensive Mathematics for Computer Scientists' because it contains the type of theory that forms the foundations of mathematics - i.e. sets and logic - without going into numerical analysis. Logic and set theory can be intuitive and fun ...


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After working as a mathematician for more than fifteen years, and as a professor for nearly ten, I must confess that all these efforts to take higher mathematics out of classrooms are somehow a waste of time. Be careful, I am not saying that higher mathematics is useless in our life, but that most people will never understand why and how it can be useful. ...


3

The axiom of choice was something people had used without always noting that there is an assumption to be made in order to justify making infinitely many choices at once. On the other hand, Cantor felt that there shouldn't be intermediate cardinals between the naturals and the reals, so he hypothesized that this is the case and spent a considerable amount ...


0

Taylor's theorem is the generalization of linear approximations. Let $f(x)$ be a differentiable function and $p$ some fixed point. Define the linear function $L(h) = f(p) + f'(p)h$. This is function approximates $f(p+h)$, when $h$ is a small number. To be more precise by what "approximates", we are saying that, $f(p+h) = L(h) + \varepsilon(h)$, where ...


0

Re: Remark 2 - producing something original. I think this is unnecessary and will create unneeded stress. There was a thesis option in the Masters Degree I did at Cambridge University in 1986. This was NOT expected to be original research. I wrote a summary of research on a particular question in Banach Algebra Functional Calculus. It was 77 pages. I did ...


-1

Firstly, as tcamp also commented, there also exists a similar construction as adjunctions in any bicategory where the 'unit' and 'counit' go in the same direction, and these are the (dual) Morita contexts, and it is also worth to study, having already wide literature, at least in its origin world of rings. (Note that adjoint equivalences are immediately ...



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